30 60 90 Triangle Calculator

Enter any known value for a 30 60 90 triangle to calculate the edge lengths, altitude, area, perimeter, inradius, and circumradius.

What is a 30 60 90 Triangle?

A 30 60 90 triangle is a special right triangle that has interior angles measuring 30°, 60°, and 90°.

You can also think of a 30 60 90 triangle as half of an equilateral triangle divided from any point to halfway along the opposite leg.

In a 30 60 90 triangle, the shortest leg will be opposite the 30° angle, and the longest leg, or hypotenuse, will be opposite the 90° angle.

Formulas to Solve a 30 60 90 Triangle

Because a 30 60 90 triangle is a right triangle, you can use formulas such as the Pythagorean theorem on them. However, there are also a few simplified formulas that can be used on a 30 60 90 triangle as well, allowing you to solve the triangle given only one side.

The legs of a 30 60 90 right triangle will always have a ratio of 1 : √3 : 2. Using the ratio of the sides, you can solve for any leg if you know just one of the other legs.

Using these ratios is a shortcut to using the known angles and the sine or cosine functions to solve the length of the legs.

Find Leg A

Leg a is the shortest side opposite the 30° angle. Given leg b or the hypotenuse c, the formulas to solve leg a are:

a = b × √3 / 3
a = c / 2

Find Leg B

Leg b is the second-longest side opposite the 60° angle. Given leg a or the hypotenuse c, the formulas to solve leg b are:

b = a√3
b = c × √3 / 2

Find the Hypotenuse

In a 30 60 90 special right triangle, the hypotenuse is the longest side opposite the right angle, and it is always equal to twice the length of the shortest leg. Thus, the formula to calculate the hypotenuse c is simply c = 2a.

Given leg a or b, the formulas to solve c are:

c = 2a
c = 2b × √3 / 3

Find the Area

Given the side lengths of a 30 60 90 triangle, you can find the area using the formula:

area = ab / 2

The area is equal to leg a times leg b, divided by 2.

Find Perimeter

Like any triangle, the perimeter is equal to the sum of the edges.

p = a + b + c

So, perimeter p is simply leg a plus leg b plus hypotenuse c.

There is a special formula to find the perimeter of a 30 60 90 triangle if you know the length of leg a:

p = a × (√3 + 3)

The perimeter p is equal to the length of leg a times the square root of 3 plus 3.

Like special right triangles? You might also like our 45 45 90 calculator.