Hypotenuse calculator

What is the hypotenuse?

The hypotenuse is the side of a right-angled triangle that lies opposite the right angle. In such triangles, the hypotenuse is always longer than the other two sides, known as the legs. In geometry and trigonometry, the hypotenuse plays a central role, especially due to the Pythagorean theorem. It is one of the most important elements of a right-angled triangle as it is opposite the right angle and typically the longest side of the triangle. Our hypotenuse calculator will help you easily determine the length of this side using various available methods.

Pythagorean theorem

The Pythagorean theorem is a key tool for determining the hypotenuse. It states that in any right-angled triangle, the square of the length of the hypotenuse ($c$) is equal to the sum of the squares of the lengths of the other two sides ($a$ and $b$):

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

Here, $a$ and $b$ are the lengths of the legs, and $c$ is the length of the hypotenuse. This method allows for easy calculation of the hypotenuse when both legs are known.

Using the angle

If one leg ($a$) and an angle ($\beta$) are known, you can use the trigonometric property of cosine to find the hypotenuse:

$c = \frac{a}{\cos(\beta)}$

Where $\beta$ is the angle adjacent to the known leg.

If one leg ($a$) and an angle ($\alpha$) are known, you can use the trigonometric property of sine to find the hypotenuse:

$c = \frac{a}{\sin(\alpha)}$

Where $\alpha$ is the angle opposite the known leg.

Area and one leg

If the area ($A$) and one leg ($a$) are known, the hypotenuse can be determined as follows:

1. Find the second leg ($b$) using the area formula: $b = \frac{2A}{a}$

2. Then use the Pythagorean theorem: $c = \sqrt{a^2 + b^2} = \sqrt{a^2 + \left(\frac{2A}{a}\right)^2}$

Examples

Example 1: Finding the hypotenuse with two legs

If the legs have lengths 3 and 4, what is the length of the hypotenuse?

Using the Pythagorean Theorem: $c = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

Example 2: Finding the hypotenuse with one leg and an angle

If one leg $a$ is 5 and the angle $\beta$ is 30°, find the hypotenuse.

Using the cosine: $c = \frac{5}{\cos(30^\circ)} \rightarrow c = \frac{5}{\sqrt{3}/2} = \frac{10}{\sqrt{3}} \approx 5.77$

Example 3: Finding the hypotenuse with area and one leg

If the area is 6 and one leg is 3, find the hypotenuse.

First, find the second leg: $b = \frac{2 \times 6}{3} = 4$

Now use the Pythagorean formula: $c = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

Notes

• Ensure that angles are expressed in radians or degrees according to the calculator settings.
• If using the area in calculations, ensure that the unit of measurement for length and area is consistent (e.g., square meters for area and meters for length).
• If you need to calculate the angles of a right triangle, you can use an angle calculator.

Frequently asked questions

How to find the hypotenuse if the legs are 6 and 8?

Using the Pythagorean Theorem: $c = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$

Why is it important to know the hypotenuse?

Knowing the hypotenuse is useful in architecture, engineering, physics, and many other disciplines where understanding the proportions and relationships of a triangle’s sides is important.

Can a calculator be used for everyday tasks?

Yes, the hypotenuse calculator can be useful in construction, design, navigation, and even in everyday tasks such as measuring distance.

Why is the hypotenuse always the longest side?

Since it is opposite the right angle, its length, according to the Pythagorean Theorem, is always greater than the two other sides in a right triangle.

Can other methods be used to find the hypotenuse?

Yes, depending on the known information, various formulas can be used, such as trigonometric ratios or area.

Find the hypotenuse of a right triangle if the legs are 3.5 and 7 cm.

Using the Pythagorean Theorem: $c = \sqrt{3.5^2 + 7^2} = \sqrt{12.25 + 49} = \sqrt{61.25} \approx 7.83$