30 60 90 Triangle Calculator

30 60 90 Triangle Calculator

Enter any known parameter of the 30-60-90 triangle, and the calculator will attempt to calculate all other related parameters.

Note:

Enter leg (a)

cm ▾

Make use of the 30 60 90 triangle calculator with radicals that helps you to solve this special type of right triangle. Except for a normal triangle that may have different measures for sides and angles, you need particular trigonometric formulas to solve for various factors related to this triangle kind.

So let’s get ahead and discuss more about this particular geometrical shape in detail.

Stay focused!

What Is 30 60 90 Triangle?

In trigonometry:

"A triangle with angles measuring 30°, 60°, and 90° is known as a 30-60-90 triangle." 30 60 90 triangle

30-60-90 Triangle Formulas:

In a 30-60-90 right triangle, the sides have a specific ratio. Let the shortest side (opposite the 30° angle) be a. Then:

Longer Side (opposite 60° angle):

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

Hypotenuse (opposite 90° angle):

$$ c = 2a $$

You can also use a hypotenuse calculator to find the hypotenuse of any triangle instantly.

Area:

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

For other triangle types or geometric shapes, you can quickly estimate the area using a general area calculator.

Perimeter:

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

You can also determine the perimeter of any geometric shape using an online perimeter calculator.

30-60-90 Triangle Side Ratios:

For a 30-60-90 triangle, if one side is given, the other sides can be determined as follows:

If the shortest side (a) is given:

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

$$ c = 2a $$

If the longer leg (b) is given:

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

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

If the hypotenuse (c) is given:

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

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

30-60-90 Triangle Ratios:

Angles:

The angles of the triangle are in the ratio:

$$ 30 : 60 : 90 $$

Sides:

The sides of a 30-60-90 triangle follow the ratio:

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

Or numerically:

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

These ratios are used in online 30-60-90 triangle calculators to compute exact side lengths, perimeter, and area for any triangle of this type.

How To Solve a 30-60-90 Triangle?

Let’s go through an example to clearly understand how to solve a special right-angled 30-60-90 triangle.

Example #1:

Given a 30-60-90 triangle with hypotenuse:

$$ c = 4 $$

Solution:

Step 1: Find the shortest side (opposite 30°):

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

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

$$ a = 2 $$

Step 2: Find the longer leg (opposite 60°):

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

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

$$ b \approx 3.464 $$

You can cross-check these calculations using a 30-60-90 triangle sides calculator.

How the 30-60-90 Triangle Calculator Works?

This free online 30-60-90 triangle solver is user-friendly and works in just a few clicks:

Input:

Enter any known value (shortest side, longer side, or hypotenuse) in its designated field
Click the "Calculate" button

Output: The calculator will automatically compute:

Length of the other sides
Hypotenuse
Area
Height
Perimeter
Inradius
Circumradius

Related Tools:

You may also try our 45-45-90 triangle calculator to solve all values in this type of right triangle. If you are dealing with any other triangle type, don’t worry—we have a detailed triangle calculator that helps determine all parameters of any triangle kind.

FAQ’s:

How do you find the ratio of a triangle?

To find the ratios of any triangle, follow these key steps:

Calculate the value of x for angles using:

$$ x = \frac{180^\circ}{a+b+c} $$

Then multiply x by each angle to get the ratio:

$$ a*x, b*x, c*x $$

You can also find the side ratios of any triangle using a Law of Sines calculator.

What is the 3-4-5 Triangle rule?

The 3-4-5 triangle rule states:

“If the length of one side is 3 and the adjacent side is 4, then the hypotenuse must be 5.”

The online 30-60-90 triangle calculator also performs similar calculations quickly.

Does 9, 12, and 15 make a right triangle?

Yes. According to the Pythagorean theorem, if the sum of the squares of the two shorter sides equals the square of the hypotenuse, the triangle is right-angled. Here, 9² + 12² = 15², so it forms a right triangle.

Does 5, 12, and 13 make a right triangle?

Yes, this set of numbers forms a right triangle because they are Pythagorean triplets.

Conclusion:

Triangles are fundamental in geometry, as many other shapes—such as polygons—can be divided into triangles to simplify calculations. The 30-60-90 triangle calculator allows quick and precise determination of triangle parameters, saving time and effort in manual calculations. It is an essential tool for students, engineers, and anyone dealing with geometry problems.

References:

From Wikipedia: Triangle, Types of Triangle, Right Triangles

From Khan Academy: Angles in Triangles, Isosceles & Equilateral Triangles Problems

From Lumen Learning: Similar Triangles and Areas