30-60-90 Triangle
Calculate sides of a 30-60-90 right triangle from one known length. Find short leg, long leg, hypotenuse, area, and perimeter using the 1:sqrt3:2 ratio.
About This Calculator
This free online 30-60-90 triangle calculator solves the triangle from any one known side. Select whether you know the short leg (opposite 30 deg), the long leg (opposite 60 deg), or the hypotenuse, enter its length, and the calculator uses the fixed side ratio of 1 : sqrt3 : 2 to find all remaining side lengths, perimeter, and area.
A 30-60-90 triangle is a special right triangle where the three angles are always 30 deg, 60 deg, and 90 deg. The side lengths are in a fixed proportional relationship: if the shortest leg is x, the long leg is xsqrt3, and the hypotenuse is 2x. This predictable ratio makes these triangles extremely useful in geometry, construction, engineering, and trigonometry -- they eliminate the need for the Pythagorean theorem in many problems.
The calculator also computes the area as half the product of the two legs and the perimeter as the sum of all three sides. Whether you are a student preparing for the SAT, ACT, or a geometry exam, or a professional needing quick triangle dimensions, this tool provides instant accurate results.
Frequently Asked Questions
What is a 30-60-90 triangle?
A 30-60-90 triangle is a special right triangle whose angles are 30 deg, 60 deg, and 90 deg. The side lengths follow the fixed ratio 1 : sqrt3 : 2, where the shortest leg is opposite the 30 deg angle, the longer leg is opposite the 60 deg angle, and the hypotenuse is opposite the 90 deg angle.
What are the side ratios of a 30-60-90 triangle?
In a 30-60-90 triangle, if the short leg (opposite 30 deg) is x, then the long leg (opposite 60 deg) is xsqrt3, and the hypotenuse is 2x. The side ratio is always 1 : sqrt3 : 2.
How do you find the sides of a 30-60-90 triangle?
Select which side you know (short leg, long leg, or hypotenuse) and enter its length. If you know the short leg, multiply by 2 for the hypotenuse and by sqrt3 for the long leg. If you know the hypotenuse, divide by 2 for the short leg. If you know the long leg, divide by sqrt3 for the short leg.
How do you find the area of a 30-60-90 triangle?
The area of a 30-60-90 triangle is calculated as 1/2 x short leg x long leg. For example, if the short leg is 5, the long leg is 5sqrt3, and the area is 0.5 x 5 x 5sqrt3 ≈ 21.65 square units.
Why are 30-60-90 triangles important?
30-60-90 triangles appear frequently in geometry, trigonometry, and standardized tests like the SAT and ACT. Their predictable side ratios allow quick problem-solving without the Pythagorean theorem in many cases.
Is this calculator free to use?
Yes, all calculators on Calculy are completely free to use with no registration required.
How do you find the perimeter of a 30-60-90 triangle?
Add all three sides together: short leg + long leg + hypotenuse. If the short leg is x, the perimeter is x + xsqrt3 + 2x = x(3 + sqrt3). For example, if the short leg is 5, the perimeter is 5 + 5sqrt3 + 10 ≈ 23.66 units.
How does a 30-60-90 triangle relate to an equilateral triangle?
A 30-60-90 triangle is exactly half of an equilateral triangle. Draw an altitude from any vertex of an equilateral triangle to the opposite side, splitting it into two congruent 30-60-90 triangles. The altitude is the long leg, half the base is the short leg, and the original side is the hypotenuse.