Right Triangle Calculator
Right Triangle
A right triangle is a type of triangle that has one angle measuring exactly 90°. These triangles are the foundation of trigonometry due to the relationships between their sides and angles.
In a right triangle, the side opposite the 90° angle is the longest and is called the hypotenuse. The other two sides are usually labeled as a and b, with the hypotenuse labeled c. Angles are often referred to using capital letters corresponding to their opposite sides: ∠A opposite side a, ∠B opposite side b, and ∠C = 90° opposite the hypotenuse c.
The symbols α (alpha) and β (beta) are used to denote the unknown angles. The height h represents the perpendicular dropped from the right angle to the hypotenuse, dividing the triangle into two smaller, similar right triangles.
Pythagorean Triangle
When all sides of a right triangle are integers, the triangle is called a Pythagorean triangle. The three sides form a Pythagorean triple. Common examples include:
- 3, 4, 5
- 5, 12, 13
- 8, 15, 17
Area and Perimeter
The perimeter is the sum of all three sides:
Perimeter = a + b + c
The area is calculated using either of the following formulas:
Area = ½ × a × b or Area = ½ × c × h
Special Right Triangles
30°-60°-90° Triangle
This triangle has angles of 30°, 60°, and 90°, and the sides follow a specific ratio of 1 : √3 : 2. Knowing one side allows you to determine the others.
Let side b = 5 (opposite the 60° angle), then:
a = b / √3 = 5 / √3
c = (2 × b) / √3 = 10 / √3
45°-45°-90° Triangle
Also known as an isosceles right triangle, it has angles of 45°, 45°, and 90°. The side ratios are 1 : 1 : √2.
Given c = 5 (hypotenuse):
a = c / √2 = 5 / √2
This triangle is useful for evaluating trigonometric functions for multiples of π/4.
Related Calculator:
Pythagorean Theorem Calculator, Surface Area Calculator, Triangle CalculatorExternal Resources:
Right Triangle Calculator on Calculator.net