Right Triangle Calculator
Calculate all sides, angles, area, and perimeter of a right triangle with our free online calculator.
Category: Calculator
Right Triangle Fundamentals
A right triangle is a triangle with one angle equal to 90 degrees (a right angle). The side opposite to the right angle is called the hypotenuse, while the other two sides are called legs or catheti.
Key Properties:
- One angle is exactly 90 degrees
- The sum of the other two angles is 90 degrees
- The Pythagorean theorem applies: a² + b² = c²
- Area = (base × height) ÷ 2
- The hypotenuse is always the longest side
Right triangles are fundamental in geometry, trigonometry, and practical applications in fields like construction, engineering, and navigation.
Common Right Triangle Ratios
| Triangle | Leg a | Leg b | Hypotenuse c | Angle α (°) | Angle β (°) |
|---|---|---|---|---|---|
| 3-4-5 | 3 | 4 | 5 | 36.87 | 53.13 |
| 5-12-13 | 5 | 12 | 13 | 22.62 | 67.38 |
| 8-15-17 | 8 | 15 | 17 | 28.07 | 61.93 |
| 7-24-25 | 7 | 24 | 25 | 16.26 | 73.74 |
| 9-40-41 | 9 | 40 | 41 | 12.68 | 77.32 |
The Pythagorean Theorem
The Pythagorean theorem is one of the fundamental relationships in Euclidean geometry. It states that in a right triangle, the square of the length of the hypotenuse equals the sum of squares of the other two sides.
The Formula:
c² = a² + b²
Where c is the hypotenuse, and a and b are the legs of the right triangle.
This theorem, attributed to the ancient Greek mathematician Pythagoras, has been proven in many different ways throughout history and remains a cornerstone of geometry and trigonometry.
Trigonometric Functions
- sin(θ)
- cos(θ)
- tan(θ)
Basic trigonometric functions are derived from right triangle relationships
Angle vs. Area Relationship
How the area of a right triangle changes with angle (keeping one leg constant at 10 units)
Applications of Right Triangles
Practical Applications
- Construction: Used for ensuring perpendicular walls and square corners in buildings
- Navigation: Calculating distances and bearings in maritime and aviation navigation
- Surveying: Measuring land boundaries and elevation changes
- Engineering: Analyzing forces in structures like bridges and trusses
- Architecture: Designing roofs, staircases, and ramps with proper angles
- Physics: Resolving vectors in mechanics and dynamics problems
Mathematical Connections
Right triangles are central to many areas of mathematics:
- Trigonometry: All trigonometric functions are based on right triangle ratios
- Coordinate Geometry: Distance formula between points derives from the Pythagorean theorem
- Complex Numbers: Can be represented as points in the complex plane forming right triangles
- Calculus: Used in derivations of fundamental theorems and formulas
- Linear Algebra: Orthogonal vectors form right angles, leading to right triangles
Special Right Triangles
Certain right triangles have special properties that make them particularly useful in mathematics and practical applications:
- 45°-45°-90° Triangle: Also called an isosceles right triangle. The two legs are equal, and both acute angles are 45°. If the legs have length 1, the hypotenuse has length √2.
- 30°-60°-90° Triangle: Formed by cutting an equilateral triangle in half. If the shorter leg has length 1, the longer leg has length √3, and the hypotenuse has length 2.
- Pythagorean Triples: Sets of three integers that satisfy the Pythagorean theorem, such as (3,4,5), (5,12,13), and (8,15,17).
Frequently Asked Questions
What is a right triangle?
A right triangle is a triangle with one angle equal to 90 degrees (a right angle). It follows the Pythagorean theorem where the square of the hypotenuse equals the sum of the squares of the other two sides.
How do you find the area of a right triangle?
The area of a right triangle is half the product of the two legs (the sides that form the right angle). Area = (a × b) / 2, where a and b are the lengths of the legs.
How do you find the hypotenuse of a right triangle?
The hypotenuse can be found using the Pythagorean theorem: c = √(a² + b²), where c is the hypotenuse and a and b are the legs of the right triangle.