Special right triangle tool
30-60-90 Triangle Calculator
Enter any one side of a 30-60-90 triangle and instantly find all three sides, with the ratio, the formula, and step-by-step working.
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Calculator
30-60-90 Triangle Side Calculator
For the general right-triangle rule behind this special case, see Pythagorean Theorem.
Ratio
The 30-60-90 Triangle Ratio
1 : √3 : 2
(short leg) : (long leg) : (hypotenuse)
A 30-60-90 triangle is a special right triangle with angles of exactly 30°, 60°, and 90°. Its
three sides are always in the fixed ratio 1 : √3 : 2, regardless of the triangle's
size. That means you only need one side to calculate the other two, which is why this triangle
shows up so often in geometry, trigonometry, and standardized tests.
The shortest side is always opposite the 30° angle. The longer leg is opposite the 60° angle and
equals the short leg multiplied by √3, about 1.732. The hypotenuse is
opposite the 90° angle and is always exactly twice the short leg.
| Known side | Short leg (a) | Long leg (b) | Hypotenuse (c) |
|---|---|---|---|
| Short leg = 1 | 1 | √3 ≈ 1.732 | 2 |
| Short leg = 5 | 5 | 5√3 ≈ 8.660 | 10 |
| Short leg = 10 | 10 | 10√3 ≈ 17.321 | 20 |
| Hypotenuse = 1 | 0.5 | √3/2 ≈ 0.866 | 1 |
| Hypotenuse = 10 | 5 | 5√3 ≈ 8.660 | 10 |
| Long leg = √3 | 1 | √3 ≈ 1.732 | 2 |
Formulas
30-60-90 Triangle Formulas
b = a√3c = 2aA = a²√3 / 2a = c / 2b = c√3 / 2A = c²√3 / 4a = b√3 / 3c = 2b√3 / 3A = b²√3 / 3
These formulas come directly from the 1 : √3 : 2 ratio. Once you know any one side,
divide or multiply by the matching ratio factor to find the others. The calculator above applies
the same formulas automatically and shows every step.
Where does the 1 : √3 : 2 ratio come from?
A 30-60-90 triangle is half of an equilateral triangle. If you start with an equilateral
triangle whose side length is 2 and draw the altitude from one vertex to the
opposite side, you split it into two congruent 30-60-90 triangles.
That altitude has length √3 by the Pythagorean theorem:
√(2² − 1²) = √3. The resulting right triangle therefore has sides
1, √3, and 2, which is the defining ratio.
Examples
Worked Examples
These three examples cover each practical starting point: knowing the short leg, the hypotenuse, or the common ramp setup that uses the short leg directly.
Example 1: Short leg = 7
Given: a = 7. Find b and c.
| Long leg | b = 7 × √3 = 7 × 1.7321 ≈ 12.124 |
|---|---|
| Hypotenuse | c = 2 × 7 = 14 |
| Area | A = 7² × √3 / 2 = 49 × 1.7321 / 2 ≈ 42.435 |
| Perimeter | P = 7 + 12.124 + 14 ≈ 33.124 |
Example 2: Hypotenuse = 12
Given: c = 12. Find a and b.
| Short leg | a = 12 / 2 = 6 |
|---|---|
| Long leg | b = 12 × √3 / 2 = 6√3 ≈ 10.392 |
| Area | A = 6 × 10.392 / 2 ≈ 31.177 |
| Perimeter | P = 6 + 10.392 + 12 ≈ 28.392 |
Example 3: A ramp at 30° has a vertical rise of 4 m
The vertical rise is the short leg, so a = 4 m. Find the horizontal run
b and the ramp length c.
| Horizontal run | b = 4 × √3 ≈ 6.928 m |
|---|---|
| Ramp length | c = 2 × 4 = 8 m |
Compare
30-60-90 vs 45-45-90 Triangle
| Feature | 30-60-90 | 45-45-90 |
|---|---|---|
| Angles | 30°, 60°, 90° | 45°, 45°, 90° |
| Side ratio | 1 : √3 : 2 | 1 : 1 : √2 |
| Legs | Unequal | Equal |
| Hypotenuse | 2 × short leg | leg × √2 |
| Origin | Half of an equilateral triangle | Half of a square |
| Common use | Ramps, roofs, 60° angles | Square diagonals, isosceles right triangles |
Both are special right triangles with fixed side ratios, so you can solve them from one known side without trigonometry. For the equal-leg case, use the 45-45-90 Triangle Calculator.
FAQ
Frequently Asked Questions
The sides are in the ratio 1 : √3 : 2. The shortest side, opposite 30°, is the
base unit. The longer leg is that base times √3, and the hypotenuse is exactly
twice the short leg.
If you know the short leg a, the hypotenuse is 2a. If you know the
long leg b, the hypotenuse is 2b√3 / 3. Enter any side into the
calculator above and it will return all three side lengths immediately.
Use the ratio 1 : √3 : 2. Short leg known: multiply by √3 for the
long leg and by 2 for the hypotenuse. Hypotenuse known: divide by
2 for the short leg and multiply by √3 / 2 for the long leg. Long
leg known: divide by √3 for the short leg and multiply by 2 / √3
for the hypotenuse.
It appears constantly in geometry, trigonometry, architecture, and test-prep work because the side ratios are exact and easy to scale. It also comes directly from splitting an equilateral triangle in half, so it is a core reference triangle for 30° and 60° angles.
If the short leg is a, the area is A = a²√3 / 2. If the
hypotenuse is c, the same area can be written as A = c²√3 / 4. The
calculator above computes the area automatically from any starting side.
Yes. It includes a 90° angle, so it is a right triangle. The Pythagorean theorem also checks
out with the ratio: 1² + (√3)² = 1 + 3 = 4 = 2².
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