Law of Cosines Calculator
Solve triangles using the law of cosines
The calculator will solve the given triangle using the law of cosines (wherever possible), with steps shown.
Related calculator: Law of Sines Calculator
Your Input
Solve the triangle, if $$$a = 7$$$, $$$b = 14$$$, $$$C = 60^0$$$.
Solution
According to the law of cosines: $$$c^{2} = a^{2} + b^{2} - 2 a b \cos{\left(C \right)}$$$.
In our case, $$$c^{2} = 7^{2} + 14^{2} - \left(2\right)\cdot \left(7\right)\cdot \left(14\right)\cdot \left(\cos{\left(60^0 \right)}\right) = 147$$$.
Thus, $$$c = 7 \sqrt{3}$$$.
According to the law of cosines: $$$a^{2} = b^{2} + c^{2} - 2 b c \cos{\left(A \right)}$$$.
In our case, $$$7^{2} = 14^{2} + \left(7 \sqrt{3}\right)^{2} - \left(2\right)\cdot \left(14\right)\cdot \left(7 \sqrt{3}\right)\cdot \left(\cos{\left(A \right)}\right)$$$.
Thus, $$$\cos{\left(A \right)} = \frac{\sqrt{3}}{2}$$$.
Hence, $$$A = 30^0$$$.
The third angle is $$$B = 180^0 - \left(A + C\right)$$$.
In our case, $$$B = 180^0 - \left(30^0 + 60^0\right) = 90^0$$$.
The area is $$$S = \frac{1}{2} a b \sin{\left(C \right)} = \left(\frac{1}{2}\right)\cdot \left(7\right)\cdot \left(14\right)\cdot \left(\sin{\left(60^0 \right)}\right) = \frac{49 \sqrt{3}}{2}.$$$
The perimeter is $$$P = a + b + c = 7 + 14 + 7 \sqrt{3} = 7 \left(\sqrt{3} + 3\right)$$$.