Right Triangle Calculator - Calculate Sides, Angles, Area & Perimeter

What is a Right Triangle?

A right triangle is a triangle in which one of the angles measures exactly 90 degrees (a right angle). The side opposite the right angle is called the hypotenuse, and it is always the longest side of the triangle. The other two sides are called legs or catheti.

a² + b² = c²

Key Properties of Right Triangles

One right angle: Exactly one angle measures 90°
Hypotenuse: The longest side, opposite the right angle
Legs: The two shorter sides that form the right angle
Pythagorean theorem: The sum of squares of legs equals square of hypotenuse
Angle sum: All three angles add up to 180°

Types of Right Triangle Calculations

1. Two Sides Known (Find Third Side)

When you know any two sides of a right triangle, you can find the third side using the Pythagorean theorem.

Example: Find hypotenuse when legs are 3 and 4

c² = a² + b² = 3² + 4² = 9 + 16 = 25

c = √25 = 5

2. Complete Triangle Analysis

When you know all three sides, you can verify it's a right triangle and calculate all angles, area, and perimeter.

Example: Verify 5-12-13 triangle

Check: 13² = 5² + 12²

169 = 25 + 144 = 169 ✓

Area = (1/2) × 5 × 12 = 30

Perimeter = 5 + 12 + 13 = 30

3. Special Right Triangles

Some right triangles have special ratios that make calculations easier and are worth memorizing.

Pythagorean Triples

3-4-5 triangle
5-12-13 triangle
8-15-17 triangle
7-24-25 triangle

Special Angle Triangles

30°-60°-90° triangle
45°-45°-90° triangle
Isosceles right triangle
Equilateral triangle (half)

Mathematical Formulas

Pythagorean Theorem

c² = a² + b²

Area of Right Triangle

A = (1/2) × a × b

Perimeter of Right Triangle

P = a + b + c

Angle Calculations

Angle A = arctan(a/b)

Angle B = arctan(b/a)

Angle C = 90°

Real-World Applications

Construction and Engineering

Roof construction: Calculating rafter lengths and angles
Foundation work: Ensuring square corners and diagonal measurements
Staircase design: Determining step dimensions and angles
Bridge construction: Calculating support beam lengths
Surveying: Measuring distances and elevations

Everyday Life

TV and monitor sizes: Finding diagonal measurements
Ladder placement: Determining safe angles and heights
Furniture placement: Calculating diagonal spaces
Garden design: Planning triangular flower beds
Sports: Calculating field dimensions and angles

Mathematics and Science

Trigonometry: Foundation for all trigonometric functions
Coordinate geometry: Distance formulas and slope calculations
Physics: Vector components and force analysis
Navigation: Bearing calculations and distance measurements
Computer graphics: 3D modeling and rendering

Special Right Triangle Ratios

30°-60°-90° Triangle

In a 30°-60°-90° triangle, the sides are in the ratio 1 : √3 : 2. The shortest side (opposite 30°) is half the hypotenuse.

45°-45°-90° Triangle

In a 45°-45°-90° triangle (isosceles right triangle), the sides are in the ratio 1 : 1 : √2. The legs are equal, and the hypotenuse is √2 times the length of each leg.

Tips for Using the Right Triangle Calculator

Always ensure the largest side is the hypotenuse
Use positive numbers for side lengths
Verify your triangle is right-angled using the Pythagorean theorem
Remember that angles in a triangle always sum to 180°
Use the special triangle ratios for quick calculations
Check your results by verifying a² + b² = c²

Common Right Triangle Examples

3-4-5 Triangle

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

Area: 6 | Perimeter: 12

5-12-13 Triangle

Sides: 5, 12, 13 | Angles: 22.62°, 67.38°, 90°

Area: 30 | Perimeter: 30

45°-45°-90° Triangle

Sides: 1, 1, √2 | Angles: 45°, 45°, 90°

Area: 0.5 | Perimeter: 2 + √2

Frequently Asked Questions

What's the difference between a right triangle and other triangles?

A right triangle has exactly one angle that measures 90 degrees, while other triangles have all angles less than 90° (acute) or one angle greater than 90° (obtuse). The Pythagorean theorem only applies to right triangles, making them special in geometry and trigonometry.

How do I know which side is the hypotenuse?

The hypotenuse is always the longest side of a right triangle and is opposite the right angle. If you're unsure, the hypotenuse will always be the side that, when squared, equals the sum of the squares of the other two sides (Pythagorean theorem).

Can I use the calculator for non-right triangles?

No, this calculator is specifically designed for right triangles. For other types of triangles, you would need different formulas and methods, such as the Law of Cosines or Law of Sines. The Pythagorean theorem only works for right triangles.

What are Pythagorean triples and why are they important?

Pythagorean triples are sets of three positive integers (a, b, c) that satisfy the equation a² + b² = c². Examples include (3, 4, 5), (5, 12, 13), and (8, 15, 17). They're important because they represent right triangles with integer side lengths, making calculations easier and more practical in real-world applications.

How accurate are the angle calculations?

The angle calculations are accurate to 2 decimal places (0.01°). For most practical purposes, this level of precision is sufficient. The calculations use the arctangent function to determine angles from side ratios, which is mathematically precise.

Can I use this calculator for 3D problems?

This calculator is designed for 2D right triangles. For 3D problems involving right triangles (like finding the diagonal of a rectangular box), you would need to apply the Pythagorean theorem multiple times or use 3D distance formulas.

What if my triangle doesn't satisfy the Pythagorean theorem?

If the sides don't satisfy a² + b² = c² (where c is the longest side), then it's not a right triangle. The calculator will show an error message. You may need to check your measurements or use a different type of triangle calculator.

How do I find the area if I only know the hypotenuse and one angle?

If you know the hypotenuse and one angle, you can use trigonometric ratios to find the legs: opposite = hypotenuse × sin(angle) and adjacent = hypotenuse × cos(angle). Then use the area formula A = (1/2) × leg1 × leg2. This calculator focuses on side-based calculations, but trigonometric methods are also valid.