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Areas of Regular Polygons and Circles – Find areas of regular polygons. – Find areas of circles. AREAS OF REGULAR POLYGONS First, some definitions: Regular Polygon – a polygon in which all segments and all angles are congruent. Center of a Polygon – the center of its circumscribed circle Radius of a polygon – the radius of its circumscribed circle, or the distance from the center to a vertex. Apothem of a polygon – distance from the center to any side of the polygon. AREAS OF REGULAR POLYGONS Example: Regular hexagon ABCDEF Center and radius Apothem B C A D F E AREAS OF REGULAR POLYGONS Example: regular hexagon Notice that triangle GFA is isosceles since all of the radii are congruent. The area of the hexagon can be determined by adding the areas of the triangles. B C G A F D E AREAS OF REGULAR POLYGONS Example: regular hexagon Since the apothem is perpendicular to the side of the hexagon, it is an altitude to ∆AGF Area of ∆AGF = ½ ba Area of the hexagon is 6(½ ba) B A C G D a b F E AREAS OF REGULAR POLYGONS Example: regular hexagon Notice that the perimeter P of the hexagon is 6b units. We can substitute P for 6b in the area formula. Area of the hexagon is 6(½ ba) Area of the hexagon is ½ Pa B A C G D a b F E Key Concept Area of a Regular Polygon If a regular polygon has an area of A square units, a perimeter of P units, and an apothem of a units, then A = ½Pa This formula can be used to find the area of any regular polygon. Example 1 Area of a Regular Polygon Find the area of a regular pentagon with a perimeter of 40 centimeters. K J L P Step 1: The internal angles of the pentagon add up to 360°, so … N Q M Example 1 Area of a Regular Polygon Find the area of a regular pentagon with a perimeter of 40 centimeters. K J L P Step 1: The measure of each angle 360° Is or 72° 5 36° N Q M PQ is the apothem of pentagon JKLMN. It bisects NPM and is a perpendicular bisector to NM. So MPQ is ½(72°) or 36°. Example 1 Area of a Regular Polygon Find the area of a regular pentagon with a perimeter of 40 centimeters. K J L P Step 2: 8 36° Since the perimeter is 40 centimeters, each side is 8 centimeters and QM is 4 centimeters. N Q 4 M Example 1 Area of a Regular Polygon K Write a trigonometric ratio to find the length of PQ J L P QM tan MPQ PQ 4 tan 36 PQ ( PQ ) tan 36 4 4 PQ tan 36 PQ 5.5 8 36° N Q 4 M Example 1 Area of a Regular Polygon K Area: 1 A Pa 2 1 (40)(5.5) 2 110 J L P 8 5.5 N Q 4 M Key Concept Area of a Circle If a circle has an area of A square units and a radius of r units, then A = πr2 r Example 2 Use Area of a Circle to Solve Real World Problems A caterer has a 48-inch table that is 34 inches tall. She wants a tablecloth that will touch the floor. Find the area of the tablecloth. A r 2 (58) 2 10,568.3 48 34 Example 3 Area of an Inscribed polygon Find the area of the shaded region. Assume the triangle is equilateral. 4