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.