A 30-60-90 triangle is half of an equilateral triangle, so its sides are always in the ratio 1 : √3 : 2. It is one of the two “special” right triangles whose trigonometric values (sin 30° = 1/2, cos 30° = √3/2) every student learns by heart.
How to use the 30 60 90 Triangle Calculator
- Short leg a (opposite 30°)
- Every result field is editable: type into Long leg b (opposite 60°) or any other result and Short leg a (opposite 30°) plus every other field update to match.
Results update as you type, and everything is calculated in your browser.
How it works
Enter the short leg a – or type the long leg, hypotenuse, area or perimeter into its field and the short leg is worked out backwards.
What each result means
- Long leg b (opposite 60°) = [Short leg a (opposite 30°)]×sqrt(3)
- Hypotenuse c = 2×[Short leg a (opposite 30°)]
- Area = [Short leg a (opposite 30°)]^2×sqrt(3)÷2
- Perimeter = [Short leg a (opposite 30°)]×(3 + sqrt(3))
- Height to the hypotenuse = [Short leg a (opposite 30°)]×sqrt(3)÷2
Example values
How the main result changes when Short leg a changes and everything else stays at the example values above:
| Short leg a (opposite 30°) (cm) | Long leg b (opposite 60°) (cm) |
|---|---|
| 2.5 | 4.3301 |
| 3.75 | 6.4952 |
| 5 | 8.6603 |
| 6.25 | 10.8253 |
| 7.5 | 12.9904 |
| 10 | 17.3205 |
Formula
a : b : c = 1 : √3 : 2
b = a√3, c = 2a, Area = a²√3/2
Worked example
Inputs — Short leg a (opposite 30°): 5 cm.
Results — Long leg b (opposite 60°) = 8.660254 cm; Hypotenuse c = 10 cm; Area = 21.650635 cm²; Perimeter = 23.660254 cm; Height to the hypotenuse = 4.330127 cm.
Key terms
- Special right triangle
- A right triangle with simple exact side ratios.
Frequently asked questions
What are the side ratios of a 30-60-90 triangle?
1 : √3 : 2.
Which side is opposite 30°?
The shortest leg, half of the hypotenuse.
Why is it called special?
Its sides and trig values have exact, easy-to-remember forms.
How is it related to an equilateral triangle?
Cutting an equilateral triangle by its altitude gives two 30-60-90 triangles.