Sectors and Circles Problems

Problems on sectors and circles with detailed solutions.

Problem 1

• The formula for the arc length is given by
  arc length AB = r * angle AOB
  
• arc length AB is given and is equal to 2r, hence
  2r = r * angle AOB
  
• Solve for angle AOB to obtain
  angle AOB = 2 radians
  
• The area of the sector is given by
  Area = (1/2) * angle AOB * r ² = (1/2) * 2 * r ² = r ²

Problem 2

• The formula for the area of a sector is
  Area = (1 / 2) * angle of sector * r ²
  
• The area and the angle are given. The angle is given in degrees and has to be converted into radians.
  20 = (1 / 2) (40 Pi / 180) * r ²
  
• Solve for radius r to obtain
  r = square root [ 2 * 20 * 180 / (40 Pi) ]
  = square root [ 180 / Pi ]
  
• The arc length is given by
  arc length of sector = angle of sector * r
  = [ 40 * Pi / 180 ] * square root [ 180 / Pi ]
  = [ Pi / 4.5 ] * square root [ 180 / Pi ] = 5.3 cm (rounded to 1 decimal place)

Problem 3

• The perimeter of ABCD is given by
  perimeter = arc AB + 6 + arc DC + 6 = 22
  
• If we let x be the size of the sector central angle, arc AB and arc DC are given by (formula for arc length of sector)
  arc AB = 2 * x and arc DC = 8 * x
  
• Substitute in the above perimeter formula
  2x + 6 + 8x + 6 = 22
  
• Solve for x
  x = 1 radian
  
• The area of figure ABCD is found by subtracting the area of the small sector from the area of the larger sector
  area of ABCD = (1 / 2) 8 ² * 1 - (1 / 2) 2 ² * 1
  = 30 cm ²

More References and Links to Geometry Tutorials and Problems