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Circumference and Area of a Circle

Circumference and Area of a Circle. Adapted from Walch Education. Key Concepts. 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 ,. Key Concepts, continued.

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Circumference and Area of a Circle

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  1. Circumference and Area of a Circle Adapted from Walch Education

  2. Key Concepts • 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 Circle

  3. Key Concepts, continued • 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 Circle

  4. Key Concepts, continued • 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 Circle

  5. Key Concepts, continued The circle in the diagram to the right has been divided into 16 equal sections. 3.5.1: Circumference and Area of a Circle

  6. Key Concepts, continued • 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 Circle

  7. Key Concepts, continued • 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 Circle

  8. Practice • 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 Circle

  9. Step 1 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 Circle

  10. Step 2 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 Circle

  11. Step 3 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 Circle

  12. Step 4 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 Circle

  13. Step 5 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 Circle

  14. Step 6 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 Circle

  15. Step 7 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 Circle

  16. Step 7, continued • The length of is 2.5 meters. 3.5.1: Circumference and Area of a Circle

  17. Step 7, continued • 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 Circle

  18. Step 8 Find the perimeter of the hexagon. • The perimeter of the hexagon is 30 meters. 3.5.1: Circumference and Area of a Circle

  19. Step 9 Compare the estimate with the calculated circumference of the circle. • Calculate the circumference. 3.5.1: Circumference and Area of a Circle

  20. Step 9, continued • 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 Circle

  21. Note: • 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 Circle

  22. Can you 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 Circle

  23. Thanks for Watching Ms. Dambreville

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