Sectors of Circles

K = area of the sector
r = radius
s = the arc length
Θ = the central angle

for

K = 1/2(r^2)Θ
K = 1/2(r)(s)
s = rΘ

Ex. 1
A sector of a circle has arc length 11 cm and central angle 2.2 radians. Find its radius and area.

• s = 11
  Θ = 2.2

• You would use the formula s = rΘ.

• Plugging into this formula, r = 11/2.2 = 5.

• To find the area, you could use either K = 1/2(r^2)Θ or K = 1/2(r)(s).
  K = 1/2(5^2)2.2 = 27.5
  or
  K = 1/2(5)(11) = 27.5

• Your answers would be r = 5 and K = 27.5

  
  

If the problem asks for apparent size, you should automatically assume to use the formula s = rΘ, where r = distance between two objects, Θ = apparent size, and s = diameter of an object.

Ex. 2
The diameter of the moon is about 3500 km. Its apparent size is about 0.0087 radians. About how far is it from Earth?

• Identify what's given.
  s = 3500
  Θ = .0087
  r = ??
  s = rΘ

• Plug into the formula.
  3500 = r( .0087)
  r = 402298.8506

• It is about 402298.8506 km from the Earth.