What is the formula of area of triangle?

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The area of a triangle represents the amount of space enclosed within its three sides. Understanding different formulas for calculating triangle area is essential for geometry, trigonometry, and practical applications in engineering, architecture, and physics.Triangle Area FormulasFormula TypeFormulaWhen to UseVariables ExplanationExampleBasic Area FormulaA = ½ × base × heightWhen base and perpendicular height are knownbase = any side length
height = perpendicular distance to baseBase = 8 cm, Height = 6 cm
Area = ½ × 8 × 6 = 24 cm²Heron's FormulaA = √[s(s-a)(s-b)(s-c)]When all three sides are knowna, b, c = side lengths
s = semi-perimeter = (a+b+c)/2Sides: 3, 4, 5 cm
s = (3+4+5)/2 = 6
Area = √[6(6-3)(6-4)(6-5)] = 6 cm²Right Triangle FormulaA = ½ × leg₁ × leg₂When two perpendicular sides are knownleg₁, leg₂ = perpendicular sides (not hypotenuse)Legs: 3 cm, 4 cm
Area = ½ × 3 × 4 = 6 cm²Equilateral Triangle FormulaA = (√3/4) × side²When all sides are equalside = length of any sideSide = 6 cm
Area = (√3/4) × 6² = 9√3 ≈ 15.59 cm²Isosceles Triangle FormulaA = (b/4) × √(4a² - b²)When two sides are equala = equal sides length
b = base lengthEqual sides = 5 cm, Base = 6 cm
Area = (6/4) × √(4×25 - 36) = 12 cm²Using Two Sides and Included AngleA = ½ × a × b × sin(C)When two sides and included angle are knowna, b = side lengths
C = angle between sides a and bSides: 4, 6 cm, Angle = 60°
Area = ½ × 4 × 6 × sin(60°) = 6√3 cm²Using Coordinates**A = ½x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)**When three vertices coordinates are knownUsing Vector Cross Product**A = ½u⃗ × v⃗**When two sides are represented as vectorsUsing CircumradiusA = (abc)/(4R)When all sides and circumradius are knowna, b, c = side lengths
R = circumradiusSides: 3, 4, 5 cm, R = 2.5 cm
Area = (3×4×5)/(4×2.5) = 6 cm²Using InradiusA = r × sWhen inradius and semi-perimeter are knownr = inradius
s = semi-perimeterInradius = 2 cm, Semi-perimeter = 6 cm
Area = 2 × 6 = 12 cm²Using Median LengthA = (4/3) × √[s_m(s_m-m_a)(s_m-m_b)(s_m-m_c)]When all three medians are knownm_a, m_b, m_c = median lengths
s_m = (m_a+m_b+m_c)/2Complex calculation - typically used in advanced problemsSpecial Triangle FormulasScalene Triangle (All sides different)Primary Formula: Use Heron's formula or coordinate methodAlternative: A = ½ × a × b × sin(C) when angle is knownRight-Angled TriangleHypotenuse Known: A = ½ × √[(a+b+c)(-a+b+c)(a-b+c)(a+b-c)]/2Using Trigonometry: A = ½ × base × height = ½ × a × b (where a, b are legs)Obtuse TriangleSame formulas apply: Use Heron's formula or sine formulaNote: One angle > 90°, but area calculations remain the sameQuick Reference for Common TrianglesTriangle TypeQuick FormulaKey Characteristics3-4-5 Right TriangleA = 6 square unitsClassic Pythagorean triple30-60-90 TriangleA = (side²√3)/4Angles: 30°, 60°, 90°45-45-90 TriangleA = side²/2Isosceles right triangleEquilateral TriangleA = (side²√3)/4All angles = 60°, all sides equalPractice ProblemsProblem 1: Basic FormulaFind the area of a triangle with base 10 cm and height 8 cm.Solution: A = ½ × 10 × 8 = 40 cm²Problem 2: Heron's FormulaFind the area of a triangle with sides 5 cm, 12 cm, and 13 cm.Solution: s = (5+12+13)/2 = 15 A = √[15(15-5)(15-12)(15-13)] = √[15×10×3×2] = 30 cm²Problem 3: Sine FormulaFind the area of a triangle with sides 6 cm and 8 cm, with an included angle of 45°.Solution: A = ½ × 6 × 8 × sin(45°) = ½ × 6 × 8 × (√2/2) = 12√2 ≈ 16.97 cm²

Formula Type	Formula	When to Use	Variables Explanation	Example
Basic Area Formula	A = ½ × base × height	When base and perpendicular height are known	base = any side length<br>height = perpendicular distance to base	Base = 8 cm, Height = 6 cm<br>Area = ½ × 8 × 6 = 24 cm²
Heron's Formula	A = √[s(s-a)(s-b)(s-c)]	When all three sides are known	a, b, c = side lengths<br>s = semi-perimeter = (a+b+c)/2	Sides: 3, 4, 5 cm<br>s = (3+4+5)/2 = 6<br>Area = √[6(6-3)(6-4)(6-5)] = 6 cm²
Right Triangle Formula	A = ½ × leg₁ × leg₂	When two perpendicular sides are known	leg₁, leg₂ = perpendicular sides (not hypotenuse)	Legs: 3 cm, 4 cm<br>Area = ½ × 3 × 4 = 6 cm²
Equilateral Triangle Formula	A = (√3/4) × side²	When all sides are equal	side = length of any side	Side = 6 cm<br>Area = (√3/4) × 6² = 9√3 ≈ 15.59 cm²
Isosceles Triangle Formula	A = (b/4) × √(4a² - b²)	When two sides are equal	a = equal sides length<br>b = base length	Equal sides = 5 cm, Base = 6 cm<br>Area = (6/4) × √(4×25 - 36) = 12 cm²
Using Two Sides and Included Angle	A = ½ × a × b × sin(C)	When two sides and included angle are known	a, b = side lengths<br>C = angle between sides a and b	Sides: 4, 6 cm, Angle = 60°<br>Area = ½ × 4 × 6 × sin(60°) = 6√3 cm²
Using Coordinates	**A = ½	x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)	**	When three vertices coordinates are known
Using Vector Cross Product	**A = ½	u⃗ × v⃗	**	When two sides are represented as vectors
Using Circumradius	A = (abc)/(4R)	When all sides and circumradius are known	a, b, c = side lengths<br>R = circumradius	Sides: 3, 4, 5 cm, R = 2.5 cm<br>Area = (3×4×5)/(4×2.5) = 6 cm²
Using Inradius	A = r × s	When inradius and semi-perimeter are known	r = inradius<br>s = semi-perimeter	Inradius = 2 cm, Semi-perimeter = 6 cm<br>Area = 2 × 6 = 12 cm²
Using Median Length	A = (4/3) × √[s_m(s_m-m_a)(s_m-m_b)(s_m-m_c)]	When all three medians are known	m_a, m_b, m_c = median lengths<br>s_m = (m_a+m_b+m_c)/2	Complex calculation - typically used in advanced problems

What is the formula of area of triangle?

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Triangle Type	Quick Formula	Key Characteristics
3-4-5 Right Triangle	A = 6 square units	Classic Pythagorean triple
30-60-90 Triangle	A = (side²√3)/4	Angles: 30°, 60°, 90°
45-45-90 Triangle	A = side²/2	Isosceles right triangle
Equilateral Triangle	A = (side²√3)/4	All angles = 60°, all sides equal