9.1.2 Developing Formulas for Circles and Regular Polygons (continued) Example 3 Finding the Area of a Regular Polygon
The area of a regular hexagon is determined in this example. The length of a side of the hexagon is given. The area of a regular polygon is one-half times the length of the apothem times the perimeter. Notice that the formula combines the areas for the six triangles that fit inside the hexagon. The perimeter is equal to the sum of the six triangle bases and the apothem is the height of each triangle. The interior angles of the hexagon have a measure of 120°, so the angles at the base of the triangle (and the third angle) have measure 60°. The height forms a 30°-60°-90° triangle, which has one leg equal to half of the hexagon side, or 5 cm, and the other leg equal to 5 3. Substitute the value for the height into the apothem value in the formula and 60 cm for the perimeter. Using a calculator the area of the hexagon is found to be 259.8 cm 2 . The area of a regular octagon is determined in this example. The length of a side of the octagon is given. The area of a regular polygon is one-half times the length of the apothem times the perimeter. The interior angles of the octagon have a measure of 135°, so the angles at the base of the triangle shown have measure 67.5°. The height of the triangle forms a triangle with a third angle equal to 22.5°. The tangent of the 22.5° angle can be used to find the length of the height (the apothem of the octagon). Substitute the value for the height into the apothem value in the formula and (3)(8) = 24 for the perimeter. Using a calculator, the area of the hexagon is found to be 43.5 ft 2 .
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