Ex. XXXIX. Find the diameters of the circles whose circumferences are respectively : (11) Find the diameter, in yards, of a circle whose circumference is a mile. (12) Required the radius of a circle whose perimeter is 220 chains. (13) How many steps did a gardener take in crossing the widest part of a circular lawn, the circumference of which is 132 yds., taking 3 ft. at every step? (14) Find, in yards, the radius of a circle whose circumference is 17 furlongs. [220 yds = 1 furlong.] (15) The iron railings round a circular shrubbery cost £44 11s., at 4s. 6d. a yard; find its diameter. (16) Suppose a coach-wheel to revolve 1920 times in a mile, what is its circumference? [Its circumference will be the distance it traverses in a single revolu. tion.] (17) If a carriage-wheel makes 360 revolutions in traversing a quarter of a mile, what is its diameter? (18) A waggon-wheel turns round 320 times in a mile; find the length of a spoke. (19) A bicycle-wheel 4 ft. 1 in. in diameter was observed to revolve 60 times in going round a circular course; required the diameter of the course. THE AREA OF A CIRCLE. 101. To find the area of a circle when the diameter and circumference are given. RULE. Multiply half the circumference by half the diameter, and the product will be the area. F Suppose we inscribe in a circle a regular polygon with a large number of sides. Then the three following facts are obvious: (1), the area of the polygon will not differ much from the area of the circle; (2) the perimeter of the polygon will not differ much from the circumference of the circle; and (3) the perpendicular drawn from the centre on a side of the polygon will not differ much from the radius of the circle. By continually increasing the number of sides, we should approach continually nearer and nearer to the circle, until at last the area of the polygon would practically become confounded with the area of the circle; the perimeter of the polygon with the circumference of the circle; and the perpendicular from the centre of the polygon with the radius of the circle. Hence, the theorem which was proved to be true for all regular polygons (Art. 92) must be true also for the circle, i.e. the area of the circle will be equal to half the rectangle contained by the circumference and the radius, or, in other words, the area of a circle is measured by half the product of the circumference and the radius. If the diameter and circumference be given, then the area will be found by multiplying half the circumference by half the diameter. Ex. What is the area of a circle whose circumference is 22 ft. and diameter 7 ft.? Area of circle} of 22 × } of 7=11×3}=381 sq. ft.=38 sq. ft. 72 sq. in. Ex. XL. (1) Find the area of a circle whose circumference is 132 yds., and diameter 42 yds. (2) The diameter of a circle is 25 yds. 2 ft., and its circumference 80 yds, 2 ft.; required its area. (3) The diameter of a circular grass-plot is 18 yds. 2 ft., and its circumference 58 yds. 2 ft.; what is its area? (4) The diameter of a circular enclosure is 77 yds., and its circumference 242 yds.; required the cost of paving it, at 6s. 8d. a square yard. (5) Find, in ac., ro., and po., the area of a circular race-course, whose diameter is 24 ch. 50 lks. and circumference 77 ch. 102. To find the area of a circle when its diameter is given. RULE. Multiply the square of the diameter by 11, and the product will be the area. Since the circumference is 2 of the diameter (Art. 99), half the circumference will be of the diameter. Now, area of circle Hence, writing" = of circum. x of diam. (Art 101). of diam." for "of circum.," we have of diam. x of diam. = = 11 of sq. of diam. Ex. 1. What is the area of a circle whose diameter is 7 Ex. 2. Find the area of a circle whose radius is 5.25. Find the areas of the circles whose diameters are re (12) 14.7 ft. Find the areas of the circles whose radii are respectively : (10) 1.925 ft. (11) 1 ft. Find, in ac. ro. po., the areas of the circles whose diameters are respectively : : (13) 2450 lks. (14) 101 ch. (15) 52 ch. 50 lks. (16) What is the area of a circle whose diameter is 196 yards? (17) What is the area of a circular plot whose diameter is 21 yards? 13. Find in se z. po.. the areas of a semicircle whose radius S 2: The Lamener of a cirzie is 364; find the area of a (a quadrant is a quarter of the arce.) 20 The Eameter of a circle is 3 ch 50 ks; find its area in mods and poles. 21 Begured in was and eva, the pressure on a circular plate 3 & 6 in dimeter, the pressure on a square in. being If the pressure of the atmosphere be 15 lbs. upon every square in what will be its pressure upon the piston of a steam-enzine whose diameter is 14 in. Express the result in evts. grs. Ite. (23) A room 25 ft. 3 in. long, and 14 ft. 6 in. wide, has a semicircular bow 21 ft. in diameter, thrown out on one side. Find the area of the whole room. (24) The shape of a plate-glass mirror is a semicircle on a square whose side is 4 ft. 8 in. Find its cost, at 6s. per sq. ft. [See £g to Ex. XXXVII, Quest. 26.] (25) In a square court whose side is 50 yds., there is a circular green whose diameter is 42 yds. How much of the court is not grass? (26) Find the number of turfs, each 25 in. by 8 in., required for a garden plot 50 ft. by 75 ft., allowing for four circular beds, each having a diameter of 5 ft. 10 in. (27) In a rectangular court 79 ft. by 39 ft., there is a circular basin 7 ft. in radius, and four rectangular grass plots, each 201 ft. by 8 ft. Find the cost of paving the remaining part of the court, at 63d. per sq. yd. (28) A square courtyard has a circular basin in the middle of it which is 28 ft. in diameter, a side of the court being 36 ft.; find what it will cost to gravel it, at d. a sq. ft. (29) The base of a conservatory is a square 18 ft. 8 in. long, on each side of which a semicircle was thrown out. How many plants will grow in it, each requiring 112 sq. in, of space? (30) The sides of a right-angled triangle are 49 ft. and 168 ft.; find, in sq. ft., the sum of the areas of the semicircles described on each of the sides and on the hypotenuse as diameters. (31) A rectangle is 107 ft. 3 in. long and 1 ft. 2 in. broad; find how many circles of 1 ft. 9 in. radius are equivalent in area to this rectangle. [In this case we must first find the area of the rectangle by Art. 50, and the area of each circle by Art. 102; then the number of times the latter is contained in the former.] (32) The side of a square, A B C D, is 5 ft. 10 in. long; a circle is inscribed in the square so as to touch all its sides; find the area between the circle and the square. [Here the diameter E F of the inscribed circle is equal to a side of the square A B C D. Find the difference between the area of the square and that of the circle.] D E C A F B (33) The area of a square is 1225; find the area of the circle inscribed in the square touching each side of it. [Find the side of the square by Art. 47; this will be the diameter of the inscribed circle.] (34) Out of a square grass-plot whose area is 7056 sq. yds., is cut the greatest possible circular flower-bed; find its area. (35) A square grass-plot 18 yds. 2 ft. long has a circular fishpond dug in its centre touching each side. Find the area of the grass surface left in the corners. 103. To find the diameter of a circle when the area is given. RULE. Divide the area by 11, and the square root of the quotient will be the diameter. Since 1x sq. of diam. area (Art. 102). = sq. of diam.=area÷1 and diam.=sq. root of (area÷11). Ex. The area of a circle is 23 sq. yds. 2 sq. ft. 88 sq. in.; find its diameter. Here, 23 sq. yds. 2 sq. ft. 88 sq. in.=30184 sq. in. Now, to divide by 11, we multiply by 11, and 30184 x 11=38416; hence, diameter sq. root of 38416-196 in. -16 ft. 4 in.5 yds. 1 ft. 4 in. Ex. XLII. Find the diameters of the circles with the following areas: (1) 154 sq. yds. (4) 1 sq. ft. 10 sq. in. (7) 3118-5 sq. Iks. (H) 1886-5 sq. F. (9) 260-26 sq I |