Areas of Circles

1
Areas of Circles

• The area formulas discovered so far in this
  chapter are for various types of polygons, all of
  which have straight sides
• Is there any way to apply these formulas to
  circles?
• What happens to a regular polygon if the apothem
  remains constant and the number of sides
  increases?
• What circle measurement corresponds to the
  apothem?
• What circle measurement corresponds to the
  perimeter?
• Use the circle measurements and the formula for
  the area of a regular polygon, A ½Pa, to
  complete the conjecture
• C-81 Circle Area Conjecture
• The area of a circle is given by the formula A
  pr2, where A is the area and r is the radius
  of the circle.

2
Areas of Circles

• Another way to derive the formula for the area of
  a circle
• Use a compass to construct a circle
• Fold the circle in half repeatedly, creasing each
  fold tightly
• Unfold the paper, and cut out each wedge
  carefully with scissors
• Arrange the wedges as shown, alternately pointing
  up and down
• What shape do the wedges appear to form?
• As more wedges are cut out and rearranged, the
  shape gets closer and closer to a parallelogram
• What is the height of the parallelogram formed?
• What is the length of the base of the
  parallelogram?
• Substitute these measurements into the formula
  for the area of a parallelogram to get the area
  of the original circle
• Once again, you get the formula A pr2