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

Q: What's the difference between degrees and radians?

Degrees split a full circle into 360 equal parts. Radians measure angles using arc length: one radian is the angle that cuts off an arc equal in length to the radius. 2π radians make a full circle, and 180^° = π radians. Radians are the natural unit for calculus and most advanced math; degrees are easier for everyday geometry.

Q: How do I convert degrees to radians?

Multiply by π/180 . Examples: 90^° × π/180 = π/2 rad. 45^° × π/180 = π/4 rad. 270^° × π/180 = 3π/2 rad. The shortcut: divide by 180 and put π on top.

Q: How do I convert radians to degrees?

Multiply by 180/π . Examples: π/3 × 180/π = 60^° ; the π cancels. 5π/6 × 180/π = 150^° . When the radian measure has no π (just a real number like 2 radians), you'll get a decimal answer: 2 × 180/π ≈ 114.59^° . The shortcut: when π appears in the radian measure, expect a clean answer; when it doesn't, expect a decimal.

Q: What's a coterminal angle?

Two angles are coterminal if they share the same initial side (positive x -axis) and same terminal side - they end up pointing the same way after rotation. To find coterminal angles, add or subtract 360^° (or 2π radians). So 60^° , 420^° , and -300^° are all coterminal.

Q: What's a reference angle?

The reference angle is the acute angle between the terminal side of an angle and the x -axis. It's always positive and between 0^° and 90^° (or 0 and π/2 ). Reference angles let you find sine, cosine, and tangent of any angle by relating it to a familiar acute angle.

Q: How do I find the reference angle?

It depends on the quadrant of the terminal side. Quadrant I: reference = angle. Quadrant II: reference = 180^° - angle. Quadrant III: reference = angle -180^° . Quadrant IV: reference = 360^° - angle. Example: for 210^° (Quadrant III), reference is 210^° - 180^° = 30^° .

Q: What are common angle measures I should memorize?

In degrees: 0, 30, 45, 60, 90, 180, 270, 360. In radians: 0, π/6 , π/4 , π/3 , π/2 , π , 3π/2 , 2π . Memorize the corresponding sine and cosine values too - they're the backbone of trig.

Q: What's the arc length formula?

Arc length s = rθ , where θ is in radians and r is the radius. Example: an arc subtending π/3 radians on a circle of radius 6 has length s = 6 × π/3 = 2π . This formula only works with radians - that's a big reason radians are useful.

Q: What's the area of a circular sector?

Area A = (1)/(2)r^2θ , again with θ in radians. Example: sector of radius 4 with central angle π/2 has area (1)/(2)(16)(π/2) = 4π . In degrees, the formula becomes A = (θ)/(360)π r^2 - longer to write, easier to forget.

Q: Where do angle conversions show up on tests?

SAT (occasionally), ACT (occasionally), CLEP Precalculus, AP Pre-Calculus, AP Calculus, and every trig class. Free-response questions on radians-to-degrees conversion, coterminal angles, reference angles, arc length, and sector area are routine. Memorize the conversion factors so you don't waste time.

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 \(=120×\frac{π}{180}=\frac{120π}{180}=\frac{2π}{3}\)

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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 \(=150×\frac{π}{180}=\frac{150π}{180}=\frac{5π}{6}\)

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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

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

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Frequently Asked Questions

What’s the difference between degrees and radians?

Degrees split a full circle into 360 equal parts. Radians measure angles using arc length: one radian is the angle that cuts off an arc equal in length to the radius. \(2\pi\) radians make a full circle, and \(180^\circ = \pi\) radians. Radians are the natural unit for calculus and most advanced math; degrees are easier for everyday geometry.

How do I convert degrees to radians?

Multiply by \(\pi/180\). Examples: \(90^\circ \times \pi/180 = \pi/2\) rad. \(45^\circ \times \pi/180 = \pi/4\) rad. \(270^\circ \times \pi/180 = 3\pi/2\) rad. The shortcut: divide by 180 and put \(\pi\) on top.

How do I convert radians to degrees?

Multiply by \(180/\pi\). Examples: \(\pi/3 \times 180/\pi = 60^\circ\); the \(\pi\) cancels. \(5\pi/6 \times 180/\pi = 150^\circ\). When the radian measure has no \(\pi\) (just a real number like 2 radians), you’ll get a decimal answer: \(2 \times 180/\pi \approx 114.59^\circ\). The shortcut: when \(\pi\) appears in the radian measure, expect a clean answer; when it doesn’t, expect a decimal.

What’s a coterminal angle?

Two angles are coterminal if they share the same initial side (positive \(x\)-axis) and same terminal side – they end up pointing the same way after rotation. To find coterminal angles, add or subtract \(360^\circ\) (or \(2\pi\) radians). So \(60^\circ\), \(420^\circ\), and \(-300^\circ\) are all coterminal.

What’s a reference angle?

The reference angle is the acute angle between the terminal side of an angle and the \(x\)-axis. It’s always positive and between \(0^\circ\) and \(90^\circ\) (or \(0\) and \(\pi/2\)). Reference angles let you find sine, cosine, and tangent of any angle by relating it to a familiar acute angle.

How do I find the reference angle?

It depends on the quadrant of the terminal side. Quadrant I: reference = angle. Quadrant II: reference = \(180^\circ – \)angle. Quadrant III: reference = angle\(-180^\circ\). Quadrant IV: reference = \(360^\circ – \)angle. Example: for \(210^\circ\) (Quadrant III), reference is \(210^\circ – 180^\circ = 30^\circ\).

What are common angle measures I should memorize?

In degrees: 0, 30, 45, 60, 90, 180, 270, 360. In radians: 0, \(\pi/6\), \(\pi/4\), \(\pi/3\), \(\pi/2\), \(\pi\), \(3\pi/2\), \(2\pi\). Memorize the corresponding sine and cosine values too – they’re the backbone of trig.

What’s the arc length formula?

Arc length \(s = r\theta\), where \(\theta\) is in radians and \(r\) is the radius. Example: an arc subtending \(\pi/3\) radians on a circle of radius 6 has length \(s = 6 \times \pi/3 = 2\pi\). This formula only works with radians – that’s a big reason radians are useful.

What’s the area of a circular sector?

Area \(A = \frac{1}{2}r^2\theta\), again with \(\theta\) in radians. Example: sector of radius 4 with central angle \(\pi/2\) has area \(\frac{1}{2}(16)(\pi/2) = 4\pi\). In degrees, the formula becomes \(A = \frac{\theta}{360}\pi r^2\) – longer to write, easier to forget.

Where do angle conversions show up on tests?

SAT (occasionally), ACT (occasionally), CLEP Precalculus, AP Pre-Calculus, AP Calculus, and every trig class. Free-response questions on radians-to-degrees conversion, coterminal angles, reference angles, arc length, and sector area are routine. Memorize the conversion factors so you don’t waste time.

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

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

What to notice first

Common student mistake

Key formulas and cues

A reliable path

Worked examples

Right-triangle sine

Unit-circle cosine

Related Topics

Step by step guide to solve angles and angle measure problems

Angles and Angle Measure – Example 1:

Angles and Angle Measure – Example 2:

Angles and Angle Measure – Example 3:

Angles and Angle Measure – Example 4:

Exercises for Solving Angles and Angle Measure

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

ِِِِDownload Angles and Angle Measure Worksheet

Recommended EffortlessMath Books

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

Frequently Asked Questions

What’s the difference between degrees and radians?

How do I convert degrees to radians?

How do I convert radians to degrees?

What’s a coterminal angle?

What’s a reference angle?

How do I find the reference angle?

What are common angle measures I should memorize?

What’s the arc length formula?

What’s the area of a circular sector?

Where do angle conversions show up on tests?

Related EffortlessMath Lessons

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Related to This Article

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