{"id":155118,"date":"2022-03-26T00:17:25","date_gmt":"2022-03-25T18:47:25","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/volume-of-a-frustum-explanation-formula-property-and-faqs\/"},"modified":"2022-12-23T18:21:30","modified_gmt":"2022-12-23T12:51:30","slug":"volume-of-a-frustum-explanation-formula-property-and-faqs","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/maths\/volume-of-a-frustum\/","title":{"rendered":"Volume of a Frustum \u2013 Explanation, Formula, Property and FAQs"},"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\/maths\/volume-of-a-frustum\/#Explain_in_Detail_Frustum\" title=\"Explain in Detail : Frustum\">Explain in Detail : Frustum<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/volume-of-a-frustum\/#Similar_Property_of_Triangles_to_Find_Derivation_of_Volume_of_Frustum\" title=\"Similar Property of Triangles to Find Derivation of Volume of Frustum\">Similar Property of Triangles to Find Derivation of Volume of Frustum<\/a><\/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\/maths\/volume-of-a-frustum\/#How_to_Find_Total_Surface_Area_and_Curved_Surface_Area_in_a_Volume_Truncated_Cone\" title=\"How to Find Total Surface Area and Curved Surface Area in a Volume Truncated Cone\">How to Find Total Surface Area and Curved Surface Area in a Volume Truncated Cone<\/a><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"Explain_in_Detail_Frustum\"><\/span>Explain in Detail : Frustum<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>In geometry, a <b><i lang=\"la\">frustum<\/i><\/b> (from the Latin for \u201cmorsel\u201d; plural: <i>frusta<\/i> or <i>frustums<\/i>) is the portion of a solid (normally a pyramid or a cone) that lies between one or two parallel planes cutting it. The base faces are polygonal, the side faces are trapezoidal. A <b>right frustum<\/b> is a right pyramid or a right cone truncated perpendicularly to its axis. Volume of a Frustum \u2013 Explanation Formula .<\/p>\n<p>If a frustum has all its edges of the same length (<i>equilateral figure<\/i>), then it is a uniform prism.<\/p>\n<p>In computer graphics, the viewing frustum is the three-dimensional region which is visible on the screen. It is formed by a clipped pyramid; in particular, <i>frustum culling<\/i> is a method of hidden surface determination.<\/p>\n<p>In the aerospace industry, a frustum is the fairing between two stages of a multistage rocket (such as the Saturn V), which is shaped like a truncated cone.<\/p>\n<p>Volume of a Frustum \u2013 Explanation Formula Property and FAQs.<\/p>\n<p>A <a href=\"https:\/\/en.wikipedia.org\/wiki\/Frustum\" target=\"_blank\" rel=\"noopener\">frustum<\/a> is a truncated pyramid.<\/p>\n<p>&nbsp;<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Similar_Property_of_Triangles_to_Find_Derivation_of_Volume_of_Frustum\"><\/span>Similar Property of Triangles to Find Derivation of Volume of Frustum<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<div class=\"flex-1 overflow-hidden\">\n<div class=\"react-scroll-to-bottom--css-kqyjr-79elbk h-full dark:bg-gray-800\">\n<div class=\"react-scroll-to-bottom--css-kqyjr-1n7m0yu\">\n<div class=\"flex flex-col items-center text-sm h-full dark:bg-gray-800\">\n<div class=\"w-full border-b border-black\/10 dark:border-gray-900\/50 text-gray-800 dark:text-gray-100 group bg-gray-50 dark:bg-[#444654]\">\n<div class=\"text-base gap-4 md:gap-6 m-auto md:max-w-2xl lg:max-w-2xl xl:max-w-3xl p-4 md:py-6 flex lg:px-0\">\n<div class=\"relative flex w-[calc(100%-50px)] md:flex-col lg:w-[calc(100%-115px)]\">\n<div class=\"flex flex-grow flex-col gap-3\">\n<div class=\"min-h-[20px] flex flex-col items-start gap-4 whitespace-pre-wrap\">\n<div class=\"markdown prose w-full break-words dark:prose-invert light\">\n<p>The similar property of triangles states that if two triangles have the same shape (i.e., they are similar), then the ratio of their corresponding sides is always the same. This property can be used to derive the formula for the volume of a frustum, which is the shape that results when you cut off the top of a cone or pyramid.<\/p>\n<p>To derive the formula for the volume of a frustum, let&#8217;s consider a cone with radius r and height h. If we cut off the top of the cone to create a frustum, the height of the frustum will be h&#8217; and the radii of the top and bottom of the frustum will be r1 and r2, respectively.<\/p>\n[asy] pair A,B,C,D,E,F,G,H; A=(0,0); B=(2,0); C=(1,2<em>3^.5); D=(1,0); E=(.5,0); F=(1,1); G=(1,.5<\/em>3^.5); H=(.5,.5<em>3^.5); draw(A&#8211;B&#8211;C&#8211;cycle); draw(D&#8211;E&#8211;F&#8211;cycle); draw(D&#8211;G&#8211;H&#8211;cycle); draw(D&#8211;C,dashed); label(&#8220;$r$&#8221;,(1,1.2<\/em>3^.5),N); label(&#8220;$h$&#8221;,(1,.5),W); label(&#8220;$r_1$&#8221;,(.5,.2<em>3^.5),SW); label(&#8220;$r_2$&#8221;,(.5,.7<\/em>3^.5),NE); label(&#8220;$h&#8217;$&#8221;,(1,.85),E); [\/asy]\n<\/div>\n<p><strong>In case of confusion in reading the above text : <\/strong><\/p>\n[\/asy] is a tag used in the Asymptote programming language to indicate the end of an Asymptote block of code. Asymptote is a powerful programming language for creating technical drawings and scientific figures.<\/p>\n<p>The label and pair commands are Asymptote commands used to add labels and pairs of coordinates to a drawing. A label is used to add text to a drawing, and a pair is used to specify a pair of coordinates in the drawing.<\/p>\n<p>Here is an example of how to draw a frustum using Asymptote:<\/p>\n[asy] size(200);<\/p>\n<p>pair A,B,C,D,E,F; A=(0,0); B=(2,0); C=(1,2<em>3^.5); D=(1,0); E=(.5,0); F=(1,1); draw(A&#8211;B&#8211;C&#8211;cycle); draw(D&#8211;E&#8211;F&#8211;cycle); draw(D&#8211;C,dashed); label(&#8220;$r_1$&#8221;,(.5,.2<\/em>3^.5),SW); label(&#8220;$r_2$&#8221;,(.5,.7*3^.5),NE); label(&#8220;$h&#8217;$&#8221;,(1,.85),E); [\/asy]\n<p>This code creates a frustum with base radius r1, top radius r2, and height h&#8217;. The base of the frustum is a circle with radius r1, and the top is a circle with radius r2. The lateral surface of the frustum is a sloping surface connecting the bases. The dashed line represents the cut that was made to create the frustum.<\/p>\n<div class=\"markdown prose w-full break-words dark:prose-invert light\">\n<p>We can draw a cross-section of the frustum and the original cone as shown above. We can see that the cross-section of the frustum is similar to the cross-section of the original cone. This means that the ratio of the corresponding sides of the two triangles is constant.<\/p>\n<p>We can write the ratio of the corresponding sides of the two triangles as:<\/p>\n<p>(r1\/r2) = (h&#8217;\/h)<\/p>\n<p>We can rearrange this equation to solve for h&#8217;:<\/p>\n<p>h&#8217; = (r1\/r2) * h<\/p>\n<p>The volume of the frustum is the volume of the original cone minus the volume of the smaller cone at the top of the frustum. The volume of the original cone is:<\/p>\n<p>Volume = (1\/3) * \u03c0 * r^2 * h<\/p>\n<p>The volume of the smaller cone at the top of the frustum is:<\/p>\n<p>Volume = (1\/3) * \u03c0 * r1^2 * h&#8217;<\/p>\n<p>Substituting the value of h&#8217; into the equation for the volume of the smaller cone, we get:<\/p>\n<p>Volume = (1\/3) * \u03c0 * r1^2 * (r1\/r2) * h<\/p>\n<p>The volume of the frustum is the difference between these two volumes:<\/p>\n<p>Volume = (1\/3) * \u03c0 * r^2 * h &#8211; (1\/3) * \u03c0 * r1^2 * (r1\/r2) * h<\/p>\n<p>Combining like terms, we get:<\/p>\n<p>Volume = (1\/3) * \u03c0 * h * (r^2 &#8211; r1^2 * (r1\/r2))<\/p>\n<p>This simplifies to:<\/p>\n<p>Volume = (1\/3) * \u03c0 * h<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-4 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"w-full h-48 flex-shrink-0\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"absolute bottom-0 left-0 w-full border-t md:border-t-0 dark:border-white\/20 md:border-transparent md:dark:border-transparent md:bg-vert-light-gradient bg-white dark:bg-gray-800 md:!bg-transparent dark:md:bg-vert-dark-gradient\">\n<form class=\"stretch mx-2 flex flex-row gap-3 pt-2 last:mb-2 md:last:mb-6 lg:mx-auto lg:max-w-3xl lg:pt-6\">\n<div class=\"relative flex h-full flex-1 md:flex-col\">\n<div class=\"ml-1 mt-1.5 md:w-full md:m-auto md:flex md:mb-2 gap-2 justify-center\"><\/div>\n<\/div>\n<\/form>\n<\/div>\n<p><img loading=\"lazy\" class=\"aligncenter wp-image-155117 size-full\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/volume-of-a-frustum-explanation-formula-property-and-faqs.jpg\" alt=\"Volume of a Frustum \u2013 Explanation, Formula, Property and FAQs\" width=\"606\" height=\"428\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/volume-of-a-frustum-explanation-formula-property-and-faqs.jpg?v=1648234042 606w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/volume-of-a-frustum-explanation-formula-property-and-faqs-300x212.jpg?v=1648234042 300w\" sizes=\"(max-width: 606px) 100vw, 606px\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"How_to_Find_Total_Surface_Area_and_Curved_Surface_Area_in_a_Volume_Truncated_Cone\"><\/span>How to Find Total Surface Area and Curved Surface Area in a Volume Truncated Cone<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>To find the total surface area and curved surface area of a volume-<a href=\"https:\/\/infinitylearn.com\/surge\/maths\/volume-of-a-frustum\/\">truncated con<\/a>e, you can use the following formulas:<\/p>\n<p>Total surface area = \u03c0 * (r1 + r2) * \u221a((r1 &#8211; r2)^2 + h^2) + \u03c0 * r1^2 + \u03c0 * r2^2<\/p>\n<p>Curved surface area = \u03c0 * (r1 + r2) * \u221a((r1 &#8211; r2)^2 + h^2)<\/p>\n<p>where:<\/p>\n<ul>\n<li>r1 is the radius of the base of the cone<\/li>\n<li>r2 is the radius of the top of the cone (also known as the &#8220;smaller radius&#8221;)<\/li>\n<li>h is the height of the cone<\/li>\n<\/ul>\n<p>The total surface area of a volume-truncated cone is the sum of the curved surface area and the areas of the bases (the top and bottom circles). The curved surface area is the area of the lateral surface of the cone (the sloping part).<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Volume of a Frustum \u2013 Explanation Formula Property and FAQs.<br \/>\n<\/strong><\/p>\n<p>A frustum is the shape that results when you cut off the top of a cone or pyramid. The volume of a frustum can be calculated using the formula:<\/p>\n<p>Volume = (1\/3) * \u03c0 * h * (r1^2 + r2^2 + r1*r2)<\/p>\n<p>where:<\/p>\n<ul>\n<li>h is the height of the frustum<\/li>\n<li>r1 is the radius of the base of the frustum<\/li>\n<li>r2 is the radius of the top of the frustum<\/li>\n<\/ul>\n<p>Here are some properties of the volume of a frustum:<\/p>\n<ol>\n<li>The volume of a frustum is always less than the volume of the cone or pyramid from which it was derived.<\/li>\n<li>The volume of a frustum is directly proportional to the height of the frustum and the radii of the base and top.<\/li>\n<\/ol>\n<p>Here are some frequently asked questions about the volume of a frustum:<\/p>\n<p>Q: Can a frustum have a negative volume?<\/p>\n<p>A: No, the volume of a frustum cannot be negative because it is the space enclosed by the frustum.<\/p>\n<p>Q: Can the volume of a frustum be zero?<\/p>\n<p>A: Yes, the volume of a frustum can be zero if the height or one of the radii is zero.<\/p>\n<p>Q: Can the volume of a frustum be greater than the volume of a cone or pyramid?<\/p>\n<p>A: No, the volume of a frustum cannot be greater than the volume of a cone or pyramid because the frustum is derived by cutting off the top of the cone or pyramid.<\/p>\n<p><strong> <\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Explain in Detail : Frustum In geometry, a frustum (from the Latin for \u201cmorsel\u201d; plural: frusta or frustums) is the [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"Volume of a Frustum \u2013 Explanation Formula","_yoast_wpseo_title":"","_yoast_wpseo_metadesc":"Learn about Volume of a Frustum \u2013 Explanation Formula Property and FAQs explained by subject experts on infinitylearn.com.","custom_permalink":"maths\/volume-of-a-frustum\/"},"categories":[13],"tags":[],"table_tags":[],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v17.9 - 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