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- Right Triangle Calculator

The right triangle calculator finds the missing area, angle, leg, hypotenuse and height of a triangle. The calculator also provides steps on how to solve the most important right triangles: the 30-60-90 triangle and the 45-45-90 triangle.

**Error: **To solve a special triangle, you need to input just one of its sides.

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Examples

ex 1:

Find the hypotenuse of a right triangle in whose legs are a=18cm
and b=13/2 cm.

ex 2:

Find the angle α of a right triangle if hypotenuse
c=8cm and leg a=4cm.

ex 3:

Find the hypotenuse c if
α=50^{o} and leg a=8.

ex 4:

Find the area of a right triangle in which β=30^{o}
and b=5/4cm.

Related calculators

Find more worked-out examples in our database of solved problems..

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

Pythagorean Theorem: |
$$ a^2 + b^2 = c^2 $$ | |

Area: |
$$ A = \frac{a b}{2} $$ | |

Trig. functions: |
$$ \sin \alpha = \frac{a}{c} $$ | |

$$ \cos \alpha = \frac{b}{c} $$ | ||

$$ \tan \alpha = \frac{a}{b} $$ |

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.

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

$$ \begin{aligned} \sin \alpha & = \frac{a}{c} \\[1 em] \sin \alpha & = \frac{8}{14} \\[1 em] \sin \alpha & = 0.5714 \\[1 em] \alpha &= \sin^{-1} (0.5714) \\[1 em] \alpha & \approx \, 39^{o} \end{aligned} $$Triangles with angles of 30^{o} – 60^{o} – 90^{o} is most common in high school
math
because 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 c
is $ 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} $.

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

RESOURCES

1. Right Angled Triangle definition with examples

2. Solve for a side in right triangles

3. Right triangles practice tests

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