The Circumference/ Diameter Ratio

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# The Circumference/ Diameter Ratio - PowerPoint PPT Presentation

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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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 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

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 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
• 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 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 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)