Isosceles Right Triangle Calculator (45-45-90)

Calculate all properties of a 45-45-90 triangle. Input any one measurement to find legs, hypotenuse, area, perimeter, and altitude.

Select Known Value

Result

Area
12.5
square units
Leg (a = b) 5
Hypotenuse (c) 7.071068
Area 12.5
Perimeter 17.071068
Altitude to Hypotenuse 3.535534
Angles 45-45-90
Inradius --
Circumradius --

Step-by-Step Solution

hypotenuse = leg * sqrt(2)

Understanding the 45-45-90 Triangle

An isosceles right triangle (also called a 45-45-90 triangle) is a special right triangle where the two legs are equal in length and the two acute angles are each 45 degrees. It is one of the most important special triangles in mathematics, appearing frequently in geometry, trigonometry, and real-world applications.

Key Formulas

Hypotenuse from Leg

The hypotenuse is always the leg multiplied by the square root of 2.

c = a * sqrt(2)

Leg from Hypotenuse

Divide the hypotenuse by the square root of 2 (or multiply by sqrt(2)/2).

a = c / sqrt(2) = c * sqrt(2) / 2

Area

Since both legs are equal, the area simplifies to half the square of one leg.

A = a^2 / 2

Altitude to Hypotenuse

The altitude from the right angle to the hypotenuse equals half the hypotenuse.

h = a * sqrt(2) / 2 = c / 2

Perimeter

Sum of two equal legs and the hypotenuse.

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

Inradius & Circumradius

Special radii for inscribed and circumscribed circles.

r = a(sqrt(2)-1), R = c/2

Why 45-45-90 Triangles Matter

The 45-45-90 triangle is one of only two special right triangles (the other being 30-60-90). Its side ratios are 1 : 1 : sqrt(2), which makes it essential for trigonometric values: sin(45) = cos(45) = sqrt(2)/2 and tan(45) = 1.

Real-World Applications

  • Architecture: Diagonal bracing in square structures creates 45-45-90 triangles.
  • Construction: Cutting a square diagonally produces two isosceles right triangles.
  • Navigation: Moving equal distances north and east creates a 45-degree bearing.
  • Design: Paper folding, tile patterns, and geometric designs frequently use this triangle.
  • Engineering: Force resolution at 45-degree angles uses these ratios.

Derivation of the Side Ratio

Starting with two equal legs of length a, by the Pythagorean theorem: a² + a² = c², so 2a² = c², giving c = a*sqrt(2). This is why the ratio is always 1 : 1 : sqrt(2).

How to Use This Calculator

  1. Select whether you know the leg, hypotenuse, or area.
  2. Enter the known value.
  3. The calculator automatically computes all other properties.