# 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)

^{ 2}= a

^{ 2}+ b

^{ 2}- 2 a b 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

_{1}= √ [ a

^{ 2}+ b

^{ 2}+ √(4 a

^{ 2}b

^{ 2}- 16 A

^{ 2}) ]

c

_{2}= √ [ 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 .