Right triangle angle calculator

What is a right triangle?

A right triangle is one of the fundamental figures in geometry. This triangle has one angle of $90^\circ$ (a right angle). Due to its simple and intuitive structure, it is widely used in various fields of science and engineering. Its properties make it easy to relate sides and angles, making it an ideal object for trigonometry study.

The basic relationship between the sides of a right triangle is defined by the Pythagorean theorem: $a^2 + b^2 = c^2$, where $a$ and $b$ are the legs, and $c$ is the hypotenuse.

Important aspects of angle calculation

Pythagorean theorem

The Pythagorean theorem is the most fundamental tool for analyzing right triangles. It not only allows us to find sides but also obtain angles using trigonometric methods. If you need to explore the application of this theorem in more detail, you can use the Pythagorean theorem calculator. It will be an indispensable assistant in solving problems related to right triangles.

Trigonometric functions

Trigonometric functions describe the relationship between angles and sides of a triangle:

• Sine ($\sin$): the ratio of the opposite leg to the hypotenuse.
• Cosine ($\cos$): the ratio of the adjacent leg to the hypotenuse.
• Tangent ($\tan$): the ratio of the opposite leg to the adjacent leg.

If two sides are known

When two sides of a right triangle are given, you can find the angles using trigonometric functions. For example, if the sides $a$ and $b$ are known, the angle $\alpha$ (opposite to side $a$) can be found as follows:

$$\alpha = \arctan\left(\frac{a}{b}\right)$$

The angle $\beta$ (opposite to side $b$) can be found as follows:

$$\beta = 90^\circ - \alpha$$

If an angle and one side are known

When one angle $\alpha$ and side $a$ are known, the other side $b$ and the hypotenuse $c$ are calculated as:

The other side $b$:

$$b = a \cdot \cot(\alpha)$$

(where $\cot(\alpha) = 1/\tan(\alpha)$)

Hypotenuse $c$:

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

Also, the angle $\beta$ can be calculated as:

$$\beta = 90^\circ - \alpha$$

If the area and one side are known

The area of a right triangle $A$ with side $a$ allows you to find the other side $b$:

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

To find the angle $\alpha$, if sides $a$ and $b$ are known (where $b$ can be explicitly expressed via $A$), use:

$$\alpha = \arctan\left(\frac{a}{b}\right)$$

And accordingly, the angle $\beta$:

$$\beta = 90^\circ - \alpha$$

If the hypotenuse and one side are known

If the hypotenuse $c$ and one of the sides $a$ are known, the other side $b$ and angles are found as:

$$b = \sqrt{c^2 - a^2}$$ $$\alpha = \arcsin\left(\frac{a}{c}\right)$$

And the angle $\beta$ is calculated as:

$$\beta = 90^\circ - \alpha$$

Another useful feature when working with right triangles is the ability to calculate the perimeter or area of the triangle. For this, you can use the right triangle calculator.

Examples

Example 1

Problem: Find the angles of a triangle if the legs $a = 3$ and $b = 4$ are given.

Solution: Hypotenuse:

$$c = \sqrt{3^2 + 4^2} = 5$$

Angles:

$$\alpha = \arctan\left(\frac{3}{4}\right) \approx 36.87^\circ$$ $$\beta = 90^\circ - \alpha = 53.13^\circ$$

Example 2

Problem: The leg $a = 5$ and angle $\beta = 30^\circ$ (adjacent to leg $a$) are known. Find the other leg and the hypotenuse.

Solution: Other leg:

$$b = 5 \cdot \tan 30^\circ \approx 2.89$$

Hypotenuse:

$$c = \frac{5}{\cos 30^\circ} \approx 5.77$$

Example 3

Problem: Find the angles and hypotenuse of a right triangle if its area is $A = 12 \, \text{sq.units}$ and leg $a = 4 \, \text{units}$.

Solution: The area of a right triangle is expressed as:

$$A = \frac{1}{2} \cdot a \cdot b$$

From which the other leg:

$$b = \frac{2A}{a} = \frac{2 \times 12}{4} = 6 \, \text{units}$$

Using the Pythagorean theorem, find the hypotenuse $c$:

$$c = \sqrt{a^2 + b^2} = \sqrt{4^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52} \approx 7.21 \, \text{units}$$

Now find the angles using trigonometric functions:

Angle $\alpha$:

$$\alpha = \arctan\left(\frac{a}{b}\right) = \arctan\left(\frac{4}{6}\right) \approx 33.69^\circ$$

Angle $\beta$:

$$\beta = 90^\circ - \alpha \approx 90^\circ - 33.69^\circ = 56.31^\circ$$

Example 4

Problem: Find the angles and the second leg of a right triangle if the hypotenuse is $c = 10 \, \text{units}$ and leg $a = 6 \, \text{units}$.

Solution: Using the Pythagorean theorem, find the second leg $b$:

$$b = \sqrt{c^2 - a^2} = \sqrt{10^2 - 6^2} = \sqrt{100 - 36} = \sqrt{64} = 8 \, \text{units}$$

Now find the angles using trigonometric functions:

Angle $\alpha$:

$$\alpha = \arcsin\left(\frac{a}{c}\right) = \arcsin\left(\frac{6}{10}\right) \approx 36.87^\circ$$

Angle $\beta$:

$$\beta = 90^\circ - \alpha \approx 90^\circ - 36.87^\circ = 53.13^\circ$$

Special recommendations

1. Calculation accuracy: Ensure your calculator is set to the correct units (degrees or radians) depending on the task.
2. Solving problems with unknowns: Always try to express unknown values through known ones before starting calculations.
3. Verification of solutions: After obtaining the values of the angles, always check that the sum of the angles in the triangle is $180^\circ$.

Frequently asked questions

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

If the hypotenuse $c$ and leg $a$ are known, the angle can be found using the arcsine:

$$\alpha = \arcsin\left(\frac{a}{c}\right)$$

Is it possible to find the angles of a triangle knowing only its area?

No, to determine the angles, you need to know at least one side or two angles.

What tools are used to solve geometry problems?

Calculators, geometric programs, and traditional tools such as a compass and protractor can be used to solve geometry problems.

How are angles related in a right triangle?

The sum of all angles in any triangle is $180^\circ$, so the two angles in a right triangle make up $90^\circ$.

Can this calculator be used for arbitrary triangles?

This calculator is intended for right triangles only. In other cases, more complex methods and formulas such as the law of sines or cosines will be required.