45-45-90 Triangle Calculator

Calculate 45-45-90 Triangle Properties

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

How to Calculate 45-45-90 Triangle Properties

A 45-45-90 triangle, also known as an isosceles right triangle, is a special right triangle with two 45° angles and one 90° angle. Its unique properties make calculations straightforward once you understand the relationships between its sides.

45-45-90 Triangle Formula

In a 45-45-90 triangle, the two legs are equal, and the hypotenuse is √2 times the length of a leg. If we denote the leg as a, then:

  • Leg: a
  • Hypotenuse: a√2

Calculation Steps

  1. Identify the known value (leg, hypotenuse, height, area, or perimeter)
  2. Calculate the leg length using the appropriate formula:
    • From leg: a = given value
    • From hypotenuse: a = c/√2
    • From height: a = h
    • From area: a = √(2A)
    • From perimeter: a = P/(2 + √2)
  3. Calculate the altitude (height) of the triangle: h = a
  4. Calculate the area: Area = (1/2) * a^2
  5. Calculate the perimeter: Perimeter = 2a + a√2
  6. Calculate the inradius: Inradius = a(√2 - 1)
  7. Calculate the circumradius: Circumradius = a√2 / 2

Example Calculation

Let's calculate the properties of a 45-45-90 triangle with a known leg of 5 units:

  1. Leg (a) = 5
  2. Hypotenuse = 5√2 ≈ 7.07
  3. Altitude = 5
  4. Area = (1/2) * 5^2 = 12.5
  5. Perimeter = 2 * 5 + 5√2 ≈ 17.07
  6. Inradius = 5(√2 - 1) ≈ 2.07
  7. Circumradius = 5√2 / 2 ≈ 3.54

Visual Representation

Here's a visual representation of a 45-45-90 triangle:

a a a√2 45° 90° 45° h = a

This diagram illustrates the key features of a 45-45-90 triangle, including its side lengths and angles.

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