2. Adapted from Walch Education

3. Pi is an irrational number that cannot be written as a repeating decimal or as a fraction. It has an infinite number of non- repeating decimal places. Therefore, 3.5.1: Circumference and Area of a Circle2

4. 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, could be calculated. This is the basis for the formula for finding the circumference of a circle. 3.5.1: Circumference and Area of a Circle3

5. The area of the circle can be derived similarly using dissection principles. Dissection involves breaking a figure down into its components. 3.5.1: Circumference and Area of a Circle4

6. The circle in the diagram to the right has been divided into 16 equal sections. 3.5.1: Circumference and Area of a Circle5

7. You can arrange the 16 segments to form a new “rectangle.” This figure looks more like a rectangle. 3.5.1: Circumference and Area of a Circle6

8. 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. 3.5.1: Circumference and Area of a Circle7

9.  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 r. 3.5.1: Circumference and Area of a Circle8

10. 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. 3.5.1: Circumference and Area of a Circle9

11. 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. 3.5.1: Circumference and Area of a Circle10

12. To find the length of, first determine the known lengths of and  Both lengths are equal to the radius of circle P, 5 meters. 3.5.1: Circumference and Area of a Circle11

13. 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.  The measure of is 60°. 3.5.1: Circumference and Area of a Circle12

14. Use trigonometry to find the length of Make a right triangle inside of by drawing a perpendicular line, or altitude, from P to. 3.5.1: Circumference and Area of a Circle13

15. Determine  bisects, or cuts in half,. Since the measure of was found to be 60°, divide 60 by 2 to determine The measure of is 30°. 3.5.1: Circumference and Area of a Circle14

16. Use trigonometry to find the length of and multiply that value by 2 to find the length of  is opposite.  The length of the hypotenuse,, is 5 meters.  The trigonometry ratio that uses the opposite and hypotenuse lengths is sine. 3.5.1: Circumference and Area of a Circle15

17.   The length of is 2.5 meters. 3.5.1: Circumference and Area of a Circle16 Substitute the sine of 30°. Multiply both sides of the equation by 5.

18.  Since is twice the length of, multiply 2.5 by 2.  The length of is 5 meters. 3.5.1: Circumference and Area of a Circle17

19. Find the perimeter of the hexagon.  The perimeter of the hexagon is 30 meters. 3.5.1: Circumference and Area of a Circle18

20. Compare the estimate with the calculated circumference of the circle.  Calculate the circumference. 3.5.1: Circumference and Area of a Circle19 Formula for circumference Substitute 5 for r.

21.  Find the difference between the perimeter of the hexagon and the circumference of the circle.  The formula for circumference gives a calculation that is 1.416 meters longer than the perimeter of the hexagon. You can show this as a percentage difference between the two values. 3.5.1: Circumference and Area of a Circle20

22.  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. 3.5.1: Circumference and Area of a Circle21

23. 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 3.5.1: Circumference and Area of a Circle22

24. Ms. Dambreville