Right Triangle Calculator - Solve Right Triangles Online

Solve any right triangle by entering two known values with our free online right triangle calculator. Find all sides, angles, area, and perimeter with clear step-by-step solutions.

Known Values

Leg a

Leg b

Hypotenuse c

Examples:

Understanding Right Triangles

A right triangle is a triangle with one 90° angle. The side opposite the right angle is called the hypotenuse - it's always the longest side. The other two sides are called legs. Right triangles are fundamental to trigonometry and appear everywhere in mathematics, science, and engineering.

What makes right triangles special is the Pythagorean theorem and the trigonometric ratios (sine, cosine, tangent). These relationships let you find any missing side or angle when you know just two pieces of information.

Right Triangle Formulas

Pythagorean Theorem

a² + b² = c²

Relates the three sides. Use when you know two sides.

Trigonometric Ratios

sin A = a/c (opposite/hypotenuse)

cos A = b/c (adjacent/hypotenuse)

tan A = a/b (opposite/adjacent)

Area

Area = ½ × a × b

The legs form the base and height.

Angle Sum

A + B + C = 180°

Since C = 90°, we have A + B = 90°

Worked Examples

Example 1: The Classic 3-4-5 Triangle

Given legs a=3 and b=4, find the hypotenuse and angles.

Using Pythagorean theorem:

c² = 3² + 4² = 9 + 16 = 25

c = √25 = 5

tan A = 3/4 = 0.75

A = arctan(0.75) ≈ 36.87°

B = 90° - 36.87° = 53.13°

Sides: 3, 4, 5 | Angles: 36.87°, 53.13°, 90°

Example 2: Finding a Missing Leg

Given leg a=5 and hypotenuse c=13, find the other leg and angles.

b² = c² - a² = 169 - 25 = 144

b = √144 = 12

sin A = 5/13 ≈ 0.3846

A = arcsin(0.3846) ≈ 22.62°

B = 90° - 22.62° = 67.38°

This is the 5-12-13 Pythagorean triple!

Example 3: Using an Angle

Given leg a=10 and angle A=30°, find the other sides.

B = 90° - 30° = 60°

sin 30° = 10/c

c = 10/sin 30° = 10/0.5 = 20

tan 30° = 10/b

b = 10/tan 30° = 10/0.577 ≈ 17.32

This is a 30-60-90 triangle!

Example 4: Real-World Application

A ladder 15 feet long leans against a wall, reaching 12 feet up. How far is the base from the wall, and what angle does the ladder make with the ground?

c = 15 ft (ladder), a = 12 ft (height)

b² = 15² - 12² = 225 - 144 = 81

b = 9 ft (distance from wall)

sin A = 12/15 = 0.8

A = arcsin(0.8) ≈ 53.13°

Base is 9 ft from wall, ladder angle is 53.13°

Quick Fact

The 3-4-5 triangle was used by ancient Egyptian "rope stretchers" to create perfect right angles for building pyramids and temples. They'd tie knots in a rope at equal intervals, forming 12 segments, then arrange it into a 3-4-5 triangle. The angle opposite the 5-segment side is guaranteed to be exactly 90°. This practical geometry predates Pythagoras by over 2000 years!

Frequently Asked Questions

What are Pythagorean triples?

Sets of three whole numbers that satisfy a² + b² = c². Common examples: 3-4-5, 5-12-13, 8-15-17, 7-24-25. Any multiple also works (6-8-10, 9-12-15). These give "nice" right triangles with integer sides.

What's special about 30-60-90 and 45-45-90 triangles?

These are special right triangles with predictable side ratios. 30-60-90: sides are 1 : √3 : 2. 45-45-90: sides are 1 : 1 : √2. Memorizing these ratios lets you solve these triangles instantly without a calculator.

Can a right triangle have equal legs?

Yes! That's a 45-45-90 triangle (isosceles right triangle). Both acute angles are 45°, and if each leg is 1, the hypotenuse is √2. This triangle appears when you cut a square diagonally.

How do I know which trig function to use?

Use SOH-CAH-TOA: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. Pick the function that uses the sides you know and the side you're looking for.

What if I only know one side?

You need at least two pieces of information to solve a right triangle: either two sides, or one side and one acute angle. With only one side, there are infinitely many possible right triangles.

Where are right triangles used in real life?

Everywhere! Construction (roof slopes, stair angles), navigation (bearing calculations), physics (force components), computer graphics (3D rendering), surveying, astronomy (parallax), and any situation involving heights, distances, or angles.

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