11.6 area of reg polygons

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11.6 area of reg polygons

  1. 1. Areas of Regular Polygons Section 11.6
  2. 2. Lesson Focus <ul><li>The focus of this lesson is on applying the formula for finding the area of a regular polygon. </li></ul>
  3. 3. Basic Terms <ul><li>Center of a Regular Polygon </li></ul><ul><li>the center of the circumscribed circle </li></ul><ul><li>Radius of a Regular Polygon </li></ul><ul><li>the distance from the center to a vertex </li></ul><ul><li>Central Angle of a Regular Polygon </li></ul><ul><li> an angle formed by two radii drawn to consecutive vertices </li></ul><ul><li>Apothem of a Regular Polygon </li></ul><ul><li>the (perpendicular) distance from the center of a regular polygon to a side </li></ul>
  4. 4. Basic Terms
  5. 5. Theorem 11-11 <ul><li>The area of a regular polygon is equal to half the product of the apothem and the perimeter. </li></ul>
  6. 6. Area of a regular polygon <ul><li>The area of a regular polygon is: </li></ul><ul><li>A = ½ Pa </li></ul><ul><li>Area </li></ul><ul><li> Perimeter </li></ul><ul><li>apothem </li></ul>
  7. 7. <ul><li>The center of circle A is: </li></ul><ul><li>A </li></ul><ul><li>The center of pentagon BCDEF is: </li></ul><ul><li>A </li></ul><ul><li>A radius of circle A is: </li></ul><ul><li>AF </li></ul><ul><li>A radius of pentagon BCDEF is: </li></ul><ul><li>AF </li></ul><ul><li>An apothem of pentagon BCDEF is: </li></ul><ul><li>AG </li></ul>B C D E F A G
  8. 8. Area of a Regular Polygon <ul><li>The area of a regular n-gon with side lengths (s) is half the product of the apothem (a) and the perimeter (P), so </li></ul><ul><li>A = ½ aP, or A = ½ a • ns. </li></ul><ul><li>NOTE: In a regular polygon, the length of each side is the same. If this length is (s), and there are (n) sides, then the perimeter P of the polygon is n • s, or P = ns </li></ul>The number of congruent triangles formed will be the same as the number of sides of the polygon.
  9. 9. More . . . <ul><li>A central angle of a regular polygon is an angle whose vertex is the center and whose sides contain two consecutive vertices of the polygon. You can divide 360 ° by the number of sides to find the measure of each central angle of the polygon. </li></ul><ul><li>360/n = central angle </li></ul>
  10. 10. Areas of Regular Polygons Center of a regular polygon: center of the circumscribed circle. Radius: distance from the center to a vertex. Apothem: Perpendicular distance from the center to a side. Example 1: Find the measure of each numbered angle. • 1 2 3 360/5 = 72 ½ (72) = 36 L 2 = 36 L 3 = 54 L 1 = 72 Example 2: Find the area of a regular decagon with a 12.3 in apothem and 8 in sides. Area of a regular polygon: A = ½ a p where a is the apothem and p is the perimeter. Perimeter: 80 in A = ½ • 12.3 • 80 A = 492 in 2 Example 3: Find the area. 10 mm • A = ½ a p p = 60 mm 5 mm LL = √3 • 5 = 8.66 a A = ½ • 8.66 • 60 A = 259.8 mm 2
  11. 11. <ul><li>But what if we are not given any angles. </li></ul>
  12. 12. Ex: A regular octagon has a radius of 4 in. Find its area. <ul><li>First, we have to find the apothem length. </li></ul><ul><li>4sin67.5 = a </li></ul><ul><li>3.7 = a </li></ul><ul><li>Now, the side length. </li></ul><ul><li>Side length=2(1.53)=3.06 </li></ul>4 a 135 o 67.5 o 3.7 x 4cos67.5 = x 1.53 = x A = ½ Pa = ½ (24.48)(3.7) = 45.288 in 2
  13. 13. Last Definition <ul><li>Central  of a polygon – an  whose vertex is the center & whose sides contain 2 consecutive vertices of the polygon. </li></ul><ul><li> Y is a central  . </li></ul><ul><li>Measure of a </li></ul><ul><li>central  is: </li></ul><ul><li>Ex: Find m  Y. </li></ul><ul><li>360/5= </li></ul><ul><li>72 o </li></ul>Y
  14. 14. Check out! http://www.mathopenref.com/polygonregulararea.html

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