30-60-90 Triangle Calculator - Special Right Triangle
Find all sides of a 30-60-90 special right triangle instantly by entering any one known side length.
Select which side you know, enter its length, and the calculator will compute the other two sides using the exact 1 : √3 : 2 ratio.
30-60-90 Triangle Calculator - Special Right Triangle
Find all sides of a 30-60-90 special right triangle instantly by entering any one known side length.
About the 30-60-90 Triangle Calculator
A 30-60-90 triangle is one of the two most important special right triangles in mathematics, the other being the 45-45-90 triangle. It takes its name from its three interior angles: 30 degrees, 60 degrees, and 90 degrees. What makes this triangle special is that its sides always maintain a fixed, predictable ratio of 1 : √3 : 2, regardless of scale. The shortest side (opposite the 30° angle) has length 1 unit, the medium side (opposite the 60° angle) has length √3 ≈ 1.732 units, and the hypotenuse (opposite the 90° angle) has length 2 units. Because of this constant ratio, knowing the length of any single side is sufficient to determine all three sides exactly.
The 30-60-90 triangle arises naturally from an equilateral triangle. If you draw the altitude from one vertex of an equilateral triangle with side length 2 to the opposite side, you split it into two congruent 30-60-90 triangles. The base half has length 1, the altitude has length √3, and the original side serves as the hypotenuse of length 2. This elegant geometric origin guarantees that the ratios are not approximations but exact algebraic values.
To find the sides from a known short side, multiply by √3 for the long side and by 2 for the hypotenuse. Given the long side, divide by √3 for the short side and multiply by 2/√3 for the hypotenuse. Given the hypotenuse, divide by 2 for the short side and multiply by √3/2 for the long side. The calculator performs these multiplications and divisions in full floating-point precision, giving you results accurate to at least 10 significant digits for any practical input.
In architecture and construction, 30-60-90 triangles appear in roof pitch design, staircase calculations, and the placement of diagonal bracing. A roof with a 30-degree pitch is structurally common in regions with moderate snowfall because it balances snow-shedding ability with wind resistance. In engineering, bolts and fasteners often have hexagonal heads, and each internal triangle of a regular hexagon is a 30-60-90 triangle. This means that knowing the wrench size (the flat-to-flat distance) immediately gives you the corner-to-corner diameter via the same ratios.
In trigonometry, the 30-60-90 triangle provides the exact values for sin(30°) = 1/2, cos(30°) = √3/2, sin(60°) = √3/2, and cos(60°) = 1/2. These values recur throughout calculus, physics, and engineering calculations, and being able to reconstruct them from the triangle ratios is a fundamental skill. In navigation and surveying, when a surveyor needs to measure an inaccessible distance, setting up a 30-60-90 triangle with known baseline allows exact computation of the far side. Solar panel installation also relies on these triangles when calculating optimal tilt angles for maximum sun exposure at a given latitude.
30-60-90 triangle examples
These examples show how the 1 : √3 : 2 ratio determines all sides from a single known measurement.
| Known Side | All Three Sides | Explanation |
|---|---|---|
| Short side = 5 | Short = 5, Long ≈ 8.660, Hypotenuse = 10 | Multiply 5 by √3 ≈ 1.7321 to get the long side; multiply 5 by 2 to get the hypotenuse. The ratio 5 : 8.660 : 10 confirms the 1 : √3 : 2 pattern. |
| Hypotenuse = 20 | Short = 10, Long ≈ 17.321, Hypotenuse = 20 | Divide the hypotenuse by 2 to find the short side; multiply the short side by √3 for the long side. The triangle scales cleanly because 20 is an even number. |
| Long side = 9 | Short ≈ 5.196, Long = 9, Hypotenuse ≈ 10.392 | Divide the long side by √3 to get the short side, then multiply the short side by 2 for the hypotenuse. Dividing by √3 ≈ 1.7321 gives a slightly irrational result. |
How to use the 30-60-90 Triangle Calculator
- Select the side you know from the Known Side dropdown: Short Side (opposite 30°), Long Side (opposite 60°), or Hypotenuse (opposite 90°).
- Enter the length of that side as a positive number in the Side Length field.
- Click Calculate to instantly see all three sides of the triangle.
- Verify the result by checking that the short side is half the hypotenuse and the long side equals the short side times √3.
- Click Reset to start a new calculation, or change the side type to cross-check your answer from a different known value.
30-60-90 Triangle FAQ
What is the side ratio of a 30-60-90 triangle?
The sides are always in the ratio 1 : √3 : 2, where 1 is the side opposite 30°, √3 (≈ 1.732) is opposite 60°, and 2 is the hypotenuse opposite 90°. This ratio is exact, not an approximation, so it holds for any size of 30-60-90 triangle.
How do I find the long side if I know the short side?
Multiply the short side by √3 (approximately 1.73205). For example, if the short side is 7, the long side is 7 × √3 ≈ 12.124 and the hypotenuse is 7 × 2 = 14. The calculator performs this multiplication for any input.
Why is √3 an exact value in a 30-60-90 triangle?
The ratios come from bisecting an equilateral triangle. An equilateral triangle with side 2 bisected by its altitude yields two 30-60-90 triangles with short side 1, hypotenuse 2, and altitude √(4−1) = √3 by the Pythagorean theorem. Because the original triangle has exact integer sides, the altitude is exactly √3, not a rounded decimal.
How does a 30-60-90 triangle differ from a 45-45-90 triangle?
A 45-45-90 triangle is isosceles — the two legs are equal and each equals the hypotenuse divided by √2. A 30-60-90 triangle has three different side lengths following the 1 : √3 : 2 ratio. The two triangles each appear in half of a square and in half of an equilateral triangle, respectively.
What are real-world uses of 30-60-90 triangles?
They appear in architecture (roof pitches, staircase design), engineering (hex bolt geometry), trigonometry (exact values of sin and cos at 30° and 60°), navigation (triangulation), and solar energy (optimising panel tilt angles). The consistent ratio makes them ideal whenever a predictable geometric relationship is needed.
Can I use this calculator for triangles that are only approximately 30-60-90?
The calculator applies the exact 1 : √3 : 2 ratio. If your physical triangle is close but not exactly 30-60-90 due to manufacturing tolerances, you can use the result as a theoretical reference and account for tolerances separately. For a non-special right triangle you would use the law of sines or the Pythagorean theorem instead.