Area of a 30 60 90 Triangle

30-60-90 Triangle Area Formula:

\[ A = \frac{\sqrt{3}}{4} \times x^2 \]

Area (square meters):

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is a 30-60-90 Triangle?

Definition: A 30-60-90 triangle is a special right triangle where the angles measure 30°, 60°, and 90°.

Properties: The sides are in the ratio 1 : √3 : 2, where:

x — Short leg (opposite 30° angle)
x√3 — Long leg (opposite 60° angle)
2x — Hypotenuse (opposite 90° angle)

2. How Does the Calculator Work?

The calculator uses the formula:

\[ A = \frac{\sqrt{3}}{4} \times x^2 \]

Where:

\( A \) — Area of the triangle (square meters)
\( x \) — Length of the short leg (meters)
\( \sqrt{3} \) — Approximately 1.732

Explanation: The formula is derived from the general triangle area formula (1/2 × base × height) using the side ratios of a 30-60-90 triangle.

3. Importance of 30-60-90 Triangle Calculations

Applications: These triangles are fundamental in geometry, trigonometry, and various engineering applications. They're commonly found in equilateral triangles split in half.

4. Using the Calculator

Tips: Simply enter the length of the short leg (x) in meters. The calculator will compute the area in square meters.

5. Frequently Asked Questions (FAQ)

Q1: What if I only know the hypotenuse?
A: Divide the hypotenuse by 2 to get the short leg (x), then use the calculator.

Q2: What if I only know the long leg?
A: Divide the long leg by √3 (≈1.732) to get the short leg (x), then use the calculator.

Q3: Why is the formula (√3/4)x²?
A: This comes from (1/2) × base × height = (1/2) × x × (x√3) = (√3/2)x²/2 = (√3/4)x².

Q4: Can I use this for other units besides meters?
A: Yes, as long as you use consistent units. The result will be in square of whatever unit you input.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact. Any inaccuracy would come from measurement of the short leg.