In 1 and 2 let R, r, and C, c, and S, s denote respectively the radii, circumferences, and surfaces of the Os. Inscribe the 2 regular polygons of the same number of sides, letting P, p and A, a denote respectively the perimeters and areas. Compare the perimeters with their radii? Write the relation. Compare the areas with their radii. Increase the number of sides indefinitely, keeping them the same in number. Does the ratio written above ever change? What does P? What does? What does A = ? a? If 2 variables are in a constant ratio, what can be said of their limits? Prove your

360.

Cor. II. (1) Write the proportion of Cor. I. by alternation.

Do you see that this proportion may be interpreted as showing that the ratio of the circumference of a to its с diameter is a constant quantity? This constant is denoted

D

by the Greek letter π. It is the initial letter of the Greek word for circumference (periphereia). It is proved by methods in higher mathematics that II is incommensurable?

(2) Prove that CD 2 R.

π

(3) Show how similar sectors are related? (How are two Os related?

EXERCISES.

347. Find in terms of the radius and diameter of the circle the perimeter of a regular inscribed hexagon.

348. Find in terms of the radius and diameter the perimeter of a regular circumscribed hexagon.

349. Solve as in Exercises 347 for the perimeter of a regular inscribed dodecagon.

350. Solve as in Exercises 348 for the perimeter of a regular circumscribed dodecagon.

351. Show that the area of a circle is four times the area of a circle described on the radius as a diameter.

352. How does the square inscribed in a semicircle compare with the area of the circle?

Let R, C, and A denote the radius, circumference, and area of the . Construct a polygon of ʼn sides. Call its per. imeter P, and apothem R, and area S. Write an expression for the area of the polygon. As the number of sides are indefinitely increased, what does S? what does P? what does PR? Now if SA or PRA and P R = CR, then how are A and C R related? Review each step of your proposition. Make it clear. State the theorem.

362.

Cor. I. Substitute the value of C in terms of R and show that A = π R2.

363.

Cor. II. Show that the area of a sector

the product of the arc by its radius.

Write the

steps

of your proof.

364.

Taking the formulæ in §346, P'=

2 PX p .P+p

and p' = vpxP',

and calling the diameter of the circle 1, can you show how we may approximate the ratio of the diameter of a circle to its circumference?

From the above formulæ the following table has been computed:

Notice that by the last result the approximate value of the ratio of the circumference to the diameter is 3.1416, correct to the 4th decimal place. That is, = 3.1416.

EXERCISES.

353. An oyster can is 4 inches in diameter and 8 inches high. How many square inches of tin are required to make it? 354. Find the length of an arc of 180° in a circle of radius 4.

355. A circus ring contains 40 square rods. Find its radius and circumference. Call π, 34.

cm.

356. The apothem of a hexagonal paving-stone is 18 Find the area of its circumscribing O.

357. How many degrees in an arc whose length = the length of the radius of the circle? This arc is called a radian and is one of the units for measuring circles.

358. A cow is tethered with a chain 15 m. long; the stake is driven 10 m. from a straight fence. Over how much ground can the cow graze?

359. A railroad fence meets a farmer's fence at an angle of 50°; the farmer tethers a cow between the fences at the corner post; the chain is 30 m. long. Over how much ground can the cow graze?

360. Construct a rt. ▲, circumscribe a circle about the ▲, and on each side, about the rt. as a diameter, describe a semicircle exterior to the A. Compare the sum of the cres

cents with the area of the A.

MAXIMA AND MINIMA OF PLANE FIGUres.

365.

When thinking of qualities of the same kind, that which is greatest is called maximum and that which is least is called minimum. What is the maximum chord in a ? What is the minimum line from a point to a line?