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

Related Topics

• How to Evaluate Trigonometric Function
• How to Find Missing Sides and Angles of a Right Triangle
• How to Solve Coterminal Angles and Reference Angles
• How to Solve Trig Ratios of General Angles

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.

Related EffortlessMath Lessons

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

• The unit circle
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