Online calculator to calculate the third side c of a triangle given its two sides a and b and area A.
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
c_{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}) ]
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).
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