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30-60-90 Triangles

Master 30-60-90 triangles with interactive lessons and practice problems! Learn the special relationship among the measures of the sides of a 30°−60°−90° triangle.

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30-60-90 Triangles

Study Guide

Key Definition

A 30°−60°−90° triangle is a special type of right triangle whose sides are in the proportion $1 : \sqrt{3} : 2$.

Important Notes

The length of the hypotenuse is twice the length of the shorter leg.
The length of the longer leg is $\sqrt{3}$ times the length of the shorter leg.
These values make the triangle a right triangle due to the Pythagorean theorem.

Mathematical Notation

$x$ represents the length of the shorter leg.

$2x$ represents the length of the hypotenuse.

$x\sqrt{3}$ represents the length of the longer leg.

Note: The square root symbol ($\sqrt{}$) indicates that we take the positive root.

Why It Works

The values satisfy the Pythagorean theorem: $x^2 + (x\sqrt{3})^2 = (2x)^2$.

Remember

In a 30°−60°−90° triangle, the sides are in the ratio $1 : \sqrt{3} : 2$.

Quick Reference

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

Understanding 30-60-90 Triangles

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Video explanation of this concept

Beginner

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Beginner Explanation

A 30°-60°-90° triangle is a special right triangle with sides in the ratio $1 : \sqrt{3} : 2$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

If the shorter leg of a 30°-60°-90° triangle is $x$, what is the length of the hypotenuse?

Real-World Problem

Question Exercise

Intermediate

Tree Shadow Scenario

A tree casts a shadow of length $x$. If the angle between the ground and the sunlight is 30°, how tall is the tree?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

If the hypotenuse of a 30°-60°-90° triangle is of length $2x$, what is the area of the triangle?

Challenge Quiz

Single Choice Quiz

Advanced

If the longer leg of a 30°-60°-90° triangle is of length $x\sqrt{3}$, what is the perimeter of the triangle?

Recap

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