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5.4 Area and Arc Length of a Sector
Cool Down: 5 minutes
Student Task Statement
Here is a sector of a circle that has a radius of 10 cm and a central angle radians. 1. Calculate the area of the sector.
2. Calculate the length of the arc enclosed by the angle.
Student Response
1. 35 square centimeters. 2. 7 centimeters.
Student Lesson Summary
When thinking about the area of a sector or the length of an arc, it is often helpful to rst think about the area or circumference of the full circle. We can then multiply the area or circumference by what fraction of the circle we have in our sector or arc. For example, suppose we have a sector with radius 5 units and central angle . Since the radius is 5 we
know the full circle would have area and it would have circumference . Since our central angle is radians, our sector must be of the entire circle because the entire circle
measures radians. The area of our sector must therefore be square units and our arc length must be units.
Determining what fraction of the circle we had was easier in this example because the radian measure was written as a fraction with numerator . It is often helpful to rewrite a radian measure as a fraction with numerator so the denominator will tell us how many copies of the sector we need to make the full circle. For example, a radian measure of could be
written instead as which lets us see more easily that we must have of a full circle.
Unit 7
Lesson 5: Angles, Sectors, and Arcs
53