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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 ▾

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Use the 30-60-90 triangle calculator with radicals to easily solve this special right triangle. Unlike regular triangles, this type follows strict side ratios, and trigonometric formulas are used to compute unknown sides and angles accurately.

Let’s dive deeper into this unique triangle type and see how to work with it effectively.

Stay attentive!

What Is a 30-60-90 Triangle?

In geometry:

"A triangle with angles of 30°, 60°, and 90° is called a 30-60-90 triangle." 30 60 90 triangle

30-60-90 Triangle Formulas

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

Long leg (opposite 60°):

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

Hypotenuse (opposite 90°):

$$ c = 2a $$

You can also use a hypotenuse calculator for any triangle.

Area:

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

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

Given one side, the others can be calculated as follows:

If shortest side (a) is known:

$$ b = a \sqrt{3}, \quad c = 2a $$

If longer leg (b) is known:

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

If hypotenuse (c) is known:

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

Angles and Sides Ratios

Angles:

$$ 30 : 60 : 90 $$

Sides:

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

Numerically:

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

These ratios are used in 30-60-90 triangle calculators for precise side lengths, perimeter, and area.

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°):

$$ b = a \sqrt{3} = 2 \sqrt{3} \approx 3.464 $$

Verify using a 30-60-90 triangle sides calculator.

How the 30-60-90 Triangle Calculator Works

Our free online tool is simple to use:

Input:

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

Output:

Other sides
Hypotenuse
Area
Height
Perimeter
Inradius
Circumradius

Related Tools:

Try our 45-45-90 triangle calculator or the general triangle calculator for other triangle types.

FAQs

How do you find a triangle’s ratio?

To calculate a triangle’s ratios:

Compute x for angles: $$ x = \frac{180^\circ}{a+b+c} $$
Multiply x by each angle: $$ a*x, b*x, c*x $$

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

What is the 3-4-5 triangle rule?

If one side = 3, adjacent = 4, then hypotenuse = 5. This is a classic Pythagorean triplet rule.

Does 9, 12, 15 form a right triangle?

Yes. 9² + 12² = 15², satisfying the Pythagorean theorem.

Does 5, 12, 13 form a right triangle?

Yes. These numbers form a Pythagorean triplet.

Conclusion

Triangles are the building blocks of geometry. The 30-60-90 triangle calculator quickly solves for all sides, angles, and area, making it an essential tool for students, engineers, and geometry enthusiasts.

References

From Wikipedia: Triangle, Right Triangles

From Khan Academy: Angles in Triangles

From Lumen Learning: Similar Triangles and Areas

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