Right triangle calculator (2024)

Special righttriangleGeneral righttriangle

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Find the hypotenuse of a right triangle in whose legs are $ a = 18~ cm $ and$ b = \dfrac{13}{2} cm $.

Find the angle $\alpha$ of a right triangle if hypotenuse $ c = 8~cm$ and leg $ a =4~cm$.

Find the hypotenuse $ ~ c ~$ if $\alpha = 50^{\circ} $ and leg $ a = 8 $.

Find the area of a right triangle in which $\beta = 30^{\circ}$ and $b =\dfrac{5}{4} cm$

Right triangle

Right triangle is a type of triangle in which the measure of one angle is 90 degrees. The side oppositethe right angle is called hypotenuse. The other two sidesare called legs. This calculator uses the following formulas to find the missing elements of a righttriangle.

Pythagorean Theorem:	$$ a^2 + b^2 = c^2 $$
Area:	$$ A = \frac{a b}{2} $$
Trig. functions:	$$ \sin \alpha = \frac{a}{c} $$
	$$ \cos \alpha = \frac{b}{c} $$
	$$ \tan \alpha = \frac{a}{b} $$

Find hypotenuse $ c $ of a right triangle if leg a = 4 cm and leg b = 8 cm.

The Pythagorean theorem is the key formula for calculating the missing sides of a right triangle.This theorem is useful when we need to find the third side if the two sides are given.

$$ \begin{aligned}c^2 &= a^2 + b^2 \\[ 1 em]c^2 &= 4^2 + 8^2 \\[ 1 em]c^2 &= 16 + 64 \\[ 1 em]c^2 &= 80 \\[ 1 em]c &= \sqrt{80} \\[ 1 em]c &= \sqrt{16 \cdot 5} \\[ 1 em]c &= 4\sqrt{5}\\\end{aligned}$$

Find the angle $\alpha$ of a right triangle if hypotenuse c = 14 cm and leg a = 8 cm.

To find the missing angle, we must use trigonometric functions. For this example,the sine function is appropriate as we have the hypotenuse and side a.

$$ \begin{aligned}\sin \alpha & = \frac{a}{c} \\[1 em]\sin \alpha & = \frac{8}{14} \\[1 em]\sin \alpha & = 0.5714 \\[1 em]\alpha &= \sin^{-1} (0.5714) \\[1 em]\alpha & \approx \, 39^{o}\end{aligned}$$

Special triangle 30-60-90

Triangles with angles of 30^o – 60^o – 90^o is most common in high schoolmathbecause it can be solved without using trigonometry.When solving this triangle, calculator uses the fact that the ratio of sides is$ 1: \sqrt{3}: 2 $ (see the picture below).

When solving 30-60-90 right trange three cases can occur:

Case 1: If the short leg is a = 10, then the side b is $ b = a \sqrt{3} = 10 \sqrt{3}$ and hypotenuse cis $ c = 2 * 10 = 20$.

Case 2: If the long leg is b = 12, then the leg a is $ a = \frac{b}{\sqrt{3}} = 12 \sqrt{3} = 4\sqrt{3} $and hypotenuse c is $ c = 2 * a = 8\sqrt{3} $.

Case 3: If the hypotenuse c = 30, then the leg a is $ a = \frac{c}{2} = 15 $ and leg b is $ b = a\sqrt{3} = 15\sqrt{3} $.

Special triangle 45-45-90

Triangle with angles of 45^o – 45^o – 90^o is the second type of specialtriangle.The ratio of sides for this triangle is$ 1: 1 : \sqrt{2} $. For example, if the shortest side a = 4, then the sideb is also 4 and hypotenuse c is $ c = a * \sqrt{2} = 4\sqrt{2} $.