Circumscribe a square about a circle. 8. Circumscribe an equilateral triangle about a circle.

PERSPECTIVE ON HISTORY The Value of ␲ In geometry, any two figures that have the same shape are described as similar. Because all circles have the same shape, we say that all circles are similar to each other. Just as a proportionality exists among the corresponding sides of similar triangles, we can demonstrate a proportionality among the circumferences distances around and diameters distances across of circles. By representing the circumferences of the circles in Figure 7.33 by C 1 , C 2 , and C 3 and their corresponding lengths of diameters by d 1 , d 2 , and d 3 , we claim that for some constant of proportionality k. C 1 d 1 = C 2 d 2 = C 3 d 3 = k We denote the constant k described above by the Greek letter ␲. Thus, in any circle. It follows that or C = ␲d ␲ = C d C 1 d 1 C 2 d 2 C 3 d 3 Figure 7.33 because d 2r in any circle. In applying these formulas for the circumference of a circle, we often leave ␲ in the answer so that the result is exact. When an approximation for the circumference and later for the area of a circle is needed, several common substitutions are used for ␲. Among these are and . A calculator may display the value . Because ␲ is needed in many applications involving the circumference or area of a circle, its approximation is often necessary; but finding an accurate approximation of ␲ was not quickly or easily done. The formula for circumference can be expressed as , but the formula for the area of the circle is . This and other area formulas will be given more attention in Chapter 8. Several references to the value of ␲ are made in literature. One of the earliest comes from the Bible; the passage from I Kings, Chapter 7, verse 23, describes the distance around a vat as three times the distance across the vat which suggests that ␲ equals 3, a very rough approximation. Perhaps no greater accuracy was needed in some applications of that time. A = ␲r 2 C = 2␲r ␲ L 3.1415926535 ␲ L 3.14 ␲ L 22 7 C = 2␲r In the content of the Rhind papyrus a document over 3000 years old, the Egyptian scribe Ahmes gives the formula for the area of a circle as . To determine the Egyptian approximation of ␲, we need to expand this expression as follows: In the formula for the area of the circle, the value of ␲ is the multiplier coefficient of . Because this coefficient is which has the decimal equivalent of 3.1604, the Egyptians had a better approximation of ␲ than was given in the book of I Kings. Archimedes, the brilliant Greek geometer, knew that the formula for the area of a circle was with C the circumference and r the length of radius. His formula was equivalent to the one we use today and is developed as follows: The second proposition of Archimedes’ work Measure of the Circle develops a relationship between the area of a circle and the area of the square in which it is inscribed. See Figure 7.34. Specifically, Archimedes claimed that the ratio of the area of the circle to that of the square was 11:14. This leads to the following set of equations and to an approximation of the value of ␲. Archimedes later improved his approximation of ␲ by showing that Today’s calculators provide excellent approximations for the irrational number ␲. We should recall, however, that ␲ is an irrational number that can be expressed as an exact value only by the symbol ␲. 3 10 71 6 ␲ 6 3 1 7 ␲ L 4 11 14 L 22 7 ␲ 4 L 11 14 ␲ r 2 4r 2 L 11 14 ␲ r 2 2r 2 L 11 14 A = 1 2 Cr = 1 2 2␲rr = ␲r 2 A = 1 2 Cr 256 81 r 2 ad - 1 9 d b 2 = a 8 9 d b 2 = a 8 9 2r b 2 = a 16 9 r b 2 = 256 81 r 2 Ad - 1 9 d B 2 r Figure 7.34