30 60 90 Triangle Calculator

Easily calculate the right triangle by putting the known value in this 30 60 90 triangle calculator. It will quickly give you the accurate value of the third side as well as the area and perimeter.

How to Use This Special Right Triangles Calculator

• Enter any one known side: short leg, long leg, or hypotenuse.
• You can use whole numbers, decimals, or fractions.
• For square roots (like √3), enter the decimal equivalent (about 1.732).
• Click "calculate." 
• The tool finds all missing sides as well as the area and perimeter.

What is a 30-60-90 triangle?

A 30 60 90 triangle is a specific kind of right triangle whose angles are 30, 60, and 90 degrees. Since it has a 90-degree angle, it always has a right angle. Therefore, it is one of the most fundamental types of triangles in all of geometry. Furthermore, its shape and properties can be relied upon. And for that reason, it shows up often in math problems.

These special right triangles 30 60 90 are also known as the result of cutting an equilateral triangle in half. In these triangles, the side opposite the 30-degree angle is the shortest, and the side opposite the 90-degree angle is the longest (this side is called the hypotenuse). Moreover, these consistent relationships make the 30 60 90 triangle very useful for solving calculations quickly.

30 60 90 Triangle Rules

The main advantage of a 30-60-90 triangle is its predictability. There is a single 30 60 90 triangle rule you can memorize, and that rule is the triangle's side lengths are always in the ratio of 1: √3 : 2. We will go over exactly how to apply this rule based on the information you have.

• If the short leg is known

If you want to get the hypotenuse, then you will have to multiply the short leg by 2. If you want to know the leg that is 60 degrees, then you will have to multiply the short leg by √3.

• If the long leg is known

To get the short leg, just divide the long leg by √3. Once you have the short leg, then the hypotenuse is easy to find. Just multiply your short leg by 2.

• If the hypotenuse is known

To find the short leg, you will have to divide the hypotenuse by 2. Then, to find the long leg, you will have to multiply the short leg by √3.

30-60-90 Triangle Examples

Take a close look at the examples provided and the details of how the special right triangle is calculated step-by-step.

Example 1: Short leg = 5 Inches

Short leg = 5 inches

Hypotenuse = 5 * 2 = 10 inches

Long leg = 5 * √3 = 5√3 = 8.66 inches

Example 2: Long Leg = 9 cm

Long Leg = 9 cm

Short leg = 9 / √3 = 3√3 = 5.2 cm

Hypotenuse = 3√3 * 2 = 6√3 = 10.39 cm

Example 3: Hypotenuse = 14 meters

Hypotenuse = 14 meters

Short leg = 14 / 2 = 7 m

Long leg = 7 * √3 = 7√3 = 12.12 m

Why is the 30-60-90 Triangle Ratio 1 : √3 : 2?

In a 30 60 90 triangle, start with an equilateral triangle where all sides are length 2 and all angles are 60°. Slice it down the middle with an altitude, and you get two identical right triangles. Each has angles of 30°, 60°, and 90°. The hypotenuse stays at 2 (the original side), and the shortest leg (opposite the 30° angle) becomes exactly half, so 1. For the remaining leg, you apply the Pythagorean theorem: 1² + b² = 2², giving b² = 3, so b = √3. That's how you get the elegant ratio of 1 : √3 : 2. It's essentially a consequence of the perfect symmetry of the equilateral triangle.

30-60-90 vs. 45-45-90 Triangles

If you see a triangle with two equal-length sides, you are dealing with a 45-45-90 triangle, not a 30-60-90. Take a glance at the comparison table to understand both triangles:

Frequently Asked Questions

Which side is opposite 30 degrees?

The shortest side.

Why is my answer rounded?

It is rounded for simplicity, and so it is easier to read.

Can the calculator give me exact values in radical form?

Yes, the calculator can give values in radical form as well as decimal form.

What is the law of a 30-60-90 triangle?

The sides always follow a set ratio.

What is the side ratio of a 30 60 90 triangle?

The side ratio of a 30 60 90 triangle is 1: √3: 2 (short: long: hypotenuse).

Can I calculate the triangle if I have only one side?

Yes, one side is enough. You can find all the other sides.

How is a 30-60-90 triangle different from a 45-45-90 triangle?

A 30-60-90 has unequal sides. Meanwhile, a 45-45-90 triangle has two equal sides.