Math
30-60-90 Triangle Calculator
A 30-60-90 triangle always has sides in the ratio 1 : √3 : 2.
Enter the short leg, the long leg, or the hypotenuse and this calculator returns the other two sides as exact radicals and decimals, plus area, perimeter, and the ratio steps.
Any positive length. The unit carries through to every answer.
Try an example
Result
Short leg
—
- Long leg
- —
- Hypotenuse
- —
- Area
- —
- Perimeter
- —
Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
Use this calculator with
Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
What a 30-60-90 triangle is
A 30-60-90 triangle is half of an equilateral triangle: cut an equilateral triangle down the middle and each half has angles 30°, 60°, and 90°. Because every 30-60-90 triangle is similar, the sides are always in the same ratio, which lets you find all three from any one.
| Known side | Short leg x | Long leg | Hypotenuse |
|---|---|---|---|
| Short leg = s | s | s√3 | 2s |
| Long leg = L | L/√3 = L√3/3 | L | 2L√3/3 |
| Hypotenuse = h | h/2 | h√3/2 | h |
How to use it
- Pick which side you know: short leg, long leg, or hypotenuse.
- Enter its length.
- Read the short leg x, then the long leg x√3 and hypotenuse 2x in exact and decimal form.
- Use the steps to show the ratio substitution in your work.
How to read the answer
Exact answers keep √3 in them because √3 is A number that cannot be written as a fraction of two whole numbers, like π or √2.; the decimal in parentheses is rounded. If the short leg is a whole number the hypotenuse is a whole number too, while the long leg stays a multiple of √3.
Common mistakes and edge cases
- Mixing up the legs: the long leg is opposite 60°, not 30°. The long leg is about 1.732 times the short leg.
- Dividing by √3 without rationalizing: L/√3 equals L√3/3, which is the form most teachers want.
- Treating the hypotenuse as √3 times the short leg — it is 2 times. Only the long leg carries the √3.
- Lengths must be positive; a side of 0 collapses the triangle.
Worked examples
Short leg 5
Short leg
5
Long leg 6
x = 6/√3 = 2√3
Short leg
2√3 (3.4641)
Hypotenuse 10
x = 5, long leg 5√3
Short leg
5
Decimal short leg 2.5
Short leg
2.5
Zero side
Short leg
—
Frequently asked questions
What are the side ratios of a 30-60-90 triangle?+
1 : √3 : 2 for short leg : long leg : hypotenuse. With short leg x the long leg is x√3 and the hypotenuse is 2x.
How do I find the short leg from the hypotenuse?+
Divide the hypotenuse by 2. For hypotenuse 14 the short leg is 7 and the long leg is 7√3 ≈ 12.12.
How do I find the short leg from the long leg?+
Divide the long leg by √3, then rationalize: L/√3 = L√3/3. For long leg 9 the short leg is 3√3 ≈ 5.196.
Why is the long leg x√3?+
From the Pythagorean theorem with hypotenuse 2x and short leg x: long² = (2x)² − x² = 3x², so long = x√3.
Is a 30-60-90 triangle half an equilateral triangle?+
Yes. Dropping a height in an equilateral triangle of side 2x creates two 30-60-90 triangles with short leg x and long leg x√3.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
Related calculators
45-45-90 Triangle Calculator
Enter the leg or hypotenuse of a 45-45-90 triangle and get the other side in exact radical form (x, x, x√2) and decimals, plus area, perimeter, and the height to the hypotenuse with steps.
Equilateral Triangle Calculator
Enter the side, height, area, perimeter, inradius, or circumradius of an equilateral triangle and get every other measurement with exact radicals (s√3/2, s²√3/4) and decimals, plus the steps.
Right Triangle Calculator
Solve a right triangle from any two values — two sides, a side and an angle, or area or perimeter with a side or angle. Get all sides, both acute angles, area, perimeter, altitude to the hypotenuse, inradius, circumradius, and steps.
Pythagorean Theorem Calculator
Solve a² + b² = c² for the hypotenuse or either leg with exact radical form (√50 = 5√2) and decimals, plus area, perimeter, both acute angles, a Pythagorean-triple check, steps, and a squares-on-sides diagram.
Hypotenuse Calculator
Find the hypotenuse of a right triangle from two legs (c = √(a² + b²)) or from one leg and an acute angle, in exact radical form and decimal, with the other leg, both angles, and area included.
Triangle Calculator
Solve any triangle from SSS, SAS, ASA, AAS, or SSA. Get all sides and angles, area, perimeter, heights, medians, inradius, circumradius, classification, a scale drawing, and both solutions in the ambiguous case.
Last updated: September 4, 2026