Areas of Regular Polygons Lesson 8.4 Areas of Regular Polygons Definition: An apothem of a regular polygon is a perpendicular segment from the center of the polygon’s circumscribed circle to a side of the polygon. You may also refer to the length of the segment as the apothem. JRLeon Geometry Chapter 8.4 HGSH

Areas of Regular Polygons Lesson 8.4 Areas of Regular Polygons You can divide a regular polygon into congruent isosceles triangles by drawing segments from the center of the polygon to each vertex. The center of the polygon is actually the center of a circumscribed circle, so these congruent segments are sometimes called the radii of a regular polygon. The area of one triangle is: 𝑎𝑠 2 Since there are 5 isosceles triangles, the total area is: 5( 𝑎𝑠 2 ) Or 5 2 (as) 7 2 (as) 6 2 (as) = 3as JRLeon Geometry Chapter 8.4 HGSH

Areas of Regular Polygons Lesson 8.4 Areas of Regular Polygons What is the perimeter of a regular polygon in terms of n and s? sn = Perimeter, where s is the side and n is the number of sides. JRLeon Geometry Chapter 8..4 HGSH

JRLeon Geometry Chapter 8..5 HGSH Areas of Circles Lesson 8.5 The polygon can be broken down into n isosceles triangles such as the one shown(where n is the number of sides). In this triangle s   is the side length of the polygon r   is the radius of the polygon and the circle h   is the height of the triangle. The area of the triangle is half the base times height: Triangle Area = 𝟏 𝟐 sh Polygon Area = ( 𝟏 𝟐 sh)n = 𝒉 𝟐 ns There are n triangles in the polygon. So: The term ns is the perimeter of the polygon (length of a side, times the number of sides). As the number of sides increase, the triangles get smaller and the polygon approaches a circle, where the value of the perimeter approaches the circle circumference, which is 2r. We substitute 2r for ns : Polygon Area = 𝒉 𝟐 2r Also, as the number of sides increases, the triangle gets narrower and narrower, and so when s approaches zero, h and r become the same length. So substituting r for h: Polygon Area = 𝒓 𝟐 2r = r2 JRLeon Geometry Chapter 8..5 HGSH

JRLeon Geometry Chapter 8..5 HGSH Areas of Circles Lesson 8.5 EXAMPLE A JRLeon Geometry Chapter 8..5 HGSH

Areas of Regular Polygons Lesson 8.5 Areas of Regular Polygons Areas of Circles Class Work / Home Work: 8.3 Pages 435 – 436 : problems 1 thru 4, 8 8.4 Pages 443 – 444 : problems 1 thru 8, 13, 14 8.5 Pages 451- 452 : problems 1 thru 10 JRLeon Geometry Chapter 8.4-8.5 HGSH