Adapted from Walch Education

Transcription

1 Adapted from Walch Education

2 Pi (p ) is an irrational number that cannot be written as a repeating decimal or as a fraction. It has an infinite number of nonrepeating decimal places. Therefore, p = circumference diameter = circumference. 2 radius Circle 2

3 A limit is the value that a sequence approaches as a calculation becomes more and more accurate. This limit cannot be reached. Theoretically, if the polygon had an infinite number of sides, p could be calculated. This is the basis for the formula for finding the circumference of a circle. Circle 3

4 The area of the circle can be derived similarly using dissection principles. Dissection involves breaking a figure down into its components. Circle 4

5 The circle in the diagram to the right has been divided into 16 equal sections. Circle 5

6 You can arrange the 16 segments to form a new rectangle. This figure looks more like a rectangle. Circle 6

7 As the number of sections increases, the rounded bumps along its length and the slant of its width become less and less distinct. The figure will approach the limit of being a rectangle. a = r pr = pr 2 Circle 7

8 Show how the perimeter of a hexagon can be used to find an estimate for the circumference of a circle that has a radius of 5 meters. Compare the estimate with the circle s perimeter found by using the formula C = 2 p r. Circle 8

9 Draw a circle and inscribe a regular hexagon in the circle. Find the length of one side of the hexagon and multiply that length by 6 to find the hexagon s perimeter. Circle 9

10 Create a triangle with a vertex at the center of the circle. Draw two line segments from the center of the circle to vertices that are next to each other on the hexagon. Circle 10

11 To find the length of BC, first determine the known lengths of PB and PC. Both lengths are equal to the radius of circle P, 5 meters. Circle 11

12 Determine The hexagon has 6 sides. A central angle drawn from P will be equal to one-sixth of the number of degrees in circle P. mðcpb = = 60 The measure of ÐCPB is 60. Circle 12

13 Use trigonometry to find the length of Make a right triangle inside of by drawing a perpendicular line, or altitude, from P to BC. BC. Circle 13

14 Determine DP ÐCPB bisects, or cuts in half,. Since the measure of ÐCPB was found to be 60, divide 60 by 2 to determine mðbpd = 30 The measure of ÐBPD is 30. Circle 14

15 Use trigonometry to find the length of and multiply that value by 2 to find the length of BC. BD is opposite ÐBPD. The length of the hypotenuse, PB, is 5 meters. BD The trigonometry ratio that uses the opposite and hypotenuse lengths is sine. Circle 15

16 sin BPD = sin 30 = BD = BD = BD Substitute the sine of 30. Multiply both sides of the equation by 5. BD = 2.5 The length of BD is 2.5 meters. Circle 16

17 Since BC is twice the length of BD, multiply 2.5 by 2. BC = = 5 The length of BC is 5 meters. Circle 17

18 Find the perimeter of the hexagon. Perimeter = BC 6 = 5 6 = 30 The perimeter of the hexagon is 30 meters. Circle 18

19 Compare the estimate with the calculated circumference of the circle. Calculate the circumference. C = 2pr C = 2p 5 C» meters Formula for circumference Substitute 5 for r. Circle 19

20 Find the difference between the perimeter of the hexagon and the circumference of the circle = meters The formula for circumference gives a calculation that is meters longer than the perimeter of the hexagon. You can show this as a percentage difference between the two values = = 4.51% Circle 20

21 From a proportional perspective, the circumference calculation is approximately 4.51% larger than the estimate that came from using the perimeter of the hexagon. If you inscribed a regular polygon with more side lengths than a hexagon, the perimeter of the polygon would be closer in value to the circumference of the circle. Circle 21

22 Show how the area of a hexagon can be used to find an estimate for the area of a circle that has a radius of 5 meters. Compare the estimate with the circle s area found by using the formula A = pr 2. Circle 22

23 Dr. Dambreville