# Triangle Calculator

## Solve triangles step by step

The calculator will try to find all sides and angles of the triangle (right triangle, obtuse, acute, isosceles, equilateral), as well as its perimeter and area, with steps shown.

### Your Input

**Solve the triangle, if $$$a = 9$$$, $$$b = 9 \sqrt{2}$$$, $$$C = 45^0$$$.**

### Solution

According to the law of cosines: $$$c^{2} = a^{2} + b^{2} - 2 a b \cos{\left(C \right)}$$$.

In our case, $$$c^{2} = 9^{2} + \left(9 \sqrt{2}\right)^{2} - \left(2\right)\cdot \left(9\right)\cdot \left(9 \sqrt{2}\right)\cdot \left(\cos{\left(45^0 \right)}\right) = 81.$$$

Thus, $$$c = 9$$$.

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

In our case, $$$\frac{9}{\sin{\left(A \right)}} = \frac{9}{\sin{\left(45^0 \right)}}$$$.

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

There are two possible cases:

$$$A = 45^0$$$

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

In our case, $$$B = 180^0 - \left(45^0 + 45^0\right) = 90^0$$$.

The area is $$$S = \frac{1}{2} a b \sin{\left(C \right)} = \left(\frac{1}{2}\right)\cdot \left(9\right)\cdot \left(9 \sqrt{2}\right)\cdot \left(\sin{\left(45^0 \right)}\right) = \frac{81}{2}$$$.

The perimeter is $$$P = a + b + c = 9 + 9 \sqrt{2} + 9 = 9 \left(\sqrt{2} + 2\right)$$$.

$$$A = 135^0$$$

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

In our case, $$$B = 180^0 - \left(135^0 + 45^0\right) = 0^0$$$.

This case is impossible, since the angle is nonpositive.

### Answer

**$$$a = 9$$$A**

**$$$b = 9 \sqrt{2}\approx 12.727922061357855$$$A**

**$$$c = 9$$$A**

**$$$A = 45^0$$$A**

**$$$B = 90^0$$$A**

**$$$C = 45^0$$$A**

**Area: $$$S = \frac{81}{2} = 40.5$$$A.**

**Perimeter: $$$P = 9 \left(\sqrt{2} + 2\right)\approx 30.727922061357855$$$A.**