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Hypotenuse Calculator

Find the hypotenuse (c), other sides (a) and (b), or Area for a right-angle triangle.

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Hypotenuse Calculator

This hypotenuse calculator helps to find the longest side of a right triangle. This tool calculates the hypotenuse by using different formulas based on the parameters that you provide. You can also find any other missing side of the right triangle with the help of our calculator.

How to Use Hypotenuse of a Triangle Calculator?

Which parameters are known? Select the method of calculation from the drop-down menu based on known values, including:

• Two sides ∟
• Angle ∡ and one side
• Area ⊿ and one side

Once selected, add values accordingly Click the 'Calculate' button to get the results.

What is the Hypotenuse of a Right Triangle?

"In the right triangle, hypotenuse is the longest side opposite to the right angle"

Other sides of the right angle triangle instead of the hypotenuse are known as legs or catheti.

Key-points About Hypotenuse:

• Longest side of a right triangle
• Opposite to the right angle
• Pythagorean Theorem “$$\ a^2 + b^2=c^2$$” used to find the hypotenuse

You can also get help from our online Pythagorean Theorem Calculator to find the unknown side of a right triangle.

Formulas for Hypotenuse:

There are different equations used by the hypotenuse leg calculator to find the length of the side that is opposite to the right angle (hypotenuse).

Condition 1 - Two side lengths are given:

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

Condition 2 - Angle & length of one side is given:

• If you have side ‘a’ and angle (α):

$$\ Hypotenuse (c) = \frac{a}{sin(α)}$$

• If you have side ‘b’ and angle (β):

$$\ Hypotenuse (c) = \frac{b}{sin(β)}$$

Condition 3 - Area & one side length are given:

• If you have area and side a:

$$\ Hypotenuse (c) = \sqrt{a^2 + \frac{area \times 2}{a^2}}$$

• If you have area and b side:

$$\ Hypotenuse (c) = \sqrt{\frac{area \times 2}{b^2} + b^2}$$ Apart from the longest length, the right triangle hypotenuse calculator also helps to find the other missing sides and area of the orthogonal triangle.

• To calculate the length of side a:

$$\ a = \frac{area \times 2}{b}$$

• To calculate the length of side b:

$$\ b = \frac{area \times 2}{a}$$

• To calculate the area of a Triangle:

$$\ area = \frac{a \times b}{2}$$

How to Find the Hypotenuse of a Right Triangle?

To find the hypotenuse, squaring the lengths of two sides that are not hypotenuse (legs) and then take a square root.

Example # 1:

Let us suppose that there's a right triangle where one leg (a) is 3cm long and the other leg (b) is 4cm long. Find the length of the longest side of this triangle (c)

• a = 3cm
• b = 4cm

Calculations:

The formula used to find the hypotenuse is:

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

Put the values into the formula:

$$\ Hypotenuse (c) = \sqrt{3^2 + 4^2}$$

$$\ Hypotenuse (c) = \sqrt{9 + 16}$$

$$\ Hypotenuse (c) = \sqrt{25}$$

$$\ Hypotenuse (c) = {5cm}$$

Example # 2:

Suppose a crane is used to lift the steel beam at a peak of the building that is under construction. The base of crane is 40 feet from the building and the arm needs to reach 30 feet above the ground at an angle of 60°. What is the required length of the crane arm (hypotenuse)?

Given Values:

• Base = ‘a’ = 40 ft
• Angle = ‘α’ = 60°

Calculations:

As we know the length of one side and the angle, so we need to use the formula to find the hypotenuse:

$$\ c = \frac{a}{sin(α)}$$

Put the values into the formula:

$$\ c = \frac{40 ft}{sin (60)}$$

$$\ c = \frac{40 ft}{0.866}$$

$$\ c = 46.1 ft$$

The required crane arm length to reach the point at a 60-degree angle is approximately 46.1 feet. Calculating the hypotenuse manually involves different formulas. No matter which set of parameters you know, simply put these parameters into the hypotenuse solver and get accurate results within the given steps.

How to Find the Hypotenuse for a 45 45 90 Right Triangle?

A 45-45-90 triangle is a special type of right triangle that has a ratio between the sides is always 1:1:√2. When one leg measures x units, the other leg is also x units in length, and the hypotenuse will be x√2 units long.

$$\ c = a\sqrt{2}$$

How to Find the Hypotenuse for a 30 60 90 Right Triangle?

The shortest leg is the opposite to the 30-degree angle and the hypotenuse is always twice the length of this leg. There are two ways to find the hypotenuse:

1. If you know the length of the shortest leg (a), you can find the hypotenuse (h)

$$\ Hypotenuse (h) = {\text {Shortest leg length} (a) \times {2}}$$

2. You can find the length of the long leg by multiplying the short leg by the square root of 3

$$\ Hypotenuse (h) = \frac{\text {Longer leg length} (b)}{\sqrt3}$$

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