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D. The Circumference/ Diameter Ratio. A circle is the set of all points on a plane equidistant from a single point The reference point is called the center of the circle A segment drawn from the center to any point on the circle is a radius

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the circumference diameter ratio

D

The Circumference/Diameter Ratio
  • A circle is the set of all points on a plane equidistant from a single point
    • The reference point is called the center of the circle
    • A segment drawn from the center to any point on the circle is a radius
    • Any segment with both endpoints on the circle is called a chord
    • Any chord that passes through the center of the circle is a diameter
    • The distance around a circle is the circumference of the circle
    • The part of a circle between two points on the circle is an arc
    • The angle between two radii of a circle is a central angle
  • Which is bigger, the circumference of a can of tennis balls or the height of the can?
    • The height is slightly greater than 3 times the diameter of a ball, or 3 times 2.5”
    • The circumference of the can is slightly greater than the circumference of a ball
the circumference diameter ratio2
The Circumference/Diameter Ratio

Circumference of a Circle

  • If C is the circumference and d is the diameter of a circle, then there is a number π such that C = πd
    • Since d = 2r where r is the radius, then C = 2πr
    • The length of an arc is the fraction of the circumference between the central angle with endpoints on the arc
  • The number π (Pi), has been known for centuries to be the ratio between the circumference and diameter of a circle
    • Mathematicians in ancient Egypt used the approximation (4/3)4.
    • Early Chinese and Hindu mathematicians used √10.
    • Other useful approximations are 22/7 or 355/113
    • As a decimal, π is approximately 3.1416
    • In 1897, the Indiana House of Representatives

passed a bill legislating a vague (and incorrect)

value for π. Fortunately the Senate tabled the measure

area of circles

r

Area of Circles
  • The area formulas 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 derive the formula

Area of a Circle

  • The area of a circle is pi times the square of its radius
    • For a circle of radius r, the formula for the area is A = πr2
area of circles4
Area 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 = πr2
any way you slice it
Any Way You Slice It
  • A sector of a circle is the region between two radii of a circle and the included arc (similar to a slice of pizza)
  • The area of a sector of a circle is found using the same approach as that used for arc length
    • The fraction of the circle covered by the sector equals the arc measure divided by 360°
    • The whole circle has an area of πr2
    • Multiplying the area of the circle by the fraction of the circle included in the sector gives the formula:

Asector = (a / 360) · πr2

any way you slice it6
Any Way You Slice It
  • A segment of a circle is the region between a chord of a circle and the included arc
  • The area of a segment of a circle is part of the area of a sector intercepted by the chord
    • The chord cuts off an arc that can be used to find the area of the corresponding sector
    • A triangle is created by the chord and the two radii to the endpoints of the chord
    • The base of the triangle is the length of the chord
    • The height of the triangle is the distance between the chord and the center of the circle
    • Subtracting the area of the triangle from the area of the sector gives the formula:

Asegment = πr2 * a / 360° – ½bh

any way you slice it7
Any Way You Slice It
  • An annulus is the region between two concentric circles (similar to a ring or washer)
  • The area of an annulus of a circle is the area between the inner and outer circles
    • The area of each concentric circle is π times the radius squared
    • The area between the circles is the area of the outer circle minus the area of the inner circle
    • Let the radius of the outer circle be R, and the radius of the inner circle be r
    • Subtracting the area of the inner circle from the area of the outer circle gives the formula:

Aannulus = πR2 – πr2or Aannulus = π(R2 – r2)