Lesson 12

Radian Sense

  • Let’s get a sense for the sizes of angles measured in radians.

12.1: Which One Doesn’t Belong: Angle Measures

Which one doesn’t belong?


Circle with a shaded sector. 1 side of the sector has length 1 centimeter. The arc of the sector is pi over 2 centimeters.


Circle with a shaded sector. The sector’s central angle measures 3 pi divided by 2 radians.


Circle with shaded central sector of 51 point 5 degrees.


Circle. Shaded sector labeled 1 rad.


12.2: Degrees Versus Radians

This double number line shows degree measurements on one line and radians on another.

Double number line. Top line is degrees, from 0 to 360 by 90’s. Bottom line is radians, with 5 unlabeled tick marks.
  1. Fill in the radian measures on the bottom line for 0\(^\circ\), 90\(^\circ\), 180\(^\circ\), 270\(^\circ\), and 360\(^\circ\).
  2. Express each radian measurement in degrees.
    1. \(\frac{\pi}{3}\) radians
    2. \(\frac{5\pi}{4}\) radians
  3. Express each degree measurement in radians.
    1. 30\(^\circ\)
    2. 120\(^\circ\)

Your boat is heading due south when you hear that you must head north-east to return home. You turn the boat, traveling in a counterclockwise circular path until you’re facing north-east. 

  1. Sketch the path of the boat.
  2. If the circle you traced had a radius of 40 feet, what distance did you travel?

12.3: Pie Coloring Contest

Your teacher will give you a set of cards with angle measures on them. Place the cards upside down in a pile. Choose 1 student to go first. This student should draw a card, then on either circle shade a sector of the circle whose central angle is the measure on the card that was drawn.

Take turns repeating these steps. If you are shading in a circle that already has a shaded sector, choose a spot next to the already-shaded sectors—don’t leave any gaps. You might have to draw additional lines to break the sectors into smaller pieces.

Continue until an angle is drawn that won’t fit in any of the sectors that are still blank.

Circle with 8 equal sectors. There is a central point and 4 lines foming 8 equal sectors.
Circle with 12 equal sections.

When you’re finished, answer these questions about each circle:

  1. What is the central angle measure for the remaining unshaded sector?
  2. What is the central angle measure for the block of shaded sectors?


We can divide circles into congruent sectors to get a sense for the size of an angle measured in radians.

Suppose we want to draw an angle that measures \(\frac{2\pi}{3}\) radians. We know that \(\pi\) radians is equivalent to 180 degrees. If we divide a sector with a central angle of \(\pi\) radians into thirds, we can shade in 2 of them to create an angle measuring \(\frac{2\pi}{3}\) radians.

Half a circle divided into 3 equal parts. 2 of the 3 parts are shaded.

Another way to understand the size of an angle measured in radians is to create a double number line with degrees on one line and radians on the other. On the double number line in the image, the degree measurements are aligned with their equivalent radian measures. For example, \(\pi\) radians is equivalent to 180\(^\circ\).

2 number lines. First labeled degrees. Starts at 0, halfway is labeled 180, ends at 360. Second starts at 0, halfway is labeled pi, ends at 2 pi.

Suppose we need to know the size of an angle that measures \(\frac{3\pi}{4}\) radians. The left half of the double number line represents \(\pi\) radians. Divide the left half of the top and bottom number lines into fourths, then count out 3 of them on the radians line to land on \(\frac{3\pi}{4}\). On the top line, each interval we drew represents 45 degrees because \(180\div 4 = 45\). If we count 3 of those intervals, we find that \(\frac{3\pi}{4}\) radians is equivalent to 135 degrees because \(45\boldcdot 3=135\).

Glossary Entries

  • radian

    The radian measure of an angle whose vertex is at the center of a circle is the ratio between the length of the arc defined by the angle and the radius of the circle.

  • sector

    The region inside a circle lying between two radii of the circle.