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.

area and two sides to find third side

The area A of a triangle given its two sides a and b making an angle α is given by:

A = (1/2) a b sin(α)


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

c 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

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 =
side a =
side b =
4 a 2 b 2 - 16 A 2 =
side c1 =
side c2 =

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