Calculate all sides, area, and perimeter of a 30-60-90 special right triangle. Enter any one side to find the others using exact ratios.
Last updated: April 2026 | By Patchworkr Team
The short leg is opposite the 30° angle, the long leg is opposite the 60° angle, and the hypotenuse is opposite the 90° angle.
A 30-60-90 triangle is a special right triangle where the three angles measure 30°, 60°, and 90°. It is one of two special right triangles (the other being 45-45-90) with exact side ratios that can be expressed using simple radicals.
The sides of a 30-60-90 triangle always follow the ratio 1 : √3 : 2, where:
This triangle appears frequently in geometry, trigonometry, and real-world applications like architecture and engineering. Its predictable ratios make calculations straightforward without needing a calculator.
Determine if you have the short leg, long leg, or hypotenuse.
Why: Different known values require different calculation paths. Identifying your starting point determines which formula to use.
Know that: short leg = x, long leg = x√3, hypotenuse = 2x for some value x.
Why: These exact ratios define the 30-60-90 triangle. They come from trigonometry and hold universally.
Use: If short leg known → multiply by √3 for long leg, multiply by 2 for hypotenuse. If long known → divide by √3 for short leg. If hypotenuse known → divide by 2 for short leg, multiply by (√3/2) for long leg.
Why: The formulas directly implement the 1:√3:2 ratio. Matrix relationships ensure consistency.
Area = (short leg × long leg) / 2; Perimeter = short leg + long leg + hypotenuse
Why: Area uses the standard triangle formula. Perimeter is simply the sum of all three sides.
Check: (short leg)² + (long leg)² = (hypotenuse)² using Pythagorean theorem.
Why: Verification catches calculation errors. All right triangles satisfy Pythagorean relation.
Ramp Construction with 30° Angle
The sides are in the ratio 1 : √3 : 2, where 1 is the short leg (opposite 30°), √3 is the long leg (opposite 60°), and 2 is the hypotenuse.
Multiply the short leg by 2. For example, if the short leg is 5, the hypotenuse is 10.
It has exact side ratios that don't require decimal approximations, making calculations faster and more precise in geometry and trigonometry.
Yes! The side ratios satisfy a² + b² = c². Try it: (x)² + (x√3)² = (2x)² simplifies to x² + 3x² = 4x².
Common in roof trusses, hexagonal structures, navigation (60° compass bearings), and equilateral triangles (which split into two 30-60-90 triangles).
Divide the long leg by √3 to get the short leg, then multiply the short leg by 2 to get the hypotenuse.
No, but if you cut an equilateral triangle in half vertically, you create two 30-60-90 triangles.
Use Area = (short leg × long leg) / 2. This is the standard triangle area formula applied to right triangles.
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