🔺 30-60-90 Triangle Calculator
Solve special right triangles with sides in ratio 1:√3:2
How to Use This Calculator
Select Which Side You Know
Choose whether you know the short side (opposite 30°), medium side (opposite 60°), or hypotenuse (opposite 90°).
Enter the Length
Input the length of the side you know. Make sure to enter a positive number.
Calculate
Click the "Calculate" button to find all remaining sides, perimeter, and area.
Review Results
See all three sides, angles (30°, 60°, 90°), perimeter, and area displayed clearly.
Formula
Ratio: 1 : √3 : 2
If shortest side = a, then: a : a√3 : 2a
Given Short Side (a):
- Medium side = a × √3
- Hypotenuse = 2a
Given Medium Side (a√3):
- Short side = Medium side ÷ √3
- Hypotenuse = 2 × Short side
Given Hypotenuse (2a):
- Short side = Hypotenuse ÷ 2
- Medium side = Short side × √3
Area:
Area = (Short side × Medium side) / 2 = (a × a√3) / 2 = (a²√3) / 2
Perimeter:
Perimeter = a + a√3 + 2a = a(3 + √3)
About 30-60-90 Triangle Calculator
The 30-60-90 triangle is a special right triangle with angles measuring 30°, 60°, and 90°. Its sides are always in the ratio 1:√3:2, making it easy to calculate unknown sides when you know just one side length.
When to Use This Calculator
- Geometry: Solve problems involving 30-60-90 triangles
- Trigonometry: Understand special angle relationships
- Construction: Calculate measurements for triangular structures
- Education: Learn about special right triangles
- Engineering: Design calculations requiring exact ratios
Why Use Our Calculator?
- ✅ Quick Calculations: Find all sides instantly from one known side
- ✅ Accurate Results: Uses exact mathematical ratios
- ✅ Complete Information: Shows all sides, angles, area, and perimeter
- ✅ Flexible Input: Works with any known side
- ✅ Educational: Helps understand special triangle properties
- ✅ Free: No registration required
Key Properties
- Side Ratio: The sides are always in the ratio 1:√3:2
- Angles: Fixed at 30°, 60°, and 90°
- Relationship: The shortest side is opposite the 30° angle, the medium side is opposite the 60° angle, and the hypotenuse is opposite the 90° angle
- Half Equilateral: A 30-60-90 triangle is half of an equilateral triangle
Example
If the shortest side is 5 units:
- Short side (opposite 30°) = 5
- Medium side (opposite 60°) = 5 × √3 ≈ 8.660
- Hypotenuse (opposite 90°) = 5 × 2 = 10
- Area = (5 × 8.660) / 2 ≈ 21.65
- Perimeter = 5 + 8.660 + 10 ≈ 23.66
Frequently Asked Questions
What is a 30-60-90 triangle?
A 30-60-90 triangle is a special right triangle with angles of 30°, 60°, and 90°. Its sides are always in the ratio 1:√3:2, where the shortest side is opposite the 30° angle.
What is the side ratio of a 30-60-90 triangle?
The sides are in the ratio 1:√3:2. If the shortest side is a, then the medium side is a√3 and the hypotenuse is 2a.
How do I find the hypotenuse of a 30-60-90 triangle?
The hypotenuse is always twice the length of the shortest side. If you know any side, calculate the shortest side first, then multiply by 2.
Is a 30-60-90 triangle always a right triangle?
Yes! A 30-60-90 triangle always has a 90° angle (right angle), making it a special case of a right triangle.
What's the difference between 30-60-90 and 45-45-90 triangles?
A 30-60-90 triangle has angles 30°, 60°, 90° with side ratio 1:√3:2. A 45-45-90 triangle has angles 45°, 45°, 90° with side ratio 1:1:√2. Both are special right triangles.
How is a 30-60-90 triangle related to an equilateral triangle?
If you split an equilateral triangle in half with an altitude, you get two 30-60-90 triangles. The altitude creates the medium side (opposite 60°), half the base is the short side (opposite 30°), and the original side is the hypotenuse (opposite 90°).
Can I use this calculator for triangles that aren't 30-60-90?
No, this calculator is specifically designed for 30-60-90 triangles. If your triangle doesn't have these exact angles, use a general triangle calculator instead.