9.1.2 Developing Formulas for Circles and Regular Polygons
Key Objectives • Develop and apply the formulas for the area and circumference of a circle. • Develop and apply the formula for the area of a regular polygon. Key Terms • A circle is the locus of points in a plane that are a fixed distance from a point called the center of the circle . • The center of a regular polygon is equidistant from the vertices. • The apothem is the distance from the center to the side of a regular polygon. • A central angle of a regular polygon has its vertex at the center, and its sides pass through consecutive vertices. Formulas • A circle with diameter d and radius r has circumference C = πd or C = 2 πr and area A = πr 2 . • The area of a regular polygon with apothem a and perimeter P is A aP . =
1 2
Example 1 Finding Measurements of Circles
The area of a circle is determined in this example. The diameter of the circle is given. The radius of a circle is equal to one-half the diameter. The radius of the circle is 12/2 = 6 inches. Substitute the value for the length of the radius into the formula for the area of a circle. The solution is A = πr 2 = π (6) 2 = 36 π in 2 .
The length of the radius of a circle is determined in this example. The circumference of the circle is given. Use the formula for the circumference of a circle, C = 2 πr . Substitute the value for the circumference into the formula and solve for the length of the radius. The solution yields r = 16 cm.
114
Made with FlippingBook - PDF hosting