45 45 90 Triangle Calculator

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

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

Triangle Properties:

leg a:
3
leg b:
3
hypotenuse c:
4.243
 
height h:
2.121
 
area:
4.5
perimeter:
10.243
 
inradius:
0.8787
circumradius:
2.121
Learn how we calculated this below


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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° right 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, 45 degree angles, and height h

In a 45 45 90 triangle, the ratio of the legs is 1 : 1 : √2. This ratio is derived by taking the sine of the 45° angle adjacent to the hypotenuse, which is equal to its opposite side divided by the hypotenuse.

Because both of the legs are equal in length, you can also take the cosine of the 45° angle, which is equal to its adjacent side divided by the hypotenuse.

The tangent of the 45° angle is equal to its adjacent side divided by the opposite side, which is equal to 1.

sin(45°) = 1 / √2
cos(45°) = 1 / √2
tan(45°) = 1

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.

Hypotenuse Formula

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.

Since the ratio of the short legs to the hypotenuse in a 45 45 90 triangle is 1 : √2, the formula can be simplified to the following:

c = a√2

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

You can also find this using our hypotenuse calculator.

Leg Length Formula

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

a = b = c√2 / 2

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

Alternatively, you can find the length of the legs if you know the hypotenuse given that the interior angles are known to be 45°. This is the formula for this method:

a = b = c × sin(45°)

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

Area Formula

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

area = 2a / 2

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

Perimeter Formula

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.