# Pythagorean Theorem (Right Triangle) Calculator

## Solve right triangles using the Pythagorean theorem

The calculator will try to find all sides of the right-angled triangle (the legs and the hypotenuse) using the Pythagorean theorem. It will also find all angles, as well as perimeter and area. The solution steps will be shown.

### Your Input

**Solve the triangle, if $$$a = 6$$$, $$$b = 6 \sqrt{3}$$$, $$$C = 90^0$$$.**

### Solution

According to the Pythagorean theorem: $$$c^{2} = a^{2} + b^{2}$$$.

In our case, $$$c^{2} = 6^{2} + \left(6 \sqrt{3}\right)^{2} = 144$$$.

Thus, $$$c = 12$$$.

According to the definition of the sine: $$$\sin{\left(A \right)} = \frac{a}{c}$$$.

Thus, $$$\sin{\left(A \right)} = \frac{1}{2}$$$.

There are two possible cases:

$$$A = 30^0$$$

The third angle is $$$B = 180^0 - \left(A + C\right)$$$.

In our case, $$$B = 180^0 - \left(30^0 + 90^0\right) = 60^0$$$.

The area is $$$S = \frac{1}{2} a b = \left(\frac{1}{2}\right)\cdot \left(6\right)\cdot \left(6 \sqrt{3}\right) = 18 \sqrt{3}$$$.

The perimeter is $$$P = a + b + c = 6 + 6 \sqrt{3} + 12 = 6 \left(\sqrt{3} + 3\right)$$$.

$$$A = 150^0$$$

The third angle is $$$B = 180^0 - \left(A + C\right)$$$.

In our case, $$$B = 180^0 - \left(150^0 + 90^0\right) = -60^0$$$.

This case is impossible, since the angle is nonpositive.