Right triangle side calculator

Right triangle

A right triangle is a geometric figure consisting of three sides, two of which (known as legs $a$ and $b$) intersect at a right angle, i.e., $90^\circ$. The third side, which is opposite the right angle, is called the hypotenuse and is denoted by the letter $c$. Such triangles possess unique properties that allow solving numerous practical problems—from construction measurements to complex engineering calculations. If you need to find the angles of a right triangle, it is recommended to use the angle calculator. For calculating the hypotenuse, the hypotenuse calculator is beneficial.

History of the right triangle

The first mention of the properties of right triangles is found in ancient Egyptian and Babylonian texts. However, they are most famously associated with the Greek mathematician Pythagoras, who formulated the famous theorem named after him. Pythagoras’ theorem states that in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Throughout centuries, this theorem has been the foundation for studying trigonometry and geometry, significantly impacting the development of mathematics.

Using the calculator

This calculator helps you determine unknown sides of a right triangle using various combinations of known information. You can calculate one of the legs if you have:

• One leg and the hypotenuse.
• One leg and an angle.
• Area and one leg.
• Hypotenuse and an angle.

Formulas

Find a leg through another leg and the hypotenuse

If the leg $a$ and the hypotenuse $c$ are known, the other leg $b$ can be found using the formula:

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

Find a leg through an angle and the hypotenuse

Knowing angle $\alpha$, which is opposite side $a$, allows finding leg $a$ through the hypotenuse $c$:

$a = c \cdot \sin\alpha$

Find a leg through an angle and another leg

If angle $\alpha$ is known, leg $a$ can be found through leg $b$:

$a = b \cdot \tan\alpha$

Find a leg through area and another leg

If the known leg $a$ and area $A$ of the triangle, the second leg $b$ can be found using:

$b = \frac{2A}{a}$

Examples

Example 1: Find a leg through another leg and the hypotenuse

Suppose the known leg is $a = 3$ and the hypotenuse $c = 5$. Use the formula to find the second leg:

$b = \sqrt{5^2 - 3^2} = \sqrt{25 - 9} = \sqrt{16} = 4$

Example 2: Find a leg through an angle and the hypotenuse

If the angle $\alpha = 30^\circ$ and the hypotenuse $c = 10$, find leg $a$:

$a = 10 \cdot \sin(30^\circ) = 10 \cdot \frac{1}{2} = 5$

Example 3: Find a leg through an angle and another leg

Suppose the known angle $\alpha = 45^\circ$ and leg $b = 7$:

$a = 7 \cdot \tan(45^\circ) = 7 \cdot 1 = 7$

Example 4: Find a leg through area and another leg

If the area $A = 6$ and leg $a = 3$, use the formula to find the other leg:

$b = \frac{2 \times 6}{3} = 4$

Notes

• Note that accurate calculations require using the angle in radians or verifying the conversion of degrees to radians.
• All trigonometric formulas assume angles are measured in the Cartesian system; auxiliary conversion is necessary for working with angles in degrees.
• This calculator is not only useful for solving school curriculum problems but also serves as a tool for engineering and scientific calculations where precision is essential.

Frequently asked questions

How to find a leg if one leg and the hypotenuse are known?

To find the other leg when you have one leg $a$ and the hypotenuse $c$, use the formula:

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

How are angles and sides related in a right triangle?

In a right triangle, the angles are related to the sides through trigonometric functions: sine, cosine, and tangent. For instance, the sine of an angle is the ratio of the opposite leg to the hypotenuse.

How to find the hypotenuse from two legs?

The hypotenuse $c$ in a right triangle can be found using the formula:

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

For quicker hypotenuse calculation, the hypotenuse calculator is helpful, although this calculator is primarily designed for finding legs.

How to calculate the triangle’s area if both sides are known?

The area of a right triangle can be calculated as one-half the product of its legs:

$A = \frac{1}{2}ab$

For a quick calculation, you can also use the right triangle calculator.