Pythagorean Theorem (Right Triangle) Calculator

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

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Your Input

Solve the triangle, if $$$a = 6$$$, $$$b = 8$$$, $$$C = 90^0$$$.

Solution

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

In our case, $$$c^{2} = 6^{2} + 8^{2} = 100$$$.

Thus, $$$c = 10$$$.

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

Thus, $$$\sin{\left(A \right)} = \frac{3}{5}$$$.

There are two possible cases:

  1. $$$A = \left(\frac{180 \operatorname{asin}{\left(\frac{3}{5} \right)}}{\pi}\right)^0$$$

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

    In our case, $$$B = 180^0 - \left(\left(\frac{180 \operatorname{asin}{\left(\frac{3}{5} \right)}}{\pi}\right)^0 + 90^0\right) = \left(\frac{- \pi \left(\frac{180 \operatorname{asin}{\left(\frac{3}{5} \right)}}{\pi} + 90\right) + 180 \pi}{\pi}\right)^0.$$$

    The area is $$$S = \frac{1}{2} a b = \left(\frac{1}{2}\right)\cdot \left(6\right)\cdot \left(8\right) = 24$$$.

    The perimeter is $$$P = a + b + c = 6 + 8 + 10 = 24$$$.

  2. $$$A = \left(\frac{- 180 \operatorname{asin}{\left(\frac{3}{5} \right)} + 180 \pi}{\pi}\right)^0$$$

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

    In our case, $$$B = 180^0 - \left(\left(\frac{- 180 \operatorname{asin}{\left(\frac{3}{5} \right)} + 180 \pi}{\pi}\right)^0 + 90^0\right) = \left(\frac{- \pi \left(90 + \frac{- 180 \operatorname{asin}{\left(\frac{3}{5} \right)} + 180 \pi}{\pi}\right) + 180 \pi}{\pi}\right)^0.$$$

    This case is impossible, since the angle is non-positive.

Answer

$$$a = 6$$$A

$$$b = 8$$$A

$$$c = 10$$$A

$$$A = \left(\frac{180 \operatorname{asin}{\left(\frac{3}{5} \right)}}{\pi}\right)^0\approx 36.869897645844021^0$$$A

$$$B = \left(\frac{- \pi \left(\frac{180 \operatorname{asin}{\left(\frac{3}{5} \right)}}{\pi} + 90\right) + 180 \pi}{\pi}\right)^0\approx 53.130102354155979^0$$$A

$$$C = 90^0$$$A

Area $$$S = 24$$$A.

Perimeter $$$P = 24$$$A.