30-60-90 Triangle Calculator
Solve any 30-60-90 triangle from a single known side using the fixed 1 : √3 : 2 ratio. Instantly find the short leg, long leg, hypotenuse, area.
30-60-90 Triangle Calculator
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Formula: If short leg = x, then long leg = x√3 and hypotenuse = 2x
Worked example — Long leg ≈ 20.7846 cm, hypotenuse = 24 cm
Formula
If short leg = x, then long leg = x√3 and hypotenuse = 2x
A 30-60-90 triangle is formed by bisecting an equilateral triangle, which locks its sides into the permanent ratio 1 : √3 : 2. The short leg sits opposite the 30° angle, the long leg (√3 times longer) sits opposite the 60° angle, and the hypotenuse is exactly twice the short leg. Knowing any one of the three sides is enough to determine the other two without trigonometric tables.
Worked Examples
Example 1: Given the short leg
Problem:A 30-60-90 triangle has a short leg of 12 cm. Find the other sides.
Solution:Short leg = 12 Long leg = 12√3 ≈ 20.7846 Hypotenuse = 2 × 12 = 24
Result:Long leg ≈ 20.7846 cm, hypotenuse = 24 cm
Example 2: Given the hypotenuse
Problem:A ladder triangle has a hypotenuse of 30 ft in a 30-60-90 setup. Find the legs.
Solution:Short leg = 30 ÷ 2 = 15 Long leg = 15√3 ≈ 25.9808
Result:Short leg = 15 ft, long leg ≈ 25.9808 ft
Frequently Asked Questions
What is a 30-60-90 triangle?
A 30-60-90 triangle is a special right triangle with interior angles of 30°, 60°, and 90°. Its side lengths always follow the fixed ratio 1 : √3 : 2.
What is the side ratio for a 30-60-90 triangle?
If the short leg is x, then the long leg is x√3 and the hypotenuse is 2x. That ratio is what makes this triangle easy to solve from any one side.
How do you find the long leg from the short leg?
Multiply the short leg by √3. For example, if the short leg is 8, the long leg is 8√3 ≈ 13.8564.
Why is the hypotenuse exactly twice the shortest side?
Because this triangle is half of an equilateral triangle. The hypotenuse is a full side of the original equilateral figure, while the side opposite 30° is the bisected base, exactly half of it. So doubling always works: a side of 8 opposite the 30° angle gives a hypotenuse of 16.
Can I use any unit?
Yes. Inches, feet, centimeters, and meters all work as long as the known side and the interpreted results use the same unit.
Why do the sides always follow the 1 : √3 : 2 ratio?
Cut an equilateral triangle of side 2x along a line of symmetry. The base is halved to x, one 60° angle survives intact, the bisected angle becomes 30°, and the cut meets the base at 90°. The hypotenuse is still the full side 2x, so the Pythagorean theorem gives the remaining leg as the square root of (2x)² − x², which is x√3. The ratio is locked in by that construction for every x.
How do you find the short leg when only the long leg is known?
Divide the long leg by √3, or equivalently multiply by √3/3 to rationalise the denominator. A long leg of 15 gives a short leg of 15/√3 = 5√3 ≈ 8.6603, and the hypotenuse is double that, about 17.3205. Multiplying when you should divide is the most common slip with this triangle, so sanity-check that the short leg came out as the smallest of the three sides.
What exact trigonometric values come out of this triangle?
Reading the side ratios from the 30° angle gives sin 30° = 1/2, cos 30° = √3/2 and tan 30° = 1/√3 = √3/3. From the 60° angle they become sin 60° = √3/2, cos 60° = 1/2 and tan 60° = √3. These are the exact values expected without a calculator in trigonometry, and they follow directly from the 1 : √3 : 2 side lengths.
How do you get the area straight from the hypotenuse?
The short leg is half the hypotenuse and the long leg is that value times √3, so with hypotenuse h the area is ½ × (h/2) × (h√3/2) = h²√3/8. For h = 24 that gives 576√3/8 ≈ 124.7077 square units, the same answer as finding the legs 12 and 20.7846 first. The matching perimeter shortcut is h(3 + √3)/2.
How do you use a 30-60-90 triangle to find the height of an equilateral triangle?
Drop an altitude from any vertex of an equilateral triangle of side s. It splits the figure into two 30-60-90 triangles whose hypotenuse is s and whose short leg is s/2, so the altitude is the long leg, s√3/2. For s = 6 the height is 3√3 ≈ 5.1962 and the area is ½ × 6 × 5.1962 ≈ 15.5885, matching the standard s²√3/4. The same altitude is the median, so the centroid sits two-thirds of the way down it.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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