In a regular polygon, each interior angle measures 135°. If

The constant of proportionality k described in the opening paragraph of this section, in Postulate 22, and in the Discover activity is represented by the Greek letter . ␲ pi GIVEN: Circle O with length of diameter d and length of radius r. See Figure 8.40. PROVE: PROOF: By Postulate 22, . Multiplying each side of the equation by d, we have . Because d 2r the diameter’s length is twice that of the radius, the formula for the circumference can be written , or . VALUE OF ␲ In calculating the circumference of a circle, we generally leave the symbol ␲ in the an- swer in order to state an exact result. However, the value of ␲ is irrational and cannot be represented exactly by a common fraction or by a terminating decimal. When an approximation is needed for ␲, we use a calculator. Approximations of ␲ that have been commonly used throughout history include , , and . Although these approximate values have been used for centuries, your calculator provides greater accuracy. A calculator will show that . ␲ L 3.141592654 ␲ L 3.1416 ␲ L 3.14 ␲ L 22 7 C = 2␲r C = ␲ 2r = C = ␲d ␲ = C d C = 2␲r Exs. 1–2 Discover Find an object of circular shape, such as the lid of a jar. Using a flexible tape measure such as a seamstress or carpenter might use, measure both the distance around circumference and the distance across length of diameter the circle. Now divide the circumference C by the diameter length d. What is your result? ANSWER The ratio should be slightly larger than 3. Technology Exploration Use computer software if available. 1. Draw a circle with center O. 2. Through O, draw diameter . 3. Measure the circumference C and length d of diameter . 4. Show that . C d L 3.14 AB AB ␲ is the ratio between the circumference C and the diameter length d of any circle; thus, in any circle. ␲ = C d DEFINITION The circumference of a circle is given by the formula C = ␲d or C = 2␲r THEOREM 8.4.1 r O Figure 8.40 EXAMPLE 1 In in Figure 8.41, OA ⫽ 7 cm. Using , a find the approximate circumference C of . b find the approximate length of the minor arc . Solution a b Because the degree of measure of is 90°, we have or of the circumference for length of AB ¬ = 90 360 44 = 1 4 44 = 11 cm 1 4 90 360 AB ¬ = 44 cm = 2 22 7 7 C = 2␲r AB ¬ }O ␲ L 22 7 }O A B O Figure 8.41 쮿 the arc length. Then In the following theorem, the lengths of the diameter and radius of the circle are rep- resented by d and r respectively. 쮿