Quick answer
In a 30-60-90 triangle the sides are in the ratio a : a√3 : 2a. With a short leg of 4, the long leg is 6.9282, the hypotenuse is 8 and the area is 13.8564.
Enter the short leg. The calculator returns the long leg, hypotenuse and area.
Related tools: Triangle Calculator: Area, perimeter, angles and type of a triangle from its three sides. 45-45-90 Triangle Calculator: Sides of a 45°-45°-90° triangle from one leg. Right Triangle Calculator: Solve a right triangle from its two legs: hypotenuse, angles, area and perimeter.
How to use the 30-60-90 Triangle Calculator
- Enter short leg length (above 0).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Short leg length | A number | > 0 |
Formula and method
Formula
short = a; long = a√3; hypotenuse = 2a
IEEE-754 double precision; shown to at most 6 decimal places.
Fixed side ratios of the special right triangle.
Worked examples
This example is verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Short leg length = 4.
| Result | Value |
|---|---|
| Short leg | 4 |
| Long leg | 6.928203 |
| Hypotenuse | 8 |
| Area | 13.856406 |
The first line is the headline result (short leg). Below it the 30-60-90 Triangle Calculator lists long leg, hypotenuse and area, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Where the ratio comes from
Cutting an equilateral triangle in half gives a 30-60-90 triangle. The hypotenuse is a side of the original (2a), the short leg is half of it (a), and the long leg follows from the Pythagorean theorem: √((2a)² − a²) = a√3.
Quick reference
Area = ½ × a × a√3 = (√3 ÷ 2) a². For a = 4: 0.5 × 4 × 6.9282 = 13.8564. If you know the hypotenuse, the short leg is half of it. The angle opposite the short leg is 30°.
How the result changes with short leg length
Holding every other input at the first worked example, this table changes only short leg length. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Short leg length | Short leg | Long leg | Hypotenuse |
|---|---|---|---|
| 2 | 2 | 3.464102 | 4 |
| 4 | 4 | 6.928203 | 8 |
| 8 | 8 | 13.856406 | 16 |
| 16 | 16 | 27.712813 | 32 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Short leg length: entering "0" shows "Short leg length must be greater than 0." beside that field and no result.
Choosing between the 30-60-90 Triangle Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Triangle Calculator](/math/triangle-calculator/) | side a, side b and side c | Area |
| [45-45-90 Triangle Calculator](/math/45-45-90-triangle-calculator/) | leg length | Leg |
| [Right Triangle Calculator](/math/right-triangle-calculator/) | leg a and leg b | Hypotenuse c |
| [Triangle Area Calculator](/math/triangle-area-calculator/) | base and height | Area |
| [SAS Triangle Calculator](/math/sas-triangle/) | side b, side c and included angle A | Side a |
Background: geometry measurements
Geometry results carry units that depend on the quantity: a perimeter or length is in linear units such as cm, an area or surface area is in square units (cm²) and a volume is in cubic units (cm³). Use one length unit for every input, because the calculator multiplies the numbers as given.
Flat (2D) shapes have an area and a perimeter; solids (3D) have a surface area and a volume. Two familiar checks: a circle of radius r has area πr² and circumference 2πr, and a sphere has volume (4/3)πr³ and surface area 4πr².
Assumptions and limitations
- Positive length
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for short leg length to confirm this.
Frequently asked questions
What is the ratio of sides?
The sides are in the ratio 1 : √3 : 2, so a short leg of 4 gives a long leg of 6.93 and a hypotenuse of 8.
What is the long leg if the hypotenuse is 10?
With hypotenuse 10 the short leg is 5 and the long leg is 5√3 ≈ 8.66.
Is it a right triangle?
Yes, with angles 30°, 60° and 90°.
What does the 30-60-90 Triangle Calculator calculate?
Sides of a 30°-60°-90° triangle from the short leg. It returns Short leg, Long leg, Hypotenuse and Area.
What formula does the 30-60-90 Triangle Calculator use?
short = a; long = a√3; hypotenuse = 2a. Fixed side ratios of the special right triangle.
What do I need to enter in the 30-60-90 Triangle Calculator?
You enter Short leg length.
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Positive length.
What is an example of a 30-60-90 Triangle result?
With short leg length 4, the short leg is 4.
Related calculators
- Triangle Calculator — Area, perimeter, angles and type of a triangle from its three sides.
- 45-45-90 Triangle Calculator — Sides of a 45°-45°-90° triangle from one leg.
- Right Triangle Calculator — Solve a right triangle from its two legs: hypotenuse, angles, area and perimeter.
- Triangle Area Calculator — Area of a triangle from base and height.
- SAS Triangle Calculator — Solve a triangle from two sides and the included angle.
- Equilateral Triangle Calculator — Area, height and perimeter of an equilateral triangle.
- Hypotenuse Calculator — Hypotenuse of a right triangle from its two legs.
- Distance Between Two Points Calculator — Straight-line distance and midpoint between two points in the plane.
- Rectangle Calculator — Area, perimeter and diagonal of a rectangle.
Sources
- OpenStax, Prealgebra 2e (Rice University): perimeter, area, volume, circumference and the Pythagorean theorem
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.