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 B C Center and radius A D Apothem F E

AREAS OF REGULAR POLYGONS Example: regular hexagon B C Notice that triangle GFA is isosceles since all of the radii are congruent. G A D The area of the hexagon can be determined by adding the areas of the triangles. F E

AREAS OF REGULAR POLYGONS Example: regular hexagon B C Since the apothem is perpendicular to the side of the hexagon, it is an altitude to ∆AGF G A D a b E F Area of ∆AGF = ½ ba Area of the hexagon is 6(½ ba)

AREAS OF REGULAR POLYGONS Example: regular hexagon B C Notice that the perimeter P of the hexagon is 6b units. G A D a b We can substitute P for 6b in the area formula. E F Area of the hexagon is 6(½ ba) Area of the hexagon is ½ Pa

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: 36° The measure of each angle Is or 72° 360° 5 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. 4 N Q M

Example 1 Area of a Regular Polygon K Write a trigonometric ratio to find the length of PQ J L P 8 36° 4 N Q M

Example 1 Area of a Regular Polygon K Area: J L P 8 5.5 4 N Q 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. 48 34

Example 3 Area of an Inscribed polygon Find the area of the shaded region. Assume the triangle is equilateral. 4