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 ? is given by:

A = (1/2) a b sin(?)

Use the cosine rule to express side c in terms of sides a and b and cos(?)

c^{ 2} = a^{ 2} + b^{ 2} - 2 a b cos (?)

Use the above equation to find an expression for cos (?)
cos (?) = (a^{ 2} + b^{ 2} - c^{ 2}) / 2 a b
Use the trigonometric identity sin(?) = √(1- cos^{ 2} (?)) to rewrite the formula for the area as
A = (1/2) a b sin(?) = (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

Note that the problem has
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).