BOOK V.

REGULAR POLYGONS AND

CIRCLES

SECTION I. THEOREMS

312. Theorem I. In any regular polygon there is a point that is equidistant from the vertices and equidistant from the sides.

Prove that the bisectors of all the angles meet in a point, and use locus.

313. Center, Radius, Apothem. That point in a regular polygon which is equidistant from the vertices, and also equidistant from the sides, is called the center of the polygon; the line from the center to a vertex is called the radius of the polygon, and a perpendicular from the center to a side is called the apothem of the polygon.

It is evident that the radius of the polygon is also the radius of the circumscribed circle, and that the apothem of the polygon is the radius of the inscribed circle.

314. COR. 1. The area of a regular polygon equals one half the product of the perimeter by the apothem.

315. COR. 2. The perimeters of regular polygons of the same number of sides are proportional to their sides, apothems, or radii.

316. COR. 3. The areas of regular polygons of the same number of sides are proportional to the squares of their sides, apothems, or radii.

Note again that the area ratio is the square of the line ratio.

491. If from a point within a polygon of n sides perpendiculars are drawn to all the sides, the sum of those perpendiculars is n times the apothem.

492. Find the area of a regular hexagon of side 10; of side a.

493. If the apothem of one regular polygon is 12, that of another of the same number of sides is 15, and the area of the first is 477.16, find the area of the second.

317. Theorem II. An equilateral polygon inscribed in a circle is regular; an equiangular polygon circumscribed about a circle is regular.

318. Variables and Limits. A constant quantity is one that keeps the same value throughout the investigation in question. A quantity may be constant in one discussion, but not in another.

A variable quantity is one that takes different successive values during an investigation.

The limit of a variable is that constant to which the variable can approach so near that the difference is less than any possible fixed quantity, but which the variable cannot equal.

If the sum of the numbers 1, 1, 1, 1, 16, 32, etc., is taken, that sum will never equal 2, no matter how large a number of terms is added. However, it is not possible to name a number less than 2, such that the sum cannot become greater than that number; that is, be nearer to 2 (its limit) than any fixed number.

It is evident from the definition that when a variable approaches its limit, the difference between the limit and the variable approaches the limit zero; and, conversely, that when the difference between a constant and a vari

able approaches zero as a limit, the variable must be approaching the constant as its limit.

319. Limit Theorems. (Given without proof; see Appendix, § 347.)

(1) If two variables approaching limits are equal for all values, their limits are equal.

(2) If a variable is approaching a limit, that variable multiplied by, or divided by, any constant will approach its limit multiplied by, or divided by, that constant.

(3) If two variables are proportional to two constants, their limits are proportional to the same constants.

320. Theorem III. If the number of sides of a regular polygon inscribed in a circle is increased indefinitely, the apothem of the polygon will approach the radius of the circle as a limit.

Show that ra<

where r, a, 8, stand for radius, apothem, and side; then show that 80, when the number of sides is increased indefinitely.

321. Circumference Axiom. The circumference of a circle is the limit which the perimeters of regular inscribed and circumscribed polygons approach when the number of sides is increased indefinitely.

322. Theorem IV. The area of a circle is the limit which the areas of regular inscribed and circumscribed polygons approach when the number of sides is increased indefinitely.

323. Theorem V. The ratio of the circumference to the diameter is the same for all circles (or, circumferences are proportional to their diameters).

SMITH'S SYL. PL. GEOM.-11

Regular inscribed polygons have perimeters proportional to the diameters; apply limit Th. III.

324. Value of the Ratio of Circumference to Diameter. The ratio of circumference to diameter is represented by the Greek letter (called pi). This letter is the initial letter of the Greek word for circumference. The value of π can be found numerically by Geometry, and the method employed is shown in the Appendix (§ 350); the value commonly used in calculations is 3.14159, or for less accurate results, 3.1416, or even 34. The last value is sufficiently accurate for many of the numerical exercises of the Geometry, but the student should become accustomed to the use of the more accurate values also.

The value of cannot be expressed exactly in the decimal system; that is, it is an incommensurable number. It has, however, been calculated to over 700 decimal places, and there is no limit to the accuracy with which a calculation can be carried out, if it is considered worth while.

325. COR. I.

The circumference of any circle equals 2 π times its radius.

494. If the radius of one circle is double the radius of a second circle, and the circumference of the second circle is 30 ft. long, how long is the circumference of the first circle?

495. What is the width of the ring between two concentric circles whose circumferences are 100 and 200 ft.

496. If one third of the circumference of one circle equals one fourth the circumference of a second circle, how do the radii compare? 497. Find the circumference of a circle of radius 10 in.

498. Find the radius of a circle of circumference 22 ft.

326. Theorem VI. The area of a circle is one half the product of the circumference by the radius.

Use a circumscribed regular polygon.

327. COR. 1.

The area of a circle equals π times the

square of the radius.

328. COR. 2. The areas of two circles are proportional to the squares of their radii.

499. Find the area of a circle of radius 10.

500. Find the radius of a circle of area 49.

501. Find the circumference of a circle of area 3.14159.

502. If the radius of one circle is four times the radius of a second circle, the area of the first is how many times the area of the second?

503. If the circumference of one circle is twice the circumference of a second, how do the areas compare?

504. What is the area of the ring between concentric circles of circumferences 100 and 200 ft.?