Right triangle side and angle calculator

What is a right triangle?

A right triangle, or right-angled triangle, is a geometric figure with one angle measuring exactly $90^\circ$. The side opposite the right angle is called the hypotenuse, and the other two sides are known as the legs (adjacent and opposite). Right triangles are foundational in trigonometry and geometry due to their unique properties, such as the Pythagorean theorem and trigonometric ratios.

Key properties:

• One angle is $90^\circ$.
• The hypotenuse is the longest side.
• The sum of the two non-right angles is $90^\circ$.
• The sides and angles follow the Pythagorean theorem and trigonometric relationships.

Key formulas for right triangles

Pythagorean theorem

For a right triangle with legs $a$ and $b$ and hypotenuse $c$: $a^2 + b^2 = c^2$

Trigonometric ratios

• Sine: $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$
• Cosine: $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
• Tangent: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$

Angle calculation

To find an angle when two sides are known: $\theta = \arctan\left(\frac{\text{Opposite}}{\text{Adjacent}}\right)$ $\theta = \arcsin\left(\frac{\text{Opposite}}{\text{Hypotenuse}}\right)$ $\theta = \arccos\left(\frac{\text{Adjacent}}{\text{Hypotenuse}}\right)$

Area of a right triangle

$\text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height}$ The base and height in a right triangle are the legs.

Step-by-step examples

Example 1: Finding the hypotenuse

Problem: A right triangle has legs measuring 5 meters and 12 meters. What is the length of the hypotenuse?

Solution:

1. Apply the Pythagorean theorem: $c^2 = 5^2 + 12^2 = 25 + 144 = 169$
2. Solve for $c$: $c = \sqrt{169} = 13 \text{ meters}$

Example 2: Calculating an angle

Problem: A right triangle has an opposite side of 7 meters and an adjacent side of 10 meters relative to angle $\theta$. What is the measure of $\theta$?

Solution:

1. Use the tangent ratio: $\tan(\theta) = \frac{7}{10} = 0.7$
2. Calculate the angle using arctangent: $\theta = \arctan(0.7) \approx 35^\circ$

Historical context

The study of right triangles dates back to ancient civilizations. The Babylonians (1800 BCE) used Pythagorean triples for land surveying, while the Egyptians employed knotted ropes to create right angles for pyramid construction. The theorem’s formal proof is attributed to Pythagoras of Samos (6th century BCE), though evidence suggests it was known earlier in India and Mesopotamia.

Applications in real life

1. Construction: Calculating roof slopes or stair angles.
2. Navigation: Determining distances using triangulation.
3. Physics: Resolving forces into perpendicular components.
4. Astronomy: Measuring star distances via parallax.

Special right triangles

1. 45°-45°-90° Triangle

• Legs are equal: $a = b$.
• Hypotenuse: $c = a\sqrt{2}$. For calculations on such a triangle, use our calculator for a 45-45-90 triangle.

2. 30°-60°-90° Triangle

• Sides follow the ratio $1 : \sqrt{3} : 2$, where the side opposite $30^\circ$ is the shortest.
• The side opposite $30^\circ$ is the shortest and equals half the hypotenuse. For calculations on such a triangle, use our calculator for a 30-60-90 triangle.

Accuracy of calculations: important notes

• The sum of angles must be $180^\circ$ (e.g., $90^\circ + 35^\circ + 55^\circ = 180^\circ$).
• Use the same units for all sides.
• Check the calculator mode (degrees or radians) when working with inverse trigonometric functions.

Frequently Asked Questions

How to calculate the hypotenuse if the legs are 9 meters and 12 meters?

1. Apply the Pythagorean theorem: $c^2 = 9^2 + 12^2 = 81 + 144 = 225$
2. Solve for $c$: $c = \sqrt{225} = 15 \text{ meters}$

What is the largest angle in a right triangle?

The largest angle is always the right angle, measuring $90^\circ$. The other two angles are acute (less than $90^\circ$).

How to find the area of a right triangle with legs 6 cm and 8 cm?

1. Use the area formula: $\text{Area} = \frac{1}{2} \times 6 \times 8 = 24 \text{ cm}^2$

Can the legs of a right triangle be equal?

Yes. In a 45°-45°-90° triangle, the legs are equal, and the hypotenuse is $a\sqrt{2}$.

Find the leg if the hypotenuse is 30 and it is known that the legs are equal?

In this case, the legs are equal $a = b = \frac{c}{\sqrt{2}}$. Let’s perform the calculation: $a = b = \frac{30}{\sqrt{2}} = 15\sqrt{2}$.

What is the hypotenuse of a right triangle?

The hypotenuse of a right triangle is equal to the leg divided by the sine of the opposite or cosine of the adjacent leg.