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Areas of Regular Polygons and Circles

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.

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Areas of Regular Polygons and Circles

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  1. Areas of Regular Polygons and Circles • Find areas of regular polygons. • Find areas of circles.

  2. 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.

  3. AREAS OF REGULAR POLYGONS Example: Regular hexagon ABCDEF B C Center and radius A D Apothem E F

  4. 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. E F

  5. 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)

  6. 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

  7. Key ConceptArea 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.

  8. Example 1Area 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 M Q

  9. Example 1Area 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 M Q PQ is the apothem of pentagon JKLMN. It bisects NPM and is a perpendicular bisector to NM. So MPQ is ½(72°) or 36°.

  10. Example 1Area 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 M Q

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

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

  13. Key ConceptArea of a Circle If a circle has an area of A square units and a radius of r units, then A = πr2 r

  14. Example 2Use 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

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

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