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Radians and Degrees Trigonometry—Grade 10

Subject: Math

Grade: 10

Lesson Objective: CCSS.MATH.CONTENT.HSG.C.B.5 – Using similarity, derive that an arc length of an angle is proportional to the radius, and define a radian as a constant proportionality.




Ask the following questions (as applicable based on answers) and allow students to answer them.

  • How do we measure angles?
  • What is the unit of measurement for angles?
  • What are acute, right, and obtuse angles?
  • How many degrees does a whole circle have?
  • What is the radius of a circle?
  • On a circle, what is an arc?

Ensure your students understand the basic concepts of angles, the radius, and the arc of a circle.



Say: There is another unit besides degrees we can measure angles with. This unit is called a radian. If the radius of a circle is laid along the edge of the circle to create an arc, and we draw lines from the ends of the arc to the center of the circle, the formed angle is one radian. 

Draw a circle, and then highlight an edge of the circle that is approximately one radius long. Draw in lines to create an angle to see the description that you just gave.

Ask: How many radians can fit in a whole circle? How many radians can fit in a half-circle? 

Use worksheet 1 and let the students work to figure out that there are approximately six radians in a circle and three radians in a half-circle.

Say: Half a circle is a little more than three radians. It is pi radians, or approximately 3.1415 radians. One-quarter of a circle is ½ pi radians. (Or pi/2 radians.)

Ask: How many radians is a whole circle? 

Make sure the students know that a whole circle is 2 pi radians. Ask if the students have any questions at this point and address any questions from material already covered.

Say: We don’t use the degree symbol when we measure in radians. We can write radians as ‘rad’ or ‘r’, or we can leave it off and just say the measurement – such as 2 pi. To convert between radians and degrees, we need to remember that ‘pi radians = 180 degrees. 

Write this relationship for the students to see.

Say: To convert from degrees to radians, we multiply the number of degrees by pi/180. 

(Write down radians = degrees x (pi/180)).

Say: To convert from radians to degrees we multiply the radians by 180 over pi. 

Write down degrees = radians x (180/pi)).

Ask if there are any questions about what you wrote down.

Say: Now, we are going to do a worksheet to review and help us practice what we learned. 

Have students work on Worksheet 2.

Feedback: Ask if there are any questions about problems on the worksheet. Ask for volunteers to show what they did to find answers for the worksheet.


Written by Lina Filcons

Education World Contributor

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