30-60-90 Triangle Calculator

Calculate 30-60-90 Triangle Properties

Enter any known value for a 30-60-90 triangle to calculate the side lengths, altitude, area, perimeter, inradius, and circumradius.

How to Calculate 30-60-90 Triangle Properties

A 30-60-90 triangle is a special right triangle with angles of 30°, 60°, and 90°. It has unique properties that make calculations straightforward once you understand the relationships between its sides.

30-60-90 Triangle Formula

The sides of a 30-60-90 triangle are in the ratio of 1 : √3 : 2. If we denote the shortest side as x, then:

  • Short side (opposite to 30° angle): x
  • Medium side (opposite to 60° angle): x√3
  • Long side (hypotenuse): 2x

Calculation Steps

  1. Identify the known side (short, medium, or long)
  2. Calculate the other two sides using the ratio 1 : √3 : 2
  3. Calculate the altitude (height) of the triangle: h = (medium side) / 2
  4. Calculate the area: Area = (1/2) * (short side) * (medium side)
  5. Calculate the perimeter: Perimeter = (short side) + (medium side) + (long side)
  6. Calculate the inradius: Inradius = Area / Semiperimeter
  7. Calculate the circumradius: Circumradius = (long side) / 2

Example Calculation

Let's calculate the properties of a 30-60-90 triangle with a known short side of 5 units:

  1. Short side (x) = 5
  2. Medium side = 5√3 ≈ 8.66
  3. Long side = 2 * 5 = 10
  4. Altitude = 8.66 / 2 ≈ 4.33
  5. Area = (1/2) * 5 * 8.66 ≈ 21.65
  6. Perimeter = 5 + 8.66 + 10 ≈ 23.66
  7. Inradius = 21.65 / (23.66 / 2) ≈ 1.83
  8. Circumradius = 10 / 2 = 5

Visual Representation

Here's a visual representation of a 30-60-90 triangle:

1 2 1 2 30° 60° 90° x (short) x√3 (medium) 2x (long)

This diagram illustrates the key features of a 30-60-90 triangle, including its side lengths and angles.

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