# 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
c1 = √ [ a 2 + b 2 + √(4 a 2 b 2 - 16 A 2) ]
c2 = √ [ a 2 + b 2 - √(4 a 2 b 2 - 16 A 2) ]
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 (c1 , c2) of the triangle if the problem has a solution (an example is done for you).

 Area = 200 side a = 40 side b = 26 4 a 2 b 2 - 16 A 2 = side c1 = side c2 =