Right Triangle Calculator

Enter any two known values for a right triangle below to calculate the edge lengths, altitude, angles, area, perimeter, inradius, and circumradius.

Solution:

sides:
a = 3
b = 4
c = 5
angles:
α = 36.87° | 0.6435 rad
β = 53.13° | 0.9273 rad
height:
h = 2.4
area:
T = 6
perimeter:
p = 12
inradius:
r = 1
circumradius:
R = 2.5
type:
scalene triangle
(3:4:5 Pythagorean triple)


What is a Right Triangle?

A right triangle, sometimes called a right-angled triangle, is a triangle with that has one angle equal to exactly 90 degrees, forming a right angle.

The edges adjacent to the right angle are referred to as legs a and b, and are also often referred to as the base and height. This calculator reserves the word height to represent the altitude of a triangle.

The edge opposite the right angle is called the hypotenuse c.

diagram showing the parts of a right triangle

There are two primary types of right triangles, isosceles and scalene.

An isosceles right triangle is one that has two angles that are 45 degrees and legs a and b of equal length. Try our isosceles triangle calculator.

A scalene right triangle is one where all the angles are unequal and all the sides are unequal.

How to Calculate Edge Lengths of a Right Triangle

To calculate one of the edge lengths in a right triangle when the other two edges are known, the Pythagorean theorem can be used.

The Pythagorean theorem states that edge a squared plus edge b squared is equal to edge c squared.

a² + b² = c²

Thus, to solve for the missing edge, enter the two known values into the formula and solve. Try our Pythagorean theorem calculator to solve.

For example, let’s calculate edge c for a triangle with edge a = 3 and edge b = 4.

3² + 4² = c²
9 + 16 = c²
25 = c²
25 = c
5 = c

Thus, c is equal to 5 for a triangle with edge a = 3 and edge b = 4.

How to Calculate the Angles of a Right Triangle

If you know the lengths of two sides of a right triangle you can calculate the angles using the trigonometry functions sine, cosine, and tangent. These are usually shortened to sin, cos, and tan.

The following equations can be used to solve an angle given two known edge lengths:

sin(θ) = opposite ÷ hypotenuse
cos(θ) = adjacent ÷ hypotenuse
tan(θ) = opposite ÷ adjacent

In a right triangle, the adjacent side is the side of the triangle that forms part of the angle but is not the hypotenuse. The opposite side is the side that does not form part of the angle.

To solve one of the angles, choose the formula that can be used given the two known sides, substitute the known values in the formula, and solve.

For example, let’s find the angle of a triangle if the adjacent side length is 7 and the hypotenuse is 15.

Start by choosing the equation using adjacent and hypotenuse:
cos(θ) = adjacent ÷ hypotenuse

cos(θ) = 7 ÷ 15
cos(θ) = 0.4667
θ = cos-1(0.4667)
θ = 62.18°

Thus, given a right triangle with an adjacent side length of 7 and hypotenuse of 15, the angle is 62.18°.

Do these formulas seem tough to remember? Try the phrase SOHCAHTOA to help represent each part of the equations.

If you split SOHCAHTOA into three parts, each part represents one of the formulas, where each letter is the first letter in the part of the equation.

SOH · CAH · TOA

SOH: sin(θ) = opposite ÷ hypotenuse
CAH: cos(θ) = adjacent ÷ hypotenuse
TOA: tan(θ) = opposite ÷ adjacent

How to Calculate Area and Perimeter

The area and perimeter of a right triangle can be solved if the lengths of each edge are known.

Area

The formula to calculate the area is:

area = 12a × b

The area of a right triangle is equal to one-half times side a times side b.

Perimeter

The formula to calculate the perimeter is:

perimeter = a + b + c

The perimeter of a right triangle is equal to side a plus side b plus side c.

Special Right Triangles

There are a few types of special right triangles, which are triangles that have specific proportions. These special right triangles also have formulas to simplify solving them.

A 30 60 90 triangle is a special right triangle with 30° and 60° interior angles adjacent to the right 90° angle.

A 45 45 90 triangle is a special right isosceles triangle with 45° interior angles adjacent to the right 90° angle.