The inscribed circle’s radius is any line segment from the center drawn

A B C F E D Q Figure 7.32 In the proof of Theorem 7.3.1, we saw that “All radii of a regular polygon are congruent.” R Y X U V W S T Q P Figure 7.31 An apothem of a regular polygon is any line segment drawn from the center of that polygon perpendicular to one of the sides. DEFINITION In regular octagon RSTUVWXY with center P see Figure 7.31, the segment is an apothem. Any regular polygon of n sides has n apothems and n radii. The proof of Theorem 7.3.1 establishes that “All apothems of a regular polygon are congruent.” PQ A central angle of a regular polygon is an angle formed by two consecutive radii of the regular polygon. DEFINITION In regular hexagon ABCDEF with center Q see Figure 7.32, angle EQD is a cen- tral angle. Due to the congruences of the triangles in the proof of Theorem 7.3.1, we see that “All central angles of a regular polygon are congruent.” This leads to Theorem 7.3.2. The measure of the central angle of a regular polygon of n sides is given by c = 360 n . THEOREM 7.3.2 We apply Theorem 7.3.2 in Example 4. EXAMPLE 4 a Find the measure of the central angle of a regular polygon of 9 sides. b Find the number of sides of a regular polygon whose central angle measures 72°. Solution a b 72 = 360 n : 72n = 360 : n = 5 sides c = 360 9 = 40° 쮿 The next two theorems follow from the proof of Theorem 7.3.1. Any radius of a regular polygon bisects the angle at the vertex to which it is drawn. THEOREM 7.3.3 Any apothem of a regular polygon bisects the side of the polygon to which it is drawn. THEOREM 7.3.4 EXAMPLE 5 Given that each side of regular hexagon ABCDEF has the length 4 in., find the length of: a Radius b Apothem Solution a By Theorem 7.3.2, the measure of is or 60°. With 䉭QED is equiangular and equilateral. Then b With apothem as shown, is a 30°-60°-90° triangle in which By Theorem 7.3.4, With opposite the 60° angle of , it follows that QG = 2 13 in. 䉭QEG QG EG = 2 in. m ∠EQG = 30°. 䉭QEG QG QE = 4 in. QE ⬵ QD, 360° 6 , ∠EQD QG QE A B C F E D G Q Exs. 9–20 쮿 Exercises 7.3

1. Describe, if possible, how

you would inscribe a circle within kite ABCD.

2. What condition must be satisfied for it to be possible to

circumscribe a circle about kite ABCD?

3. Describe, if possible, how

you would inscribe a circle in rhombus JKLM.

10. In a regular polygon with each side of length 6.5 cm, the

perimeter is 130 cm. How many sides does the regular polygon have?

11. If the perimeter of a regular dodecagon 12 sides is

99.6 cm, how long is each side?

12. If the apothem of a square measures 5 cm, find the

perimeter of the square.

13. Find the lengths of the apothem and the radius of a square

whose sides have length 10 in.

14. Find the lengths of the apothem and the radius of a regular

hexagon whose sides have length 6 cm.

15. Find the lengths of the side and the radius of an

equilateral triangle whose apothem’s length is 8 ft.

16. Find the lengths of the side and the radius of a regular

hexagon whose apothem’s length is 10 m.

17. Find the measure of the central angle of a regular polygon of

a 3 sides. c 5 sides. b 4 sides. d 6 sides.

18. Find the measure of the central angle of a regular polygon of

a 8 sides. c 9 sides. b 10 sides. d 12 sides.

19. Find the number of sides of a regular polygon that has a

central angle measuring a 90°. c 60°. b 45°. d 24°.

20. Find the number of sides of a regular polygon that has a

central angle measuring a 30°. c 36°. b 72°. d 20°. A C D B Exercises 1, 2 M L K J

4. What condition must be

satisfied for it to be possible to circumscribe a circle about trapezoid RSTV? In Exercises 5 to 8, perform constructions.

5. Inscribe a regular octagon within a circle. 6. Inscribe an equilateral triangle within a circle.

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

9. Find the perimeter of a regular octagon if the length of

each side is 3.4 in. R S T V

21. Find the measure of each interior angle of a regular

polygon whose central angle measures a 40°. c 60°. b 45°. d 90°.

22. Find the measure of each exterior angle of a regular

polygon whose central angle measures a 30°. c 45°. b 40°. d 120°.

23. Find the number of sides for a regular polygon in which

the measure of each interior angle is 60° greater than the measure of each central angle.

24. Find the number of sides for a regular polygon in which

the measure of each interior angle is 90° greater than the measure of each central angle.

25. Is there a regular polygon for which each central angle

measures a 40°? c 60°? b 50°? d 70°?

26. Given regular hexagon ABCDEF with each side of

length 6, find the length of diagonal . HINT: With G on , draw . BG ⬜ AC AC AC A B C F E D G R Y X U V W S T T V S R Q P Exercises 28, 29

27. Given regular octagon RSTUVWXY with each side of

length 4, find the length of diagonal HINT: Extended sides, as shown, form a square. RU .

28. Given that RSTVQ is a regular pentagon and is

equilateral in the figure below, determine a the type of triangle represented by . b the type of quadrilateral represented by TVPS.

29. Given: Regular pentagon RSTVQ with equilateral

Find: m ∠VPS 䉭PQR 䉭VPQ 䉭PQR

30. Given: Regular pentagon JKLMN not shown with

diagonals and Find:

31. Prove: If a circle is divided into n congruent arcs

, the chords determined by joining consecutive endpoints of these arcs form a regular polygon.

32. Prove: If a circle is divided into n congruent arcs

, the tangents drawn at the endpoints of these arcs form a regular polygon. n Ú 3 n Ú 3 m ∠LNK KN LN 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