Unit 7, Lesson 13
Using Radians
Geometry
13.1
Warm-up
A circle with radius 24 inches has a sector with central angle radians.
- What fraction of the whole circle is represented by this sector?
- Find the area of the sector.
Sign in to view assessments and invite other educators
Sign in using your existing Kendall Hunt account. If you don’t have one, create an educator account.
Lin and Elena are trying to find the area of the shaded sector in the image.
Lin says, “We’ve found sector areas when the central angles are given in degrees, but here the central angle is in radians. Should I start by finding the area of the full circle?”
Elena says, “I saw someone using the formula , where is the measure of the angle in radians, and is the radius. But I don’t know where that came from.”
A city is designing a new park. One feature of the park is a garden designed for walking meditation. The space they have chosen is 14 yards long on one edge. There will be alternating circular arcs of plants and paved paths.
Image of the city plan for a new park shaped like circular arcs. From point T, two straight lines extend in two directions and opening to the right. Horizontal line T D has points A, B, C and D. The length of segment T A is 2 yards, A B is 4 yards, B C is 5 yards and C D is 3 yards. At points A. B, C and D, arcs extend up to the left to the other straight line. The arc length at A is 3 yards, at B is 9 yards, at C is 16 point 5 yards and the arc length at D is 21 yards.
Suppose we want to find the area of a sector of a circle whose central angle is radians and whose radius is units. First we find the area of the whole circle, square units. Then we find the fraction of the circle represented by the sector. The complete circle measures radians. So the fraction is . Multiply the fraction by the area to get . This can be rewritten as .
For example, let the radius be 25 units and the radian measure of the sector’s central angle be radians. Substitute these values into the formula we just created to get . This can be rewritten as , which is about 589 square units. Using the formula was a bit quicker than going through the process of finding the specific fraction of the circle represented by this sector and multiplying by the circle’s area.
Additionally, we can write a formula for arc length based on the definition of radian measure: , where is the central angle measure in radians, is the arc length, and is the radius. Rewrite the definition to solve for arc length. We get .