30-60-90 Triangle Calculator

What is a 30-60-90 Triangle?

A 30-60-90 triangle is a special right triangle with angles measuring 30°, 60°, and 90°. This triangle is half of an equilateral triangle and has unique proportions that make it valuable in geometry and trigonometry.

Key Properties of 30-60-90 Triangles

• The shortest side is opposite the 30° angle
• The longest side (hypotenuse) is opposite the 90° angle
• The medium-length side is opposite the 60° angle
• The ratio of side lengths is always 1 : √3 : 2

Formulas for 30-60-90 Triangles

Let \(x\) be the length of the shortest side. Then:

1. Short side (opposite 30°): \(a = x\)
2. Long side (opposite 60°): \(b = x\sqrt{3}\)
3. Hypotenuse (opposite 90°): \(c = 2x\)
4. Area: \(A = \frac{x^2\sqrt{3}}{4}\)
5. Perimeter: \(P = x(2 + \sqrt{3})\)

Step-by-Step Calculations

Let's calculate these properties for a 30-60-90 triangle with shortest side \(x = 2\) units:

1. Short side: \[a = x = 2 \text{ units}\]
2. Long side: \[b = x\sqrt{3} = 2\sqrt{3} \approx 3.46 \text{ units}\]
3. Hypotenuse: \[c = 2x = 2(2) = 4 \text{ units}\]
4. Area: \[A = \frac{x^2\sqrt{3}}{4} = \frac{2^2\sqrt{3}}{4} = \sqrt{3} \approx 1.73 \text{ square units}\]
5. Perimeter: \[P = x(2 + \sqrt{3}) = 2(2 + \sqrt{3}) \approx 7.46 \text{ units}\]

Visual Representation

This diagram illustrates a 30-60-90 triangle with the calculated dimensions and angles.

Example

In a 30-60-90 triangle, if the shortest side is 5 units long, find the lengths of the other sides and the area of the triangle.

Solution:

1. Short side: \(a = 5\) units
2. Long side: \(b = 5\sqrt{3} \approx 8.66\) units
3. Hypotenuse: \(c = 2(5) = 10\) units
4. Area: \(A = \frac{5^2\sqrt{3}}{4} = \frac{25\sqrt{3}}{4} \approx 10.83\) square units

This example demonstrates how the unique proportions of a 30-60-90 triangle allow us to easily calculate all sides and the area when given just one side length.