Know one side of a 30-60-90 triangle? Enter it to get the other two sides in exact radical form (like 5√3) and as decimals, along with the area, perimeter, altitude and a labeled drawing. You can also start from the area, perimeter or altitude.
Last updated:
Enter your values to see the result.
Advertisement
Advertisement
How to use the 30-60-90 Triangle Calculator
Choose which measurement you know: short leg, long leg, hypotenuse, altitude, area or perimeter.
Enter its value; whole numbers, decimals and fractions such as 7.5 all work.
Pick a unit if you want labeled answers (optional).
Read the sides in exact radical form and as decimals.
Check the steps and the drawing, or download the table as CSV.
30-60-90 triangle formulas
Call the short leg a. Everything else follows from the 1 : √3 : 2 ratio, so one measurement is enough to solve the whole triangle.
You know
Short leg a
Long leg b
Hypotenuse c
Short leg a
a
a√3
2a
Long leg b
b√3/3
b
2b√3/3
Hypotenuse c
c/2
c√3/2
c
Altitude h (to c)
2h√3/3
2h
4h√3/3
Perimeter P
P(3 − √3)/6
P(√3 − 1)/2
P(3 − √3)/3
Area = (√3/2)·a² and perimeter = a(3 + √3). The altitude from the right angle splits the hypotenuse into pieces of a/2 and 3a/2, creating two smaller 30-60-90 triangles.
Common 30-60-90 triangles
Short leg
Long leg
Hypotenuse
Area
1
√3 ≈ 1.732
2
√3/2 ≈ 0.866
2
2√3 ≈ 3.464
4
2√3 ≈ 3.464
3
3√3 ≈ 5.196
6
9√3/2 ≈ 7.794
4
4√3 ≈ 6.928
8
8√3 ≈ 13.856
5
5√3 ≈ 8.660
10
25√3/2 ≈ 21.651
√3 ≈ 1.732
3
2√3 ≈ 3.464
3√3/2 ≈ 2.598
2√3 ≈ 3.464
6
4√3 ≈ 6.928
6√3 ≈ 10.392
Tips for homework and tests
Find the short leg first. Whatever you are given, convert it to the short leg, then multiply by √3 or 2.
Rationalize the denominator. 6/√3 is usually written as 2√3: multiply top and bottom by √3 and simplify.
Check the order. Short leg < long leg < hypotenuse, because 1 < 1.732 < 2.
Connect it to trig. The ratios give the exact values sin 30° = 1/2, cos 30° = √3/2 and tan 60° = √3 used on the unit circle.
Frequently asked questions
What is the ratio of the sides in a 30-60-90 triangle?
The sides are always in the ratio 1 : √3 : 2. If the short leg (opposite 30°) is a, the long leg (opposite 60°) is a√3 and the hypotenuse is 2a. For a = 5 that gives 5, 5√3 ≈ 8.66 and 10.
How do I find the short leg from the hypotenuse?
Divide the hypotenuse by 2. A 30-60-90 triangle with hypotenuse 14 has a short leg of 7 and a long leg of 7√3 ≈ 12.12.
How do I find the short leg from the long leg?
Divide the long leg by √3, then rationalize: a = b/√3 = b√3/3. A long leg of 6 gives a short leg of 2√3 ≈ 3.464 and a hypotenuse of 4√3 ≈ 6.928.
What is the area of a 30-60-90 triangle?
Area = ½ × short leg × long leg = ½ × a × a√3 = (√3/2) × a². With a short leg of 5 the area is 25√3/2 ≈ 21.65 square units.
Why does a 30-60-90 triangle have these ratios?
It is exactly half of an equilateral triangle. Cutting an equilateral triangle with side 2a down the middle gives a hypotenuse of 2a and a short leg of a; the Pythagorean theorem then gives the long leg √(4a² − a²) = a√3.
What does exact radical form mean?
It keeps square roots as symbols instead of rounding them, for example 4√3 instead of 6.9282. Teachers usually ask for this form because it is precise; the calculator shows both.
Which side is opposite the 60° angle?
The long leg. The smallest side is always opposite the smallest angle (30°), and the hypotenuse, the longest side, is opposite the 90° angle.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.