Right Triangle Calculator
Enter two sides of a right triangle to find the hypotenuse, area, and perimeter. Works from two legs or from one leg and the hypotenuse.
Right Triangle Fundamentals
Right triangles are the foundation of trigonometry and practical geometry. The right angle creates a perfect reference point, and the Pythagorean theorem a² + b² = c² connects the three sides through a simple equation. This relationship holds for all right triangles regardless of size or orientation.
The two legs can be thought of as base and height for area purposes, so the area formula simplifies to A = (a × b) / 2. No need to calculate a separate height; the perpendicular legs provide it directly.
The perimeter is the sum of all three sides: P = a + b + c. Once you know two of the three sides, the Pythagorean theorem gives you the third, and from there you can compute area and perimeter automatically.
Practical Applications of Right Triangles
Construction and carpentry rely on right triangles for squaring corners and cutting materials. The 3-4-5 method is a staple: measure 3 units along one edge, 4 units along the other, and adjust until the diagonal measures exactly 5 units. That guarantees a perfect right angle.
In navigation, right triangles model vertical and horizontal components of travel. Pilots calculate altitude changes and ground distance using right triangle geometry. Surveyors use it to determine slopes, elevations, and horizontal distances from angular measurements.
Even screen dimensions are often specified as diagonal measurements. If you know the aspect ratio (say, 16:9), you can treat the width and height as the legs of a right triangle and the diagonal as the hypotenuse to find the actual dimensions.
Special Right Triangles: 45-45-90 and 30-60-90
The 45-45-90 triangle is an isosceles right triangle. Both legs are equal, and the hypotenuse is the leg length multiplied by √2 (approximately 1.414). This triangle appears in square diagonals and in situations requiring equal perpendicular components.
The 30-60-90 triangle has sides in the ratio 1 : √3 : 2. If the short leg (opposite the 30-degree angle) is 1, the long leg is √3, and the hypotenuse is 2. These triangles arise when you bisect an equilateral triangle.
Both special cases are right triangles, so this calculator handles them. Enter the known sides and the Pythagorean theorem will confirm the ratios. Memorizing these ratios speeds up many geometry and trigonometry problems.
Frequently Asked Questions
What defines a right triangle?
A right triangle has one 90-degree angle. The two sides forming the right angle are called legs, and the side opposite the right angle is the hypotenuse.
How do you find the hypotenuse?
Use the Pythagorean theorem: c = √(a² + b²), where a and b are the legs and c is the hypotenuse.
What is the area formula for a right triangle?
Area = (leg₁ × leg₂) / 2. The two legs serve as base and height, making the calculation straightforward.
Can I find a leg if I know the hypotenuse and the other leg?
Yes. Use b = √(c² - a²), where c is the hypotenuse, a is the known leg, and b is the unknown leg. Switch to the 'From Leg and Hypotenuse' mode in this calculator.
What are the most common right triangle ratios?
The 3-4-5 triangle and the 5-12-13 triangle are Pythagorean triples. The 45-45-90 triangle has legs in ratio 1:1:√2, and the 30-60-90 triangle has sides in ratio 1:√3:2.