The Circumference/ Diameter Ratio

1. 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

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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)