Find third side of a triangle Given its area and two sides - Geometry Calculator
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 calculator
Enter the area, side a and side b and press "enter". The output is the third side c (c 1 , c 2 ) of the triangle if the problem has a solution (an example is done for you).
More References and links
Area of Triangles .Cosine Law Problems .
Online Geometry Calculators and Solvers .