45-45-90 Triangle Calculator

Enter a leg or the hypotenuse of a 45-45-90 isosceles right triangle to find all sides, area, and perimeter.

Enter a Known Side

Result

All Three Sides
7 : 7 : 9.8995
units
Leg a (opposite 45°)7
Leg b (opposite 45°)7
Hypotenuse c (opposite 90°)9.899495
Area24.5
Perimeter23.899495

Step-by-Step Solution

Ratio: 1 : 1 : sqrt(2) => 7 : 7 : 9.8995

The 45-45-90 Special Right Triangle

A 45-45-90 triangle is a special right triangle that is also isosceles, meaning its two legs are equal in length. The angles are 45 degrees, 45 degrees, and 90 degrees. The sides follow the fixed ratio 1 : 1 : √2. If each leg has length a, the hypotenuse is a√2.

Side Ratios

Both legs are equal; the hypotenuse is √2 times a leg.

a : a : a*sqrt(2)

Area Formula

Since both legs are equal, the area simplifies.

A = a^2 / 2

Perimeter

Sum of two equal legs and the hypotenuse.

P = 2a + a*sqrt(2) = a(2 + sqrt(2))

How to Solve a 45-45-90 Triangle

You only need one measurement to solve the entire triangle:

  • Given a leg (a): The other leg equals a, and the hypotenuse = a√2
  • Given the hypotenuse (c): Each leg = c / √2 = c√2 / 2

Real-World Applications

The 45-45-90 triangle appears when you cut a square diagonally in half. It is commonly used in construction for diagonal bracing, in navigation for equal-distance turns, and in design for creating balanced proportions. Baseball diamonds and chess boards both contain implicit 45-45-90 triangles.