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Areas of Regular Polygons

Areas of Regular Polygons. Lesson Focus. The focus of this lesson is on applying the formula for finding the area of a regular polygon. Basic Terms. Center of a Regular Polygon the center of the circumscribed circle Radius of a Regular Polygon the distance from the center to a vertex

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Areas of Regular Polygons

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  1. Areas of Regular Polygons

  2. Lesson Focus The focus of this lesson is on applying the formula for finding the area of a regular polygon.

  3. Basic Terms Center of a Regular Polygon the center of the circumscribed circle Radius of a Regular Polygon the distance from the center to a vertex Central Angle of a Regular Polygon an angle formed by two radii drawn to consecutive vertices Apothem of a Regular Polygon the (perpendicular) distance from the center of a regular polygon to a side

  4. Basic Terms

  5. Theorem 11-11 The area of a regular polygon is equal to half the product of the apothem and the perimeter.

  6. Area of a regular polygon The area of a regular polygon is: A = ½ Pa Area Perimeter apothem

  7. B The center of circle A is: A The center of pentagon BCDEF is: A A radius of circle A is: AF A radius of pentagon BCDEF is: AF An apothem of pentagon BCDEF is: AG F C A G E D

  8. Area of a Regular Polygon • The area of a regular n-gon with side lengths (s) is half the product of the apothem (a) and the perimeter (P), so A = ½ aP, or A = ½ a • ns. 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 The number of congruent triangles formed will be the same as the number of sides of the polygon.

  9. More . . . • 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. • 360/n = central angle

  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. 3 2 L2 = 36 • ½ (72) = 36 360/5 = 72 1 54 L1 = 72 L3 = Area of a regular polygon: A = ½ a p where a is the apothem and p is the perimeter. Example 2: Find the area of a regular decagon with a 12.3 in apothem and 8 in sides. A = 492 in2 Perimeter: 80 in A = ½ • 12.3 • 80 Example 3: Find the area. 10 mm A = ½ a p p = 60 mm • LL = √3 • 5 = 8.66 a 5 mm A = ½ • 8.66 • 60 A = 259.8 mm2

  11. But what if we are not given any angles.

  12. Ex: A regular octagon has a radius of 4 in. Find its area. First, we have to find the apothem length. 4sin67.5 = a 3.7 = a Now, the side length. Side length=2(1.53)=3.06 67.5o x 4 a 3.7 4cos67.5 = x 135o 1.53 = x A = ½ Pa = ½ (24.48)(3.7) = 45.288 in2

  13. Last Definition Central  of a polygon – an  whose vertex is the center & whose sides contain 2 consecutive vertices of the polygon. Y is a central . Measure of a central  is: Ex: Find mY. 360/5= 72o Y

  14. Check out!http://www.mathopenref.com/polygonregulararea.html

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