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Introduction

Introduction You have used the formulas for finding the circumference and area of a circle. In this lesson, you will prove why the formulas for circumference and area work. You will see how the ratio of can be proven . Key Concepts

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Introduction

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  1. Introduction You have used the formulas for finding the circumference and area of a circle. In this lesson, you will prove why the formulas for circumference and area work. You will see how the ratio of can be proven. 3.5.1: Circumference and Area of a Circle

  2. Key Concepts You know that the circumference of a specific circle divided by its diameter is the ratio pi, written as . 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. We know that the circumference of a circle = • diameter or 2 • radius. Therefore, 3.5.1: Circumference and Area of a Circle

  3. Key Concepts, continued Long ago, mathematicians didn’t yet know the value of pi. Archimedes, a great mathematician from ancient Greece, used inscribed polygons to determine the value of pi. He started by inscribing a regular hexagon in a circle. 3.5.1: Circumference and Area of a Circle

  4. Key Concepts, continued 3.5.1: Circumference and Area of a Circle

  5. Key Concepts, continued He determined that each side of the hexagon equals the radius of the circle. AB = BD = DE = EF = FG = GA = CE Archimedes realized that if the perimeter of the hexagon were equal to the circumference of the circle, then both would equal 6r. This would mean that = 3. However, the circumference is larger than the hexagon; therefore, Archimedes thought, must be larger than 3. 3.5.1: Circumference and Area of a Circle

  6. Key Concepts, continued Next, Archimedes inscribed a regular dodecagon—a 12-sided polygon— in the circle. The perimeter of the dodecagon was much closer to the actual perimeter of the circle. 3.5.1: Circumference and Area of a Circle

  7. Key Concepts, continued He calculated the perimeter of the dodecagon to be approximately 6.21166. This means ≈ 3.10583. However, the circumference of the circle is still larger than the dodecagon, so must be greater than 3.10583. Next, Archimedes inscribed a 24-sided regular polygon and calculated its perimeter. This polygon’s perimeter is even closer to the circumference of a circle. Archimedes found that the ratio of the perimeter to the diameter is closer to the value of . 3.5.1: Circumference and Area of a Circle

  8. Key Concepts, continued Archimedes kept going with this process until he had inscribed a 48-sided polygon. As the number of sides of a polygon increases, the polygon looks more and more like a circle. As he worked, the number for the ratio of became more and more accurate. The more sides an inscribed polygon has, the closer its perimeter is to the actual circumference of the circle. Therefore, Archimedes determined that as the number of sides of a polygon inside a circle increases, the calculation approaches the limit for the value of . 3.5.1: Circumference and Area of a Circle

  9. 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. Increasing the number of side lengths for the inscribed polygon causes the polygon’s perimeter to get closer and closer to the length of the circumference of the circle. 3.5.1: Circumference and Area of a Circle

  10. Key Concepts, continued The area of the circle can be derived similarly using dissection principles. Dissection involves breaking a figure down into its components. In the diagram below, a circle has been divided into four equal sections. 3.5.1: Circumference and Area of a Circle

  11. Key Concepts, continued If you cut the four sections from the circle apart, you can arrange them to resemble a rectangle. The width of the “rectangle” equals the radius, r, of the original circle. The length is equal to half of the circumference, or • r. 3.5.1: Circumference and Area of a Circle

  12. 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

  13. 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

  14. 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. The formula for the area of a rectangle is l • w = a. The length of the rectangle made out of the circle segments is • r. The width is r. Thus, the area of the circle is . This proof is a dissection of the circle. 3.5.1: Circumference and Area of a Circle

  15. Key Concepts, continued Remember that a sector is the part of a circle that is enclosed by a central angle. A central angle has its vertex on the center of the circle. A sector will have an angular measure greater than 0° and less than 360°. 3.5.1: Circumference and Area of a Circle

  16. Common Errors/Misconceptions not realizing that there is more than one way to prove a formula using the wrong formula for area, circumference, or the area of a sector using the diameter in a formula instead of the radius 3.5.1: Circumference and Area of a Circle

  17. Guided Practice Example 1 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

  18. Guided Practice: Example 1, continued 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

  19. Guided Practice: Example 1, continued 3.5.1: Circumference and Area of a Circle

  20. Guided Practice: Example 1, continued 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

  21. Guided Practice: Example 1, continued 3.5.1: Circumference and Area of a Circle

  22. Guided Practice: Example 1, continued 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

  23. Guided Practice: Example 1, continued 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

  24. Guided Practice: Example 1, continued 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

  25. Guided Practice: Example 1, continued 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

  26. Guided Practice: Example 1, continued 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

  27. Guided Practice: Example 1, continued The length of is 2.5 meters. 3.5.1: Circumference and Area of a Circle

  28. Guided Practice: Example 1, 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

  29. Guided Practice: Example 1, continued Find the perimeter of the hexagon. The perimeter of the hexagon is 30 meters. 3.5.1: Circumference and Area of a Circle

  30. Guided Practice: Example 1, continued Compare the estimate with the calculated circumference of the circle. Calculate the circumference. 3.5.1: Circumference and Area of a Circle

  31. Guided Practice: Example 1, 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

  32. Guided Practice: Example 1, continued 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

  33. Guided Practice: Example 1, continued 3.5.1: Circumference and Area of a Circle

  34. Guided Practice Example 2 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

  35. Guided Practice: Example 2, continued Inscribe a hexagon into a circle and divide it into 6 equal triangles. 3.5.1: Circumference and Area of a Circle

  36. Guided Practice: Example 2, continued Use the measurements from Example 1 to find the area of one of the six triangles. 3.5.1: Circumference and Area of a Circle

  37. Guided Practice: Example 2, continued First, determine the formula to use. 3.5.1: Circumference and Area of a Circle

  38. Guided Practice: Example 2, continued From Example 1, the following information is known: You need to find the height, . In , the height, , is the adjacent side length. 3.5.1: Circumference and Area of a Circle

  39. Guided Practice: Example 2, continued Since the hypotenuse, , is a radius of the circle, it is 5 meters. Since the measure of and the hypotenuse are known, use the cosine of 30º to find 3.5.1: Circumference and Area of a Circle

  40. Guided Practice: Example 2, continued Now that the length of is known, use that information to find the area of using the formula determined earlier. 3.5.1: Circumference and Area of a Circle

  41. Guided Practice: Example 2, continued Find the area of the hexagon. Multiply the area of one triangle times 6, the number of triangles in the hexagon. The area of hexagon is about 64.95 m2. 3.5.1: Circumference and Area of a Circle

  42. Guided Practice: Example 2, continued Compare the area of the hexagon with the area of the circle. Find the area of the circle. The actual area of circle P is about 78.54 m2. 3.5.1: Circumference and Area of a Circle

  43. Guided Practice: Example 2, continued Find the difference between the area of the hexagon and the area of the circle. The actual area of the circle is approximately 13.59 m2 greater than the hexagon’s area. Show the difference as a percentage. 3.5.1: Circumference and Area of a Circle

  44. Guided Practice: Example 2, continued The actual area of the circle is about 17.30% larger than the estimate found by using the area of the hexagon. The estimate of a circle’s area calculated by using an inscribed polygon can be made closer to the actual area of the circle by increasing the number of side lengths of the polygon. ✔ 3.5.1: Circumference and Area of a Circle

  45. Guided Practice: Example 2, continued 3.5.1: Circumference and Area of a Circle

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