Sec. 10 – 3 Area of Regular Polygons Objective: 1) To find the area of a regular polygon.

A = Area of 1 triangle • # of triangles = ( ½ • apothem • side length s) • # of sides = ½ • apothem • # of sides • side length s = ½ • apothem • perimeter of a polygon This approach can be used to find the area of any regular polygon. F A H a E G B D C Hexagon ABCDEF with center G, radius GA, and apothem GH

** Regular Polygon – All sides  All s  Vertex s Regular Pentagon inscribed inside of a circle. Center of circle and the reg. pentagon. Radius of circle and the reg. pentagon. ** 5  isosceles Δs Apothem -  distance from the center to a side.

Find the missing  measures. # of sides Vertex  = 360 n m1 = 360/n = 360/5 = 72° 1 2 Apothem bisects vertex  of the Is. Δ formed by the radii. 3 m2 + m3 + 90 = 180 36 + m3 +90 = 180 m3 = 54 m2 = ½ m1 m2 = ½ (72) m2 = 36

Finding the area of a reg. polygon? Suppose reg. n-agon w/ side s The radii divides the figure into n  isos Δs each w/ area = ½ as s Apothem Side p = ns Area of n-gon = n • ½ as Perimeter of n-gon = ns Perimeter of reg. Polygon # of sides Length of sides

The area formula for any reg. Polygon A = ½ ap Apothem Area of any reg. Polygon Perimeter of reg. polygon

Example 2: find the area of a regular decagon with a 12 Example 2: find the area of a regular decagon with a 12.3in apothem and an 8in side. p = ns = (10)(8cm) = 80cm A = ½ ap = ½ (12.3in)(80in) = 492in2

Example 3: Find the apothem of a reg. hexagon with sides of 10mm. 60 2 3 a = √3 • short side a = √3 • (5mm) a = 5√3 = 8.66 30 30-60-90 Δ shortcut a 60 Could you find the area of it now?? 5mm

Example 4: Find the area of a regular pentagon with 7 Example 4: Find the area of a regular pentagon with 7.2ft sides and a 6.1ft radius. p = ns = (5)(7.2ft) = 36ft A = ½ ap = ½ (4.92ft)(36ft) = 88.6ft2 6.1ft 6.1ft a a2 + b2 = c2 a2 + 3.62 = 6.12 a2 + 12.96 = 37.21 a2 = 24.25 a = 4.92 3.6 ft

What is the difference between the radius and the apothem? The radius is longer because it is from the center all the way to the edge of the circle but the apothem is just from the center to the middle of the side What is the formula for the area of a regular polygon? A = ½asn A = ½aP