Find third side of a triangle Given its area and two sides - Geometry Calculator
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| Online calculator to calculate the third side c of a triangle given its two sides a and b and area A. Formulas
The area A of a triangle is given its two sides a and b making an angle α is given by: Use the cosine rule to express side c in terms of sides a and b and cos(α) Use the above equation to find an expression for cos (α) cos (α) = (a 2 + b 2 - c 2) / 2 a b Use the trigonometric identity sin(α) = √(1- cos 2 (α)) to rewrite the formula for the area as A = (1/2) a b sin(α) = (1/2) a b √(1- ((a 2 + b 2 - c 2) / 2 a b) 2) Simplify to obtain A = (1 / 4) √(4 a 2 b 2 - (a 2 + b 2 - c 2) 2) Square both sides and solve for the third side c to obtain two possible solutions c2 = √ [ a 2 + b 2 - √(4 a 2 b 2 - 16 A 2) ] 1) two solutions if 4 a 2 b 2 - 16 A 2 > 0 2) one solution (or two equal solutions) if 4 a 2 b 2 - 16 A 2 = 0 3) no solutions if 4 a 2 b 2 - 16 A 2 < 0 How to use the calculatorEnter the area, side a and side b and press "enter". The output is the third side c (c1 , c2) of the triangle if the problem has a solution (an example is done for you).
More References and linksArea of Triangles.Cosine Law Problems. Online Geometry Calculators and Solvers. |
