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 $$$a = 6$$$, $$$b = 6 \sqrt{3}$$$, $$$C = 90^{\circ}$$$.
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:
$$$A = 30^{\circ}$$$
The third angle is $$$B = 180^{\circ} - \left(A + C\right)$$$.
In our case, $$$B = 180^{\circ} - \left(30^{\circ} + 90^{\circ}\right) = 60^{\circ}$$$.
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 $$$P = a + b + c = 6 + 6 \sqrt{3} + 12 = 6 \left(\sqrt{3} + 3\right)$$$.
$$$A = 150^{\circ}$$$
The third angle is $$$B = 180^{\circ} - \left(A + C\right)$$$.
In our case, $$$B = 180^{\circ} - \left(150^{\circ} + 90^{\circ}\right) = -60^{\circ}$$$.
This case is impossible, since the angle is nonpositive.
Answer
$$$a = 6$$$A
$$$b = 6 \sqrt{3}\approx 10.392304845413264$$$A
$$$c = 12$$$A
$$$A = 30^{\circ}$$$A
$$$B = 60^{\circ}$$$A
$$$C = 90^{\circ}$$$A
Area: $$$S = 18 \sqrt{3}\approx 31.176914536239791$$$A.
Perimeter: $$$P = 6 \left(\sqrt{3} + 3\right)\approx 28.392304845413264$$$A.