30 60 90 Special Right Triangle Calculator

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

Solution:

sides:
a = 3
b = 5.196
c = 6
height:
h = 2.598
area:
T = 7.794
perimeter:
p = 14.196
inradius:
r = 1.098
circumradius:
R = 3


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.

diagram of a special right 30 60 90 triangle showing legs a and b, hypotenuse c, 30 & 60 degree angles, and height h

Formulas to Solve a 30 60 90 Triangle

Because a 30 60 90 triangle is a right triangle the formulas for a right triangle can also be used on them. However, there are also a few simplified formulas that can be used on a 30 60 90 triangle as well.

Solve Leg A

Given leg b, the formula to solve leg a is:

a = b × √33

Thus, leg a is equal to leg b times the square root of 3, divided by 3.

Solve Leg B

Given leg a, the formula to solve leg b is:

b = a√3

So, leg b is equal to leg a times the square root of 3.

Calculate the Hypotenuse

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

Find the Area

Given the side lengths of a 30 60 90 triangle, the formula to find the area is:

T = ab2

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

Calculate 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.

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