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Hypotenuse Calculator

Find the hypotenuse (c), other sides (a) and (b), or Area:

Calculate From:

a

cm ▾

b

cm ▾

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Hypotenuse Calculator

The Hypotenuse Calculator helps you find the longest side of a right triangle. You can calculate the hypotenuse based on different known parameters and also determine any missing side of the triangle.

How to Use the Hypotenuse Calculator

Select the known parameters from the drop-down menu, which may include:

Two sides (legs) ∟
Angle ∡ and one side
Area ⊿ and one side

After selecting the method, enter the values and click Calculate to get the results.

What is the Hypotenuse of a Right Triangle?

The hypotenuse is the longest side of a right triangle, located opposite the right angle. The other sides are called legs or catheti.

Longest side of a right triangle
Opposite the right angle
Calculated using the Pythagorean Theorem: \(a^2 + b^2 = c^2\)

Use our Pythagorean Theorem Calculator to find unknown sides of a triangle.

Formulas for Hypotenuse

Condition 1 – Two sides are known:

\( c = \sqrt{a^2 + b^2} \)

Condition 2 – One side and an angle are known:

If side \(a\) and angle \(\alpha\) are known: \( c = \frac{a}{\sin(\alpha)} \)
If side \(b\) and angle \(\beta\) are known: \( c = \frac{b}{\sin(\beta)} \)

Condition 3 – Area and one side are known:

Given area and side \(a\): \( c = \sqrt{a^2 + \frac{2 \times \text{area}}{a^2}} \)
Given area and side \(b\): \( c = \sqrt{\frac{2 \times \text{area}}{b^2} + b^2} \)

Other Triangle Calculations:

Length of side \(a\): \( a = \frac{2 \times \text{area}}{b} \)
Length of side \(b\): \( b = \frac{2 \times \text{area}}{a} \)
Area of the triangle: \( \text{area} = \frac{a \times b}{2} \)

Example #1: Find Hypotenuse from Two Legs

Right triangle with legs:

a = 3 cm
b = 4 cm

Formula: \( c = \sqrt{a^2 + b^2} \)

Calculation:

\( c = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \text{ cm} \)

Example #2: Find Hypotenuse from Side and Angle

A crane lifts a steel beam 30 ft above the ground. Base distance from the building is 40 ft. Angle of elevation = 60°.

Base \(a = 40 \text{ ft}\)
Angle \(\alpha = 60^\circ\)

Formula: \( c = \frac{a}{\sin(\alpha)} \)

Calculation:

\( c = \frac{40}{\sin(60)} = \frac{40}{0.866} \approx 46.1 \text{ ft} \)

The crane arm must be approximately 46.1 ft to reach the top.

Special Right Triangles

45°-45°-90° Triangle

For a 45°-45°-90° triangle, the sides are in the ratio \(1:1:\sqrt{2}\). If one leg = \(x\), hypotenuse = \( c = x\sqrt{2} \)

30°-60°-90° Triangle

The shortest leg is opposite 30°; the hypotenuse is twice this length: \( c = 2 \times \text{short leg} \)

If the longer leg is known: \( c = \frac{\text{long leg}}{\sqrt{3}} \)

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