ASV - Arc Length and Sector Area Lesson

Arc Length and Sector Area

It is likely that you remember finding the circumference and area of a circle in previous math courses. Now, we are going to talk about finding arc length, which is a portion of a circle's circumference, and sector area, which is a portion of a circle's area. Download the Arc Length and Sector Area Investigation Handout by clicking here, and be prepared with the following materials: a round cookie (or any other circular object that you can cut in half), string or ribbon, a ruler, a protractor, paper, pencil, and a calculator. Links to an external site. Once you have these materials, please watch the videos below and follow along with a fun activity!

Now that you have learned all about arc length and sector area, let's try some practice problems!

There's one more thing we need to discuss when we talk about arc length and sector area. You are likely very familiar with measuring angles in degrees. However, there is another unit of angle measure that exists. What is a radian? Let's find out ...

For this activity, you need: a compass, 3 sheets of paper, & string

• With this activity in mind, describe what a radian is. (Recall that the measure of arc AB is equal to the measure of <AOB.) 
  • Hint: A radian is an angle measure with an arc length equal to one radius length.
• Circumference can be represented in terms of radians. Take your string (whose measure is one radian) and see how many pieces it would take to go around the circle. In terms of radians, what is the circumference of the circle? 
  • Solution: 6.28, or about 6 radians
• Remember the formula for the circumference of a circle is $\Pi d\:or\:2\Pi r$. Compare this to your answer for the previous question. 
  • Solution: $2\Pi$ is about 6.28

Now that you know what a radian is...

• Approximately how many radians are in a half of a circle? 
  • Solution: 3.14, or about 3, there are $\Pi$ radians in half of a circle
• Approximately, how many radians are in a quarter of a circle?
  • Solution: 1.57, or about 1.5. To be exact there are $\frac{\Pi}{2}$ radians in a quarter or a circle
• How many radians are in 30°, 60°, 90°? 
  • Solution: $\frac{\Pi}{6}$,$\frac{\Pi}{3}$,$\frac{\Pi}{2}$
• How could you find the number of radians in any angle?
  • Solution: Divide the angle measure (in degrees) by 360 degrees and multiple by $2\Pi$. In a mathematical formula, this looks like $\frac{\theta}{360}\cdot2\Pi=\frac{2\Pi\theta}{360}=\frac{\Pi\theta}{180}$. We can multiply the angle to measure (in degrees) by $\frac{\Pi}{180}$.

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