Quadrilateral Circle Problems

Question

1. A sector of a circle is radius 7 cm has an area of 7π / 2 cm^2. Find the central angle of the sector.
2. Consider a circle of area 81 π cm^2.

a.) Find the radius of the circle

b.) Consider a central angle of θ = π / 3, find the arc length subtends by θ

Answer 1

1 a circle's sector has radius 7 cm

and the sector's area = 7pi/2 cm^2

the ratio of the central angle to 2pi = the ratio of the sector's area to the circle's area

theta = central angle

theta/2pi = sector area/circle area

theta/2pi = (7pi/2)/pi(7)^2

=(7pi/2)/(49pi)

=1/14

theta = 2pi/14

theta = pi/7 = the central angle

pi/7 radians = (180/pi)(pi/7)

= 180/7 = about 25.7 degrees

another way to look at this problem is find the area of the full circle, pi(r^2) = 49pi

divide the area of the sector by the circle's full area to get (7pi/2)/49pi = 1/14

1/14 of 2pi = 2pi/14 = pi/7 = theta= the central angle of the sector

2 a A cirlce with area= 81pi cm^2 has radius = 9 cm

The area = pi(r)^2 where r = the radius r^2 = 81, r = sqr81 = 9

2 b A central angle of pi/3 has arc length = radius times the central angle when measured in radians

= radius times radians

= 9(pi/3)

= 3pi cm = arc length