How to Solve Angles and Angle Measure? (+FREE Worksheet!)

Picture two people describing the same turn. One says ninety degrees, the other says a quarter of pi. They are both right, just using different units. Angles can be measured in degrees (a full circle is 360) or in radians (a full circle is 2 pi), and the bridge between them is 180 degrees equals pi radians. Coterminal angles are ones that land in the same spot after full spins. Get comfortable converting back and forth, and your trig formulas stop feeling like two unrelated languages.

Key takeaways:

A full circle is \(360^\circ\) or \(2\pi\) radians.
Convert degrees to radians: multiply by \(\pi/180\). Convert radians to degrees: multiply by \(180/\pi\).
Positive angles rotate counterclockwise; negative angles rotate clockwise.
Coterminal angles share the same terminal side - they differ by integer multiples of \(360^\circ\) (or \(2\pi\)).
Reference angle: the acute angle between the terminal side and the \(x\)-axis (always positive, always \(<90^\circ\)).

Learn how to convert degrees to radians or radians to degrees by following a step-by-step guide.

How to Solve Angles and Angle Measure? (+FREE Worksheet!)

Tutor-style math help

Solve Angles and Angle Measure: what to notice and how to work it

Trigonometry skill

Trigonometry connects an angle to a triangle ratio, a unit-circle coordinate, or a repeating graph. Choosing the right picture makes the problem much easier.

What to notice first

Decide whether the problem is triangle-based, circle-based, or graph-based. Then use the matching definition.

Common student mistake

Do not mix degrees and radians. The angle unit must match the formula, graph scale, or calculator setting.

Key formulas and cues

\(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\)

\(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\)

\(\tan\theta=\frac{\sin\theta}{\cos\theta}\)

\(\sin^2\theta+\cos^2\theta=1\)

A reliable path

Choose the modelUse a right triangle, the unit circle, or a transformed graph.
Track unitsConvert degrees and radians when needed.
Use identitiesReplace complicated trig expressions with equivalent simpler ones.

Worked examples

Right-triangle sine

Example: opposite = 5, hypotenuse = 13

Sine is opposite over hypotenuse.
Substitute 5 and 13.
Leave the ratio simplified.

Answer: \(\sin\theta=\frac5{13}\)

Unit-circle cosine

Example: \(\cos(0)\)

At angle 0, the point is (1, 0).
Cosine is the x-coordinate.
Read the x-value.

Answer: \(1\)

Open pop-up practice

Try one before moving on

Try: In a right triangle, tangent equals which ratio?

Answer: Opposite over adjacent.

Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.

Step by step guide to solve angles and angle measure problems

To convert degrees to radians, use this formula: \(\color{blue}{Radians = Degrees \ × \frac{π}{180}}\)
To convert radians to degrees, use this formula: \(\color{blue}{Degrees =Radians ×\frac{180}{π}}\)

Angles and Angle Measure – Example 1:

Convert \(120\) degrees to radians.

Solution:

Use this formula: \(Radians\) \(=\) \(Degrees\) \(×\frac{π}{180}\)
Radians \(=120×\frac{π}{180}=\frac{120π}{180}=\frac{2π}{3}\)

The Absolute Best Books to Ace Pre-Algebra to Algebra II

Angles and Angle Measure – Example 2:

Convert \(\frac{\pi}{3}\) to degrees.

Solution:

Use this formula: \(Degrees\) \(=\) \(Radians\) \(×\frac{180}{π}\)
Radians \(=\frac{π}{3}×\frac{180}{π}=\frac{180π}{3π}=60\)

Angles and Angle Measure – Example 3:

Convert \(150\) degrees to radians.

Solution:

Use this formula: \(Radians\) \(=\) \(Degrees\) \(×\frac{π}{180}\)
Radians \(=150×\frac{π}{180}=\frac{150π}{180}=\frac{5π}{6}\)

The Best Book to Help You Ace Pre-Algebra

Angles and Angle Measure – Example 4:

Convert \(\frac{2π}{3}\) to degrees.

Solution:

Use this formula: \(Degrees\) \(=\) \(Radians\) \(×\frac{ 180}{ π }\)
Radians \(=\frac{2π}{3}×\frac{180}{π}=\frac{360π}{3π}=120\)

Exercises for Solving Angles and Angle Measure

Convert each degree measure into radians and convert each radian measure into degrees.

\(\color{blue}{-150^\circ}\)
\(\color{blue}{420^\circ}\)
\(\color{blue}{300^\circ}\)
\(\color{blue}{\frac{5π}{9}=}\)
\(\color{blue}{-\frac{π}{3}=}\)
\(\color{blue}{\frac{13π}{6}=}\)

ِِِِDownload Angles and Angle Measure Worksheet

\(\color{blue}{-\frac{5π}{6}}\)
\(\color{blue}{\frac{7π}{3}}\)
\(\color{blue}{\frac{5π}{3}}\)
\(\color{blue}{100^\circ}\)
\(\color{blue}{-60^\circ}\)
\(\color{blue}{390^\circ}\)

The Greatest Books for Students to Ace the Algebra

Recommended EffortlessMath Books

For a workbook that covers every right-triangle and unit-circle topic with worked examples, the Trigonometry for Beginners (resource currently unavailable) builds the whole subject up from scratch. If you’re combining trig with function work for pre-calc, the Pre-Calculus for Beginners ties the ideas into a full pre-calc course.

Trigonometry for Beginners The Ultimate Step by Step Guide to Acing Trigonometry

Frequently Asked Questions

What are angle measures?

An angle measure tells how far one ray turns from another around their shared vertex. It is usually given in degrees, where a full turn is 360°, or in radians, where a full turn is 2π. A right angle measures 90° or π/2 radians, and a straight angle measures 180° or π radians. Counterclockwise turns are positive and clockwise turns are negative.

Can angles be measured in radians?

Yes. Radians measure an angle by arc length: one radian cuts off an arc equal to the circle’s radius, and 180° = π radians. To convert degrees to radians, multiply by π/180, so 120° × π/180 = 2π/3. To convert back, multiply by 180/π, so 5π/4 × 180/π = 225°. Radians are standard in trigonometry and calculus.

Can angles be measured in minutes?

Yes. A degree can be split into 60 minutes (′), and each minute into 60 seconds (″), a system used in navigation, astronomy and surveying. For example, 30°15′ means 30 and 15/60 degrees, or 30.25°. To convert a decimal like 47.6° into minutes, multiply the decimal part by 60: 0.6 × 60 = 36, so 47.6° = 47°36′.

Can angle measures be congruent?

Angles are congruent when they have the same measure, so we say the angles are congruent and their measures are equal. For example, two 65° angles are congruent. Coterminal angles are a related idea: 40° and 400° end at the same position because they differ by 360°, but their measures are different, so they are not congruent.

Is measuring angles geometry?

Yes. Measuring angles is a core geometry skill, used to classify angles as acute (less than 90°), right (90°), obtuse (between 90° and 180°) or straight (180°) and to work with triangles and polygons. It carries into trigonometry, where angles move onto the unit circle and are often measured in radians, such as π/6 for 30°.

Related EffortlessMath Lessons

If a topic on this page feels rusty, these short lessons go deeper:

Effortless Math Team

1 week ago

Effortless Math Team

Related to This Article

Angles and Angle Measure How to Solve Angles and Angle Measure

What people say about "Angles and Angle Measure"?

No one replied yet.

How to Solve Angles and Angle Measure? (+FREE Worksheet!)