# 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. ## FormulasThe 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 calculatorEnter 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 linksArea of Triangles.Cosine Law Problems. Online Geometry Calculators and Solvers. |