30 60 90 Triangle Calculator

Choose Known Side:

Enter leg (a):

cm ▾

This 30-60-90 triangle calculator finds missing side lengths and related properties of a special right triangle with fixed angles of 30°, 60°, and 90°.

What Is a 30-60-90 Triangle?

“A triangle having measures of angles equal to 30, 60, and 90 degrees is known as 30 60 90 triangle”

30 60 90 triangle

30-60-90 Triangle Formulas:

As in a right triangle, we have three sides of different lengths. Now if the longest one is hypotenuse, then we are left with two sides only. Among these two, let’s suppose the shorter side is “a”. We have the following formulas then:

Long leg:

$$ b = \sqrt{3} $$

Hypotenuse:

$$ c = 2a $$

You can also use a hypotenuse calculator for any triangle.

Area:

$$ \text{Area} = a^2 \frac{\sqrt{3}}{2} $$

General area calculators can also estimate the area for any triangle type.

Perimeter:

$$ \text{Perimeter} = a (3 + \sqrt{3}) $$

Or use an online perimeter calculator for any polygon or triangle.

Side Ratios of a 30-60-90 Triangle:

As we have three sides of any triangle, the same is the case in the current situation. Below here we have:

If length a is given:

$$ b = a \sqrt{3} $$

$$ \quad c = 2a $$

If length of longer side is given that is b:

$$ a = \frac{b}{\sqrt{3}} $$

$$ \quad c = \frac{2b}{\sqrt{3}} $$

If only hypotenuse length is given:

$$ a = \frac{c}{2} $$

$$ \quad b = \frac{c\sqrt{3}}{2} $$

Angles and Sides Ratios:

In case of Angles, the standard 30 60 90 triangle formula ratios are given as follows:

Angles:

$$ 30 : 60 : 90 $$

$$ 1 : 2 : 3 $$

Sides:

In case of sides, 30 60 90 triangle side ratios are as follows:

$$ a : a\sqrt{3} : 2a $$

$$ 1 : \sqrt{3} : 2 $$

The online 30 60 90 triangle calculator also takes into consideration these ratios so as to calculate exact results for this particular geometrical figure.

Example of Solving a 30-60-90 Triangle:

Consider a 30-60-90 triangle with hypotenuse:

$$ c = 4 $$

Solution:

Step 1: Shortest side (opposite 30°):

$$ a = \frac{c}{2} = \frac{4}{2} = 2 $$

Step 2: Longer leg (opposite 60°):

$$ b = c \frac{\sqrt{3}}{2} $$

$$ b = 4 \frac{\sqrt{3}}{2} $$

$$ b = 4∗0.866 $$

$$ b = \approx 3.464 $$

How to use the 30 60 90 Triangle Calculator?

This free 30 60 90 triangle solver is very simple to use. Follow these steps:

Input:

Enter any known side (shortest, longer leg, or hypotenuse)
Click "Calculate"

Output:

The online 30-60-90 triangle calculator does the following computations:

Length of second side
Hypotenuse
Area
Height
Perimeter
Inradius
Circumradius

Related Tools:

You may also try our 45 45 90 triangle calculator to solve for each and every value in this right triangle. But what to do if you are dealing with any other triangle type? Do not worry as we have the best solution for this. Yes, we are talking about nothing but a detailed triangle calculator that helps you to determine all parameters of any triangle kind.

FAQ's:

How do you find a triangle’s ratio?

To find the ratios of any triangle, you need to follow up the following key points in your mind:

First of all, calculate the value of x for angles by following the expression:
$$ x = \frac{180^\circ}{a+b+c} $$
Then find the product of x with all angle sone by or as follows:
$$ a*x, b*x, c*x $$

Side ratios can also be found using a Law of Sines calculator.

What is the 3-4-5 triangle rule?

The 3 4 5 rule of triangle states that:

“If the length of one side of triangle is 3 and of adjacent side is 4, then the length of the diagonal joining them must be 5”

The online 30 60 90 triangle calculator with radicals also does the same calculations but saving you a lot of time.

Does 9, 12, 15 form a right triangle?

Yes!

According to the pythagorean theorem, if the sum of the squares of the short sides becomes equal to the square of the hypotenuse length (9² + 12² = 15²), then the set of sides makes a perfect right triangle. And 9 12 and 15 make a perfect right angled triangle.

Does 5, 12, 13 form a right triangle?

Yes, three of these numbers make a right triangle. This is because they are pythagorean triplets.

Conclusion:

Triangles are very crucial to understand and resolve for their parameters. The reason is that many other geometrical figures such as polygons having 4, 5, or even 6 sides can be reduced to triangle, this provides an instant approach to the manual calculations of the triangle thus further creating ease in simplifying the polygons. And rest when fast calculations are concerned, the 30 60 90 triangle calculator is the only best choice you are left with. So stop thinking and start making use of this calculator.

References:

From Wikipedia: Triangle, Right Triangles.
From Khan Academy: Angles in Triangles.
From Lumen Learning: Similar Triangles and Areas.