45-45-90 Triangle Calculator

Last Updated on March 2, 2026 | 1 : 45 pm by Anas Brittany

Use this 45-45-90 triangle calculator to find the missing sides of an isosceles right triangle. Enter any one side (leg or hypotenuse) and the calculator will automatically compute the other sides using the special 45-45-90 triangle ratio.

What Is a 45-45-90 Triangle?

A 45-45-90 triangle is a special right triangle where both acute angles are 45°. Because the two angles are equal, the triangle is also an isosceles right triangle, meaning the two legs have the same length.

This triangle appears frequently in geometry, trigonometry, construction, and engineering.

45-45-90 Triangle Ratio

The sides of a 45-45-90 triangle always follow a fixed ratio:

Leg : Leg : Hypotenuse = 1 : 1 : √2

If the leg is x:

• Other leg = x
• Hypotenuse = x√2

If the hypotenuse is c:

• Each leg = c / √2

45-45-90 Triangle Formulas

Hypotenuse:
c = a√2

Legs:
a = b = c / √2

Area:
Area = a² / 2

Perimeter:
P = 2a + a√2

Example

If a leg is 5:

Hypotenuse = 5√2 ≈ 7.07
Area = 12.5
Perimeter ≈ 17.07

When 45-45-90 Triangles Are Used

These triangles are commonly used in:

• Square diagonals
• Roof and ramp design
• Engineering layouts
• Computer graphics and design
• Geometry and trigonometry problems

Because the ratio is fixed, calculations are faster than using the full Pythagorean theorem.

Frequently Asked Questions

Why are the legs equal?

The two angles are both 45°, so the triangle is isosceles. Equal angles always have equal opposite sides.

Is this the same as using the Pythagorean theorem?

Yes. The relationship comes from:
a² + a² = c² → 2a² = c² → c = a√2

Can this triangle be scaled?

Yes. The ratio stays the same no matter the size, so the formulas work for any measurement units.