{"id":626278,"date":"2023-06-22T12:37:18","date_gmt":"2023-06-22T07:07:18","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/?p=626278"},"modified":"2025-07-25T18:08:17","modified_gmt":"2025-07-25T12:38:17","slug":"area-of-triangles-with-3-sides","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/articles\/area-of-triangles-with-3-sides\/","title":{"rendered":"Area of Triangles with 3 Sides"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_37 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" style=\"display: none;\"><label for=\"item\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/label><input type=\"checkbox\" id=\"item\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' style='display:block'><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/area-of-triangles-with-3-sides\/#What_is_the_Perimeter_of_a_triangle\" title=\"What is the Perimeter of a triangle?\">What is the Perimeter of a triangle?<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/area-of-triangles-with-3-sides\/#What_is_the_Base_and_Height_of_a_triangle\" title=\"What is the Base and Height of a triangle?\">What is the Base and Height of a triangle?<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/area-of-triangles-with-3-sides\/#What_is_the_Area_of_a_triangle\" title=\"What is the Area of a triangle?\">What is the Area of a triangle?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/area-of-triangles-with-3-sides\/#Area_of_Triangle_with_3_Sides_Formula\" title=\"Area of Triangle with 3 Sides Formula\">Area of Triangle with 3 Sides Formula<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/area-of-triangles-with-3-sides\/#How_is_the_formula_to_find_the_area_of_a_triangle_obtained\" title=\"How is the formula to find the area of a triangle obtained?\">How is the formula to find the area of a triangle obtained?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/area-of-triangles-with-3-sides\/#Solved_Examples_on_Area_of_Triangles_with_3_Sides\" title=\"Solved Examples on Area of Triangles with 3 Sides\">Solved Examples on Area of Triangles with 3 Sides<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/area-of-triangles-with-3-sides\/#Solved_Examples_on_Area_of_Triangle_with_3_Sides\" title=\"Solved Examples on Area of Triangle with 3 Sides\">Solved Examples on Area of Triangle with 3 Sides<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/area-of-triangles-with-3-sides\/#Question_1_Find_the_area_of_a_triangle_with_sides_of_lengths_7_cm_8_cm_and_9_cm\" title=\"Question 1 Find the area of a triangle with sides of lengths 7 cm, 8 cm, and 9 cm.\">Question 1 Find the area of a triangle with sides of lengths 7 cm, 8 cm, and 9 cm.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/area-of-triangles-with-3-sides\/#Question_2_A_triangle_has_side_lengths_of_5_m_12_m_and_13_m_Calculate_its_area\" title=\"Question 2 A triangle has side lengths of 5 m, 12 m, and 13 m. Calculate its area.\">Question 2 A triangle has side lengths of 5 m, 12 m, and 13 m. Calculate its area.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/area-of-triangles-with-3-sides\/#Question_3_Calculate_the_area_of_a_triangle_with_sides_measuring_10_cm_14_cm_and_16_cm\" title=\"Question 3 Calculate the area of a triangle with sides measuring 10 cm, 14 cm, and 16 cm.\">Question 3 Calculate the area of a triangle with sides measuring 10 cm, 14 cm, and 16 cm.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/area-of-triangles-with-3-sides\/#Question_4_A_triangle_has_side_lengths_of_9_in_40_in_and_41_in_Find_its_area\" title=\"Question 4 A triangle has side lengths of 9 in, 40 in, and 41 in. Find its area.\">Question 4 A triangle has side lengths of 9 in, 40 in, and 41 in. Find its area.<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/area-of-triangles-with-3-sides\/#FAQs_on_Area_of_Triangles_with_3_Side\" title=\"FAQs on Area of Triangles with 3 Side\">FAQs on Area of Triangles with 3 Side<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/area-of-triangles-with-3-sides\/#How_can_I_find_the_length_of_an_unknown_side_in_a_triangle\" title=\"How can I find the length of an unknown side in a triangle?\">How can I find the length of an unknown side in a triangle?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/area-of-triangles-with-3-sides\/#How_can_I_find_the_angles_of_a_triangle_if_I_know_the_lengths_of_its_sides\" title=\"How can I find the angles of a triangle if I know the lengths of its sides?\">How can I find the angles of a triangle if I know the lengths of its sides?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/area-of-triangles-with-3-sides\/#What_is_the_perimeter_of_a_triangle_and_how_do_I_calculate_it\" title=\"What is the perimeter of a triangle and how do I calculate it? \">What is the perimeter of a triangle and how do I calculate it? <\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/area-of-triangles-with-3-sides\/#Can_I_find_the_area_of_a_triangle_without_knowing_its_height\" title=\"Can I find the area of a triangle without knowing its height? \">Can I find the area of a triangle without knowing its height? <\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<p>A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C.The length of the sides of a triangle may be same or different. If all the 3 sides of a triangle are equal, then it is an equilateral triangle. If 2 sides of a triangle are equal, it is an isosceles triangle.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"What_is_the_Perimeter_of_a_triangle\"><\/span>What is the Perimeter of a triangle?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The Perimeter of a triangle is the total length around the triangle. Consider a triangle with sides of lengths \u2018a\u2019, \u2018b\u2019, and \u2018c\u2019.<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-626279\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/06\/perimeter-of-triangle.png\" alt=\"Perimeter of triangle\" width=\"286\" height=\"281\" \/><\/p>\n<p>Thus, the perimeter of the triangle = a + b + c.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"What_is_the_Base_and_Height_of_a_triangle\"><\/span>What is the Base and Height of a triangle?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>The Base of a triangle is the side on which the triangle rests. Any one of the three sides of the triangle can be considered as the base. The perpendicular drawn from a base to the opposite vertex of a triangle is known as the corresponding Height of the triangle.<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-626280\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/06\/Base-and-Height-of-a-triangle.png\" alt=\"Base and Height of a triangle\" width=\"921\" height=\"244\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/06\/Base-and-Height-of-a-triangle.png?v=1687415222 921w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/06\/Base-and-Height-of-a-triangle-300x79.png?v=1687415222 300w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/06\/Base-and-Height-of-a-triangle-768x203.png?v=1687415222 768w\" sizes=\"(max-width: 921px) 100vw, 921px\" \/><\/p>\n<p>For the above triangles, \u210e1 is the corresponding height of base b1, \u210e2 is the corresponding height of base b2, \u210e3 is the corresponding height of base b3.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"What_is_the_Area_of_a_triangle\"><\/span>What is the Area of a triangle?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ul>\n<li>To find the area of the triangle, a base and its corresponding height are considered. Consider a triangle with base \u2018b\u2019 and corresponding height \u2018h\u2019.<\/li>\n<\/ul>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-626282\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/06\/area-of-triangle.png\" alt=\"area of triangle\" width=\"254\" height=\"163\" \/><\/p>\n<p>Then, the area of the triangle = \u00bd \u00d7 b \u00d7 \u210e<\/p>\n<p>That is, area of a triangle is \u00bd times the product of its base and the corresponding height.<\/p>\n<ul>\n<li>If the height and base of a triangle is not given and the lengths of all three sides of the triangle are known, Heron&#8217;s formula can be used to calculate the area. Heron&#8217;s formula is a general formula for calculating the area of any triangle using the lengths of its sides.<\/li>\n<\/ul>\n<p><strong>Heron\u2019s Formula:<\/strong> Let&#8217;s say the lengths of the three sides of the triangle are a, b, and c.<\/p>\n<p>The semi-perimeter (s) of the triangle is calculated as (a + b + c) \/ 2.<\/p>\n<p>Then, the area (A) of the triangle can be found using the formula:<\/p>\n<p>A = \u221a(s x (s &#8211; a) x (s &#8211; b) x (s &#8211; c))<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Area_of_Triangle_with_3_Sides_Formula\"><\/span>Area of Triangle with 3 Sides Formula<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The area of a triangle with three sides was first discovered by the Greek mathematician Heron of Alexandria. He created a formula that allows us to find the area of a triangle without needing to know the angles or any other distances, just the lengths of its sides.<\/p>\n<p>Let ABC be a triangle such that the length of the 3 sides of the triangle is AB = c, BC = a and CA = b.<\/p>\n<p>The semi-perimeter of triangle ABC = s = (a + b + c)\/2<\/p>\n<p>Then, the area of triangle ABC = \u221a[s \u00d7 (s \u2013 a) \u00d7 (s \u2013 b) \u00d7 (s \u2013 c)].<\/p>\n<h2><span class=\"ez-toc-section\" id=\"How_is_the_formula_to_find_the_area_of_a_triangle_obtained\"><\/span>How is the formula to find the area of a triangle obtained?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Consider a triangle as shown below:<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-626285\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/06\/area-of-a-triangle.png\" alt=\"area of a triangle\" width=\"287\" height=\"175\" \/><\/p>\n<p>If the base and the corresponding height of this triangle are b and h respectively, then the area of this triangle is 1\/2 \u00d7 b \u00d7 \u210e. Let us verify this using the following steps.<\/p>\n<ul>\n<li><strong>Step 1:<\/strong> Consider a copy of the same triangle along with the original one. Since, both the triangles are congruent, they will have the same area.<\/li>\n<li><strong>Step 2<\/strong>: The second triangle is flipped horizontally and then vertically, it will be as shown below.<\/li>\n<\/ul>\n<p><img loading=\"lazy\" class=\"alignnone wp-image-626288\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/06\/triangles.png\" alt=\"Triangles\" width=\"320\" height=\"150\" \/><\/p>\n<ul>\n<li><strong>Step 3:<\/strong> Combining these two triangles, we give us a quadrilateral as shown below.<\/li>\n<\/ul>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-626289\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/06\/quadrilateral.png\" alt=\"quadrilateral\" width=\"513\" height=\"333\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/06\/quadrilateral.png?v=1687416760 513w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/06\/quadrilateral-300x195.png?v=1687416760 300w\" sizes=\"(max-width: 513px) 100vw, 513px\" \/><\/p>\n<p>The area of this quadrilateral will be the sum of the areas of the two triangles. But areas of both the triangles are the same.<\/p>\n<p>So, Area of the quadrilateral = 2 x Area of a triangle. &#8230; (i)<\/p>\n<ul>\n<li><strong>Step 4:<\/strong> Now, the quadrilateral is fragmented as shown below<\/li>\n<\/ul>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-626290\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/06\/quadrilaterals.png\" alt=\"quadrilaterals\" width=\"316\" height=\"155\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/06\/quadrilaterals.png?v=1687416859 316w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/06\/quadrilaterals-300x147.png?v=1687416859 300w\" sizes=\"(max-width: 316px) 100vw, 316px\" \/><\/p>\n<p>Area of rectangle = L x B<\/p>\n<p>So, area of rectangle = b x h \u2026. (ii)<\/p>\n<p>Also, the area of the rectangle = area of the quadrilateral from step 3 \u2026.. (iii)<\/p>\n<p>From (i), (ii), and (iii),<\/p>\n<p>2 x Area of triangle = b x h<\/p>\n<p>Hence, Area of triangle = \u00bd \u00d7 b \u00d7 \u210e<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Solved_Examples_on_Area_of_Triangles_with_3_Sides\"><\/span>Solved Examples on Area of Triangles with 3 Sides<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Example 1. If the base and the corresponding height of a triangle is 10 cm and 4 cm respectively, then its area is given by,<\/p>\n<p>Solution: Area of the Triangles = 1\/2 x b x h<\/p>\n<p>= 1\/2 x 10 x 4<\/p>\n<p>= 1\/2  x 40<\/p>\n<p>= 20 cm<sup>2<\/sup><\/p>\n<p><span class=\"TextRun SCXW102172382 BCX0\" lang=\"EN-US\" xml:lang=\"EN-US\" data-contrast=\"auto\"><span class=\"NormalTextRun SCXW102172382 BCX0\">So, the area of the triangle is 20<\/span><\/span><span class=\"TextRun SCXW102172382 BCX0\" lang=\"EN-US\" xml:lang=\"EN-US\" data-contrast=\"auto\"><span class=\"NormalTextRun SCXW102172382 BCX0\">\ud835\udc50\ud835\udc5a<\/span><\/span><sup><span class=\"TextRun SCXW102172382 BCX0\" lang=\"EN-US\" xml:lang=\"EN-US\" data-contrast=\"auto\"><span class=\"NormalTextRun Superscript SCXW102172382 BCX0\" data-fontsize=\"12\">2<\/span><\/span><\/sup><span class=\"TextRun SCXW102172382 BCX0\" lang=\"EN-US\" xml:lang=\"EN-US\" data-contrast=\"auto\"><span class=\"NormalTextRun SCXW102172382 BCX0\">.<\/span><\/span><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Solved_Examples_on_Area_of_Triangle_with_3_Sides\"><\/span>Solved Examples on Area of Triangle with 3 Sides<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3><span class=\"ez-toc-section\" id=\"Question_1_Find_the_area_of_a_triangle_with_sides_of_lengths_7_cm_8_cm_and_9_cm\"><\/span>Question 1 <strong>Find the area of a triangle with sides of lengths 7 cm, 8 cm, and 9 cm.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<ol>\n<li>Identify the side lengths:<br \/>\n<span class=\"katex\"><span class=\"katex\"><span class=\"katex-mathml\"> <\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mn>7<\/mn><mtext>\u2009<\/mtext><mtext>cm<\/mtext><mo separator=\"true\">,<\/mo><mtext>\u2009<\/mtext><mi>b<\/mi><mo>=<\/mo><mn>8<\/mn><mtext>\u2009<\/mtext><mtext>cm<\/mtext><mo separator=\"true\">,<\/mo><mtext>\u2009<\/mtext><mi>c<\/mi><mo>=<\/mo><mn>9<\/mn><mtext>\u2009<\/mtext><mtext>cm<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">a = 7 \\, \\text{cm}, \\, b = 8 \\, \\text{cm}, \\, c = 9 \\, \\text{cm}<\/annotation><\/semantics><\/math>\n<p><\/span><\/span><\/span>&nbsp;<\/p>\n<p><span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">7<\/span><span class=\"mord text\"><span class=\"mord\">cm<\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">b<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">8<\/span><span class=\"mord text\"><span class=\"mord\">cm<\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">9<\/span><span class=\"mord text\"><span class=\"mord\">cm<\/span><\/span><\/span><\/span><\/span><\/li>\n<li>Calculate the semi-perimeter (s):&nbsp;<br \/>\n<math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>s<\/mi><mo>=<\/mo><mfrac><mrow><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mo>+<\/mo><mi>c<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mn>2<\/mn><\/mfrac><mo>=<\/mo><mfrac><mrow><mo stretchy=\"false\">(<\/mo><mn>7<\/mn><mo>+<\/mo><mn>8<\/mn><mo>+<\/mo><mn>9<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><mn>2<\/mn><\/mfrac><mo>=<\/mo><mfrac><mn>24<\/mn><mn>2<\/mn><\/mfrac><mo>=<\/mo><mn>12<\/mn><mtext>\u2009<\/mtext><mtext>cm<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">s = \\frac{(a + b + c)}{2} = \\frac{(7 + 8 + 9)}{2} = \\frac{24}{2} = 12 \\, \\text{cm}<\/annotation><\/semantics><\/math>\n<p><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">s<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\">2<span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mbin\">+<\/span><span class=\"mord mathnormal\">b<\/span><span class=\"mbin\">+<\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mclose\">)<\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\">2<span class=\"mopen\">(<\/span>7<span class=\"mbin\">+<\/span>8<span class=\"mbin\">+<\/span>9<span class=\"mclose\">)<\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\">224<\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">12<\/span><span class=\"mord text\"><span class=\"mord\">cm<\/span><\/span><\/span><\/span><\/li>\n<li>Use Heron\u2019s formula to find the area:&nbsp;<br \/>\n<math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mtext>Area<\/mtext><mo>=<\/mo><msqrt><mrow><mi>s<\/mi><mo>\u00d7<\/mo><mo stretchy=\"false\">(<\/mo><mi>s<\/mi><mo>\u2212<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u00d7<\/mo><mo stretchy=\"false\">(<\/mo><mi>s<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u00d7<\/mo><mo stretchy=\"false\">(<\/mo><mi>s<\/mi><mo>\u2212<\/mo><mi>c<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Area} = \\sqrt{s \\times (s &#8211; a) \\times (s &#8211; b) \\times (s &#8211; c)}<\/annotation><\/semantics><\/math>\n<p><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord text\"><span class=\"mord\">Area<\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"svg-align\"><span class=\"mord\"><span class=\"mord mathnormal\">s<\/span><span class=\"mbin\">\u00d7<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">s<\/span><span class=\"mbin\">\u2212<\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mclose\">)<\/span><span class=\"mbin\">\u00d7<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">s<\/span><span class=\"mbin\">\u2212<\/span><span class=\"mord mathnormal\">b<\/span><span class=\"mclose\">)<\/span><span class=\"mbin\">\u00d7<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">s<\/span><span class=\"mbin\">\u2212<\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo>=<\/mo><msqrt><mrow><mn>12<\/mn><mo>\u00d7<\/mo><mo stretchy=\"false\">(<\/mo><mn>12<\/mn><mo>\u2212<\/mo><mn>7<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u00d7<\/mo><mo stretchy=\"false\">(<\/mo><mn>12<\/mn><mo>\u2212<\/mo><mn>8<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u00d7<\/mo><mo stretchy=\"false\">(<\/mo><mn>12<\/mn><mo>\u2212<\/mo><mn>9<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">= \\sqrt{12 \\times (12 &#8211; 7) \\times (12 &#8211; 8) \\times (12 &#8211; 9)}<\/annotation><\/semantics><\/math>\n<p><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"svg-align\"><span class=\"mord\">12<span class=\"mbin\">\u00d7<\/span><span class=\"mopen\">(<\/span>12<span class=\"mbin\">\u2212<\/span>7<span class=\"mclose\">)<\/span><span class=\"mbin\">\u00d7<\/span><span class=\"mopen\">(<\/span>12<span class=\"mbin\">\u2212<\/span>8<span class=\"mclose\">)<\/span><span class=\"mbin\">\u00d7<\/span><span class=\"mopen\">(<\/span>12<span class=\"mbin\">\u2212<\/span>9<span class=\"mclose\">)<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo>=<\/mo><msqrt><mrow><mn>12<\/mn><mo>\u00d7<\/mo><mn>5<\/mn><mo>\u00d7<\/mo><mn>4<\/mn><mo>\u00d7<\/mo><mn>3<\/mn><\/mrow><\/msqrt><mo>=<\/mo><msqrt><mn>720<\/mn><\/msqrt><mo>\u2248<\/mo><mn>26.83<\/mn><mtext>\u2009<\/mtext><msup><mtext>cm<\/mtext><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">= \\sqrt{12 \\times 5 \\times 4 \\times 3} = \\sqrt{720} \\approx 26.83 \\, \\text{cm}^2<\/annotation><\/semantics><\/math>\n<p><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"svg-align\"><span class=\"mord\">12<span class=\"mbin\">\u00d7<\/span>5<span class=\"mbin\">\u00d7<\/span>4<span class=\"mbin\">\u00d7<\/span>3<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"svg-align\"><span class=\"mord\">720<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><span class=\"mrel\">\u2248<\/span><\/span><span class=\"base\"><span class=\"mord\">26.83<\/span><span class=\"mord\"><span class=\"mord text\">cm<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/li>\n<\/ol>\n<h3><span class=\"ez-toc-section\" id=\"Question_2_A_triangle_has_side_lengths_of_5_m_12_m_and_13_m_Calculate_its_area\"><\/span>Question 2 <strong>A triangle has side lengths of 5 m, 12 m, and 13 m. Calculate its area.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<ol>\n<li>Identify the side lengths:<br \/>\n<span class=\"katex\"><span class=\"katex\"><span class=\"katex-mathml\"> <\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mn>5<\/mn><mtext>\u2009<\/mtext><mtext>m<\/mtext><mo separator=\"true\">,<\/mo><mtext>\u2009<\/mtext><mi>b<\/mi><mo>=<\/mo><mn>12<\/mn><mtext>\u2009<\/mtext><mtext>m<\/mtext><mo separator=\"true\">,<\/mo><mtext>\u2009<\/mtext><mi>c<\/mi><mo>=<\/mo><mn>13<\/mn><mtext>\u2009<\/mtext><mtext>m<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">a = 5 \\, \\text{m}, \\, b = 12 \\, \\text{m}, \\, c = 13 \\, \\text{m}<\/annotation><\/semantics><\/math>\n<p><\/span><\/span><\/span>&nbsp;<\/p>\n<p><span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">5<\/span><span class=\"mord text\"><span class=\"mord\">m<\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">b<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">12<\/span><span class=\"mord text\"><span class=\"mord\">m<\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">13<\/span><span class=\"mord text\"><span class=\"mord\">m<\/span><\/span><\/span><\/span><\/span><\/li>\n<li>Calculate the semi-perimeter (s):&nbsp;<br \/>\n<math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>s<\/mi><mo>=<\/mo><mfrac><mrow><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mo>+<\/mo><mn>12<\/mn><mo>+<\/mo><mn>13<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><mn>2<\/mn><\/mfrac><mo>=<\/mo><mfrac><mn>30<\/mn><mn>2<\/mn><\/mfrac><mo>=<\/mo><mn>15<\/mn><mtext>\u2009<\/mtext><mtext>m<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">s = \\frac{(5 + 12 + 13)}{2} = \\frac{30}{2} = 15 \\, \\text{m}<\/annotation><\/semantics><\/math>\n<p><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">s<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\">2<span class=\"mopen\">(<\/span>5<span class=\"mbin\">+<\/span>12<span class=\"mbin\">+<\/span>13<span class=\"mclose\">)<\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\">230<\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">15<\/span><span class=\"mord text\"><span class=\"mord\">m<\/span><\/span><\/span><\/span><\/li>\n<li>Use Heron\u2019s formula:&nbsp;<br \/>\n<math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mtext>Area<\/mtext><mo>=<\/mo><msqrt><mrow><mn>15<\/mn><mo>\u00d7<\/mo><mo stretchy=\"false\">(<\/mo><mn>15<\/mn><mo>\u2212<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u00d7<\/mo><mo stretchy=\"false\">(<\/mo><mn>15<\/mn><mo>\u2212<\/mo><mn>12<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u00d7<\/mo><mo stretchy=\"false\">(<\/mo><mn>15<\/mn><mo>\u2212<\/mo><mn>13<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Area} = \\sqrt{15 \\times (15 &#8211; 5) \\times (15 &#8211; 12) \\times (15 &#8211; 13)}<\/annotation><\/semantics><\/math>\n<p><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord text\"><span class=\"mord\">Area<\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"svg-align\"><span class=\"mord\">15<span class=\"mbin\">\u00d7<\/span><span class=\"mopen\">(<\/span>15<span class=\"mbin\">\u2212<\/span>5<span class=\"mclose\">)<\/span><span class=\"mbin\">\u00d7<\/span><span class=\"mopen\">(<\/span>15<span class=\"mbin\">\u2212<\/span>12<span class=\"mclose\">)<\/span><span class=\"mbin\">\u00d7<\/span><span class=\"mopen\">(<\/span>15<span class=\"mbin\">\u2212<\/span>13<span class=\"mclose\">)<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo>=<\/mo><msqrt><mrow><mn>15<\/mn><mo>\u00d7<\/mo><mn>10<\/mn><mo>\u00d7<\/mo><mn>3<\/mn><mo>\u00d7<\/mo><mn>2<\/mn><\/mrow><\/msqrt><mo>=<\/mo><msqrt><mn>900<\/mn><\/msqrt><mo>=<\/mo><mn>30<\/mn><mtext>\u2009<\/mtext><msup><mtext>m<\/mtext><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">= \\sqrt{15 \\times 10 \\times 3 \\times 2} = \\sqrt{900} = 30 \\, \\text{m}^2<\/annotation><\/semantics><\/math>\n<p><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"svg-align\"><span class=\"mord\">15<span class=\"mbin\">\u00d7<\/span>10<span class=\"mbin\">\u00d7<\/span>3<span class=\"mbin\">\u00d7<\/span>2<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"svg-align\"><span class=\"mord\">900<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">30<\/span><span class=\"mord\"><span class=\"mord text\">m<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/li>\n<\/ol>\n<h3><span class=\"ez-toc-section\" id=\"Question_3_Calculate_the_area_of_a_triangle_with_sides_measuring_10_cm_14_cm_and_16_cm\"><\/span>Question 3 <strong>Calculate the area of a triangle with sides measuring 10 cm, 14 cm, and 16 cm.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<ol>\n<li>Identify the side lengths:<br \/>\n<span class=\"katex\"><span class=\"katex\"><span class=\"katex-mathml\"> <\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mn>10<\/mn><mtext>\u2009<\/mtext><mtext>cm<\/mtext><mo separator=\"true\">,<\/mo><mtext>\u2009<\/mtext><mi>b<\/mi><mo>=<\/mo><mn>14<\/mn><mtext>\u2009<\/mtext><mtext>cm<\/mtext><mo separator=\"true\">,<\/mo><mtext>\u2009<\/mtext><mi>c<\/mi><mo>=<\/mo><mn>16<\/mn><mtext>\u2009<\/mtext><mtext>cm<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">a = 10 \\, \\text{cm}, \\, b = 14 \\, \\text{cm}, \\, c = 16 \\, \\text{cm}<\/annotation><\/semantics><\/math>\n<p><\/span><\/span><\/span>&nbsp;<\/p>\n<p><span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">10<\/span><span class=\"mord text\"><span class=\"mord\">cm<\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">b<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">14<\/span><span class=\"mord text\"><span class=\"mord\">cm<\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">16<\/span><span class=\"mord text\"><span class=\"mord\">cm<\/span><\/span><\/span><\/span><\/span><\/li>\n<li>Calculate the semi-perimeter (s):&nbsp;<br \/>\n<math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>s<\/mi><mo>=<\/mo><mfrac><mrow><mo stretchy=\"false\">(<\/mo><mn>10<\/mn><mo>+<\/mo><mn>14<\/mn><mo>+<\/mo><mn>16<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><mn>2<\/mn><\/mfrac><mo>=<\/mo><mfrac><mn>40<\/mn><mn>2<\/mn><\/mfrac><mo>=<\/mo><mn>20<\/mn><mtext>\u2009<\/mtext><mtext>cm<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">s = \\frac{(10 + 14 + 16)}{2} = \\frac{40}{2} = 20 \\, \\text{cm}<\/annotation><\/semantics><\/math>\n<p><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">s<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\">2<span class=\"mopen\">(<\/span>10<span class=\"mbin\">+<\/span>14<span class=\"mbin\">+<\/span>16<span class=\"mclose\">)<\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\">240<\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">20<\/span><span class=\"mord text\"><span class=\"mord\">cm<\/span><\/span><\/span><\/span><\/li>\n<li>Use Heron\u2019s formula:&nbsp;<br \/>\n<math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mtext>Area<\/mtext><mo>=<\/mo><msqrt><mrow><mn>20<\/mn><mo>\u00d7<\/mo><mo stretchy=\"false\">(<\/mo><mn>20<\/mn><mo>\u2212<\/mo><mn>10<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u00d7<\/mo><mo stretchy=\"false\">(<\/mo><mn>20<\/mn><mo>\u2212<\/mo><mn>14<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u00d7<\/mo><mo stretchy=\"false\">(<\/mo><mn>20<\/mn><mo>\u2212<\/mo><mn>16<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Area} = \\sqrt{20 \\times (20 &#8211; 10) \\times (20 &#8211; 14) \\times (20 &#8211; 16)}<\/annotation><\/semantics><\/math>\n<p><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord text\"><span class=\"mord\">Area<\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"svg-align\"><span class=\"mord\">20<span class=\"mbin\">\u00d7<\/span><span class=\"mopen\">(<\/span>20<span class=\"mbin\">\u2212<\/span>10<span class=\"mclose\">)<\/span><span class=\"mbin\">\u00d7<\/span><span class=\"mopen\">(<\/span>20<span class=\"mbin\">\u2212<\/span>14<span class=\"mclose\">)<\/span><span class=\"mbin\">\u00d7<\/span><span class=\"mopen\">(<\/span>20<span class=\"mbin\">\u2212<\/span>16<span class=\"mclose\">)<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo>=<\/mo><msqrt><mrow><mn>20<\/mn><mo>\u00d7<\/mo><mn>10<\/mn><mo>\u00d7<\/mo><mn>6<\/mn><mo>\u00d7<\/mo><mn>4<\/mn><\/mrow><\/msqrt><mo>=<\/mo><msqrt><mn>4800<\/mn><\/msqrt><mo>\u2248<\/mo><mn>69.28<\/mn><mtext>\u2009<\/mtext><msup><mtext>cm<\/mtext><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">= \\sqrt{20 \\times 10 \\times 6 \\times 4} = \\sqrt{4800} \\approx 69.28 \\, \\text{cm}^2<\/annotation><\/semantics><\/math>\n<p><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"svg-align\"><span class=\"mord\">20<span class=\"mbin\">\u00d7<\/span>10<span class=\"mbin\">\u00d7<\/span>6<span class=\"mbin\">\u00d7<\/span>4<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"svg-align\"><span class=\"mord\">4800<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><span class=\"mrel\">\u2248<\/span><\/span><span class=\"base\"><span class=\"mord\">69.28<\/span><span class=\"mord\"><span class=\"mord text\">cm<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/li>\n<\/ol>\n<h3><span class=\"ez-toc-section\" id=\"Question_4_A_triangle_has_side_lengths_of_9_in_40_in_and_41_in_Find_its_area\"><\/span>Question 4 <strong>A triangle has side lengths of 9 in, 40 in, and 41 in. Find its area.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<ol>\n<li>Identify the side lengths:<br \/>\n<span class=\"katex\"><span class=\"katex\"><span class=\"katex-mathml\"> <\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mn>9<\/mn><mtext>\u2009<\/mtext><mtext>in<\/mtext><mo separator=\"true\">,<\/mo><mtext>\u2009<\/mtext><mi>b<\/mi><mo>=<\/mo><mn>40<\/mn><mtext>\u2009<\/mtext><mtext>in<\/mtext><mo separator=\"true\">,<\/mo><mtext>\u2009<\/mtext><mi>c<\/mi><mo>=<\/mo><mn>41<\/mn><mtext>\u2009<\/mtext><mtext>in<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">a = 9 \\, \\text{in}, \\, b = 40 \\, \\text{in}, \\, c = 41 \\, \\text{in}<\/annotation><\/semantics><\/math>\n<p><\/span><\/span><\/span>&nbsp;<\/p>\n<p><span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">9<\/span><span class=\"mord text\"><span class=\"mord\">in<\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">b<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">40<\/span><span class=\"mord text\"><span class=\"mord\">in<\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">41<\/span><span class=\"mord text\"><span class=\"mord\">in<\/span><\/span><\/span><\/span><\/span><\/li>\n<li>Calculate the semi-perimeter (s):&nbsp;<br \/>\n<math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>s<\/mi><mo>=<\/mo><mfrac><mrow><mo stretchy=\"false\">(<\/mo><mn>9<\/mn><mo>+<\/mo><mn>40<\/mn><mo>+<\/mo><mn>41<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><mn>2<\/mn><\/mfrac><mo>=<\/mo><mfrac><mn>90<\/mn><mn>2<\/mn><\/mfrac><mo>=<\/mo><mn>45<\/mn><mtext>\u2009<\/mtext><mtext>in<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">s = \\frac{(9 + 40 + 41)}{2} = \\frac{90}{2} = 45 \\, \\text{in}<\/annotation><\/semantics><\/math>\n<p><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">s<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\">2<span class=\"mopen\">(<\/span>9<span class=\"mbin\">+<\/span>40<span class=\"mbin\">+<\/span>41<span class=\"mclose\">)<\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\">290<\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">45<\/span><span class=\"mord text\"><span class=\"mord\">in<\/span><\/span><\/span><\/span><\/li>\n<li>Use Heron\u2019s formula:&nbsp;<br \/>\n<math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mtext>Area<\/mtext><mo>=<\/mo><msqrt><mrow><mn>45<\/mn><mo>\u00d7<\/mo><mo stretchy=\"false\">(<\/mo><mn>45<\/mn><mo>\u2212<\/mo><mn>9<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u00d7<\/mo><mo stretchy=\"false\">(<\/mo><mn>45<\/mn><mo>\u2212<\/mo><mn>40<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u00d7<\/mo><mo stretchy=\"false\">(<\/mo><mn>45<\/mn><mo>\u2212<\/mo><mn>41<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Area} = \\sqrt{45 \\times (45 &#8211; 9) \\times (45 &#8211; 40) \\times (45 &#8211; 41)}<\/annotation><\/semantics><\/math>\n<p><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord text\"><span class=\"mord\">Area<\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"svg-align\"><span class=\"mord\">45<span class=\"mbin\">\u00d7<\/span><span class=\"mopen\">(<\/span>45<span class=\"mbin\">\u2212<\/span>9<span class=\"mclose\">)<\/span><span class=\"mbin\">\u00d7<\/span><span class=\"mopen\">(<\/span>45<span class=\"mbin\">\u2212<\/span>40<span class=\"mclose\">)<\/span><span class=\"mbin\">\u00d7<\/span><span class=\"mopen\">(<\/span>45<span class=\"mbin\">\u2212<\/span>41<span class=\"mclose\">)<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo>=<\/mo><msqrt><mrow><mn>45<\/mn><mo>\u00d7<\/mo><mn>36<\/mn><mo>\u00d7<\/mo><mn>5<\/mn><mo>\u00d7<\/mo><mn>4<\/mn><\/mrow><\/msqrt><mo>=<\/mo><msqrt><mn>8100<\/mn><\/msqrt><mo>=<\/mo><mn>90<\/mn><mtext>\u2009<\/mtext><msup><mtext>in<\/mtext><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">= \\sqrt{45 \\times 36 \\times 5 \\times 4} = \\sqrt{8100} = 90 \\, \\text{in}^2<\/annotation><\/semantics><\/math>\n<p><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"svg-align\"><span class=\"mord\">45<span class=\"mbin\">\u00d7<\/span>36<span class=\"mbin\">\u00d7<\/span>5<span class=\"mbin\">\u00d7<\/span>4<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"svg-align\"><span class=\"mord\">8100<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">90<\/span><span class=\"mord\"><span class=\"mord text\">in<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/li>\n<\/ol>\n<h2><span class=\"ez-toc-section\" id=\"FAQs_on_Area_of_Triangles_with_3_Side\"><\/span>FAQs on Area of Triangles with 3 Side<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"How_can_I_find_the_length_of_an_unknown_side_in_a_triangle\"><\/span>How can I find the length of an unknown side in a triangle?<span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tTo find the length of an unknown side in a triangle, you can use various formulas, such as the Pythagorean Theorem for right triangles or the law of cosines for non-right triangles. The specific formula used depends on the information given, such as the lengths of other sides and angles of the triangle. \t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"How_can_I_find_the_angles_of_a_triangle_if_I_know_the_lengths_of_its_sides\"><\/span>How can I find the angles of a triangle if I know the lengths of its sides?<span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tTo find the angles of a triangle when the lengths of its sides are known, you can use the law of cosines or the law of sines. The law of cosines is applicable to all triangles, while the law of sines is specifically useful for solving triangles that are not right triangles. \t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"What_is_the_perimeter_of_a_triangle_and_how_do_I_calculate_it\"><\/span>What is the perimeter of a triangle and how do I calculate it? <span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\the perimeter of a triangle is the sum of the lengths of its three sides. To calculate the perimeter, you simply add up the lengths of the three sides. \t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"Can_I_find_the_area_of_a_triangle_without_knowing_its_height\"><\/span>Can I find the area of a triangle without knowing its height? <span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tYes, you can find the area of a triangle without knowing its height if you have the lengths of its sides. By using Heron's formula, you can calculate the area based on the lengths of the sides alone, without explicitly needing the height.\t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\n<script type=\"application\/ld+json\">\n\t{\n\t\t\"@context\": \"https:\/\/schema.org\",\n\t\t\"@type\": \"FAQPage\",\n\t\t\"mainEntity\": [\n\t\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"How can I find the length of an unknown side in a triangle?\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"To find the length of an unknown side in a triangle, you can use various formulas, such as the Pythagorean Theorem for right triangles or the law of cosines for non-right triangles. The specific formula used depends on the information given, such as the lengths of other sides and angles of the triangle.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"How can I find the angles of a triangle if I know the lengths of its sides?\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"To find the angles of a triangle when the lengths of its sides are known, you can use the law of cosines or the law of sines. The law of cosines is applicable to all triangles, while the law of sines is specifically useful for solving triangles that are not right triangles.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"What is the perimeter of a triangle and how do I calculate it? \",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"he perimeter of a triangle is the sum of the lengths of its three sides. To calculate the perimeter, you simply add up the lengths of the three sides.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"Can I find the area of a triangle without knowing its height? \",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"Yes, you can find the area of a triangle without knowing its height if you have the lengths of its sides. By using Heron's formula, you can calculate the area based on the lengths of the sides alone, without explicitly needing the height.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t\t\t\t]\n\t}\n<\/script>\n\n","protected":false},"excerpt":{"rendered":"<p>A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. [&hellip;]<\/p>\n","protected":false},"author":52,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"Area of Triangles with 3 Sides","_yoast_wpseo_title":"Area of Triangles with 3 Sides | Formula, Definition and Solved Examples","_yoast_wpseo_metadesc":"A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C.The length of the sides of a triangle may be same or different.","custom_permalink":"articles\/area-of-triangles-with-3-sides\/"},"categories":[8442,8443],"tags":[],"table_tags":[],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v17.9 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Area of Triangles with 3 Sides | Formula, Definition and Solved Examples<\/title>\n<meta name=\"description\" content=\"A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. 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