Trigonometry Practice

Special Right Triangles Practice Worksheet

Practice 45°-45°-90° and 30°-60°-90° triangles using exact side ratios, radicals, and step-by-step solutions.

Start Practicing Review the Special Right Triangles Guide

Try each problem on your own first. Then click Show solution to check the triangle ratio, setup, exact value, and reasoning.

Keep These Ratios Nearby

Special Right Triangle Essentials

45°-45°-90°

\[ x,\quad x,\quad x\sqrt{2} \]

The two legs are equal. The hypotenuse is \(x\sqrt{2}\).

30°-60°-90°

\[ x,\quad x\sqrt{3},\quad 2x \]

Short leg, long leg, hypotenuse.

Opposite 30°

The shortest side is always opposite the \(30^\circ\) angle.

Opposite 90°

The hypotenuse is always opposite the right angle.

Download Practice PDF Open Special Right Triangles Reference

Level 1

Recognize the Ratios

Use the basic 45°-45°-90° and 30°-60°-90° side relationships to find missing sides.

Problem 1

45°-45°-90°

Find the missing side lengths.

8 ? ? 45° 45°

Step-by-Step Solution

A 45°-45°-90° triangle has the side ratio

\[ x:x:x\sqrt{2}. \]

Since one leg is \(8\), the other leg is also \(8\).

\[ x=8 \]

Multiply the leg by \(\sqrt{2}\) to find the hypotenuse.

\[ 8\sqrt{2} \]
Answer: \[ \boxed{ 8 \text{ and } 8\sqrt{2} } \]

Problem 2

30°-60°-90°

Find the missing side lengths.

5 ? ? 60° 30°

Step-by-Step Solution

The side opposite the \(30^\circ\) angle is the short leg, so

\[ x=5. \]

Use the 30°-60°-90° ratio

\[ x:x\sqrt{3}:2x. \]

The long leg is

\[ 5\sqrt{3}. \]

The hypotenuse is

\[ 2(5)=10. \]
Answer: \[ \boxed{ 5\sqrt{3} \text{ and } 10 } \]

Problem 3

45°-45°-90°

The hypotenuse is \(12\sqrt{2}\). Find the length of each leg.

\[ h=12\sqrt{2} \]

Step-by-Step Solution

In a 45°-45°-90° triangle, the hypotenuse is

\[ x\sqrt{2}. \]

Set this equal to the given hypotenuse.

\[ x\sqrt{2} = 12\sqrt{2} \]

Therefore,

\[ x=12. \]

Both legs of a 45°-45°-90° triangle are equal.

Answer: \[ \boxed{ 12 \text{ and } 12 } \]

Level 2

Find Missing Sides

Work backward from the hypotenuse or long leg and simplify exact radical values.

Problem 4

30°-60°-90°

The hypotenuse is \(18\). Find the lengths of both legs.

\[ h=18 \]

Step-by-Step Solution

In a 30°-60°-90° triangle, the hypotenuse is

\[ 2x. \]

Set \(2x\) equal to the given hypotenuse.

\[ 2x=18 \]

Divide by \(2\).

\[ x=9 \]

So the short leg is \(9\). Multiply by \(\sqrt{3}\) to find the long leg.

\[ 9\sqrt{3} \]
Answer: \[ \boxed{ 9 \text{ and } 9\sqrt{3} } \]

Problem 5

30°-60°-90°

The long leg is \(7\sqrt{3}\). Find the short leg and hypotenuse.

\[ x\sqrt{3}=7\sqrt{3} \]

Step-by-Step Solution

The long leg in a 30°-60°-90° triangle is

\[ x\sqrt{3}. \]

Compare the given side to the ratio.

\[ x\sqrt{3} = 7\sqrt{3} \]

Therefore,

\[ x=7. \]

The short leg is \(7\), and the hypotenuse is twice the short leg.

\[ 2x=2(7)=14 \]
Answer: \[ \boxed{ 7 \text{ and } 14 } \]

Problem 6

45°-45°-90°

The hypotenuse is \(10\). Find the length of each leg in simplified radical form.

\[ x\sqrt{2}=10 \]

Step-by-Step Solution

In a 45°-45°-90° triangle, the hypotenuse is

\[ x\sqrt{2}. \]

Set the expression equal to \(10\).

\[ x\sqrt{2}=10 \]

Divide by \(\sqrt{2}\).

\[ x = \frac{10}{\sqrt{2}} \]

Rationalize the denominator.

\[ \begin{aligned} x &= \frac{10}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} \\[6pt] &= \frac{10\sqrt{2}}{2} \\[6pt] &= 5\sqrt{2} \end{aligned} \]

Both legs are equal.

Answer: \[ \boxed{ 5\sqrt{2} \text{ and } 5\sqrt{2} } \]

Level 3

Apply Special Right Triangles

Recognize special right triangles inside larger geometric figures and use the correct ratio to solve.

Problem 7

Square Diagonal

A square has side length \(9\). Find the length of its diagonal.

9 9 ?

Step-by-Step Solution

The diagonal of a square divides it into two congruent 45°-45°-90° triangles.

Each leg has length \(9\), so use the ratio

\[ x:x:x\sqrt{2}. \]

Substitute \(x=9\).

\[ d = 9\sqrt{2} \]
Answer: \[ \boxed{9\sqrt{2}} \]

Problem 8

Equilateral Triangle

An equilateral triangle has side length \(12\). Find its height.

12 12 ? 6 6

Step-by-Step Solution

Drawing the altitude splits the equilateral triangle into two 30°-60°-90° triangles.

The altitude also bisects the base, so the short leg is

\[ \frac{12}{2}=6. \]

In a 30°-60°-90° triangle, the long leg is

\[ x\sqrt{3}. \]

Substitute \(x=6\).

\[ h=6\sqrt{3} \]
Answer: \[ \boxed{6\sqrt{3}} \]

Problem 9

Mixed Recognition

A right triangle has a \(30^\circ\) angle and a hypotenuse of \(20\). Find both legs.

20 60° 30° ? ?

Step-by-Step Solution

Because the triangle has angles \(30^\circ\), \(60^\circ\), and \(90^\circ\), use the ratio

\[ x:x\sqrt{3}:2x. \]

The hypotenuse corresponds to \(2x\).

\[ 2x=20 \]

Solve for \(x\).

\[ x=10. \]

Therefore, the short leg is \(10\), and the long leg is

\[ 10\sqrt{3}. \]
Answer: \[ \boxed{ 10 \text{ and } 10\sqrt{3} } \]

Before You Check the Reference

Use the Same Process Every Time

  1. Identify whether the triangle is 45°-45°-90° or 30°-60°-90°.
  2. Label the sides using the correct special-triangle ratio.
  3. Match the given side to the correct part of the ratio.
  4. Solve for \(x\).
  5. Use \(x\) to find any remaining missing sides.

Continue Learning

Related Special Right Triangle Resources

Review the lesson, keep the printable reference nearby, or continue building the geometry and trigonometry skills connected to special right triangles.

Special Right Triangles Guide

Review the 45°-45°-90° and 30°-60°-90° ratios with diagrams, examples, and step-by-step explanations.

Special Right Triangles Reference

Keep the two special triangle ratios and side relationships nearby while practicing.

Unit Circle Guide

See how special right triangles create the exact coordinates and trigonometric values on the unit circle.

Unit Circle Practice

Practice exact sine, cosine, and tangent values using the special triangle relationships.

Radians and Degrees

Review common angle measures and degree-radian conversions used throughout trigonometry.

Trigonometry Resources

Browse more trigonometry lessons, practice pages, worksheets, and interactive learning tools.

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