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.

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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:

  1. $$$A = 30^0$$$

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

  2. $$$A = 150^0$$$

    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.

Answer

$$$a = 6$$$A

$$$b = 6 \sqrt{3}\approx 10.392304845413264$$$A

$$$c = 12$$$A

$$$A = 30^0$$$A

$$$B = 60^0$$$A

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

Area: $$$S = 18 \sqrt{3}\approx 31.176914536239791$$$A.

Perimeter: $$$P = 6 \left(\sqrt{3} + 3\right)\approx 28.392304845413264$$$A.