30-60-90 Triangle Calculator
Find the missing side lengths, area, and perimeter of a 30-60-90 triangle. Enter any one known side to calculate the other sides using the 1 : √3 : 2 ratio.
Calculate Triangle Sides
Enter one known side. The other values are calculated automatically.
Triangle Results
30-60-90 Triangle Diagram
Side lengths update as you enter a value
How to Use the 30-60-90 Triangle Calculator
Enter any one known side of the triangle. The calculator uses the 1 : √3 : 2ratio to find the remaining sides, then calculates the triangle's area and perimeter.
- Enter the known short leg, long leg, or hypotenuse.
- The calculator automatically finds the other two side lengths.
- Check the diagram to see the 30°, 60°, and 90°angles and corresponding side values.
- Use the calculated areaand perimeterto solve your geometry problem.
30-60-90 Triangle Calculator
The 30-60-90 triangle calculator helps you quickly find the sides, area, and perimeter of a special right triangle whose angles are 30°, 60°, and 90°. By entering any one side, the calculator automatically computes the remaining sides using the well-known 1 : √3 : 2 ratio.
This triangle appears frequently in geometry, trigonometry, architecture, and engineering, so understanding its properties can make many math problems easier to solve.
Short side = x
Long side = x√3
Hypotenuse = 2x
What Is a 30-60-90 Triangle?
A 30-60-90 triangle is a special right triangle where the three interior angles measure:
- 30°
- 60°
- 90°
Since it contains a right angle, the triangle follows thePythagorean theorem.
30-60-90 Triangle Formula
A 30-60-90 triangle is a special right triangle where the side lengths always follow a fixed ratio.
- Short leg (a) – opposite 30°
- Long leg (b) = a × √3
- Hypotenuse (c) = 2a
Example Calculation
If the short leg is a = 5
b = a × √3 = 5 × 1.732 = 8.66
c = 2a = 2 × 5 = 10
Area = ½ × a × b = ½ × 5 × 8.66 = 21.65
30-60-90 Triangle Formulas for Each Known Side
You can solve a 30-60-90 triangle when any one side is known. Use the following formulas to calculate the remaining sides.
Short Leg Known
Long leg=a√3
Hypotenuse=2a
Long Leg Known
Short leg=b / √3
Hypotenuse=2b / √3
Hypotenuse Known
Short leg=c / 2
Long leg=c√3 / 2
30-60-90 Triangle Side Ratio
The sides follow this fixed ratio:
1 : √3 : 2
| Side | Description |
|---|---|
| a | Short leg (opposite 30°) |
| b | Long leg (opposite 60°) |
| c | Hypotenuse (opposite 90°) |
Step-by-Step Calculation
Given short side=10
Long side=10 × √3=17.32
Hypotenuse=2 × 10=20
Area=½ × 10 × 17.32=86.60
Area and Perimeter
Area Formula
Area=½ × a × b
Area=(√3 / 2)a²
Perimeter Formula
Perimeter=a + b + c
Perimeter=a(3 + √3)
Common 30-60-90 Triangle Values
| Short leg | Long leg | Hypotenuse |
|---|---|---|
| 1 | √3 ≈ 1.732 | 2 |
| 2 | 2√3 ≈ 3.464 | 4 |
| 5 | 5√3 ≈ 8.660 | 10 |
| 10 | 10√3 ≈ 17.321 | 20 |
FAQ on 30-60-90 Triangle Calculator
The sides of a 30-60-90 triangle always follow the fixed ratio 1 : √3 : 2, where the shortest side is opposite 30°, the longest side (hypotenuse) is opposite 90°, and the middle side is opposite 60°.
The area is calculated using:
Area=(√3 / 2) × a²where ais the shortest side.
You can solve a 30-60-90 triangle by using the fixed ratio 1 : √3 : 2. If you know one side, you can calculate the other two sides using simple multiplication.
These triangles are widely used in geometry, trigonometry, engineering, architecture, construction, and RF planningbecause of their predictable side ratios.
The long leg (opposite 60°) is found by multiplying the short leg by √3.
Formula: b=a√3
This ratio comes from dividing an equilateral triangle into two right triangles, which creates angles of 30°, 60°, and 90°.