45 45 90 Special Right Triangle Calculator

Enter any known value for a 45 45 90 triangle to calculate the edge lengths, altitude, area, perimeter, inradius, and circumradius.

Solution:

sides:
a = 3
b = 3
c = 4.243
height:
h = 2.121
area:
T = 4.5
perimeter:
p = 10.243
inradius:
r = 0.8787
circumradius:
R = 2.121


What is a 45 45 90 Triangle?

A 45 45 90 triangle is a special right triangle that has two 45° interior angles and one 90° interior angle. A 45 45 90 triangle is a right triangle and is also an isosceles triangle.

You can also think of a 45 45 90 triangle as half of a square divided from one corner to the opposite corner.

diagram of a special right 45 45 90 triangle showing legs a and b, hypotenuse c, 30 & 60 degree angles, and height h

45 45 90 Triangle Formulas

Since a 45 45 90 triangle is a special right triangle, the formulas used to calculate parts of a right triangle can be used, substituting the angles measurements.

Solve the Hypotenuse

Given a known leg length, the hypotenuse can be solved using the Pythagorean theorem. Thus, the formula to solve the hypotenuse is:

c = a² + b²

The hypotenuse c is equal to the square root of leg a squared plus leg b squared. Note that in a 45 45 90 triangle legs a and b are the same length.

This formula can be simplified a bit to the following:

c = a√2

The hypotenuse c is equal to leg a times the square root of 2.

Solve the Leg

Given the hypotenuse, the legs can be solved using the Pythagorean theorem in a slightly different way.

a = b = c√22

The legs a and b are equal to the hypotenuse c times the square root of 2, divided by 2.

Alternatively, the legs can be solved using 45° angle and hypotenuse with the formula below.

a = b = c × sin(45°)

The legs a and b are equal to the hypotenuse c times the sin of the 45° angle.

Solve the Area

The area of a 45 45 90 triangle can be solved with the formula:

T = 2a2

Thus, the area T is equal to 2 times leg a, divided by 2

Solve the Perimeter

Use this formula to calculate the perimeter:

p = 2a + c

Thus, the perimeter p is equal to 2 times leg a plus hypotenuse c.

For more special right triangle formulas, try our 30 60 90 calculator.