(9) What will it cost to turf an oval piece of ground whose diameters are 8 ch. 75 lks. and 7 ch., at the rate of £20 per acre? (10) What is the difference between the area of a circle whose diameter is 196 yds. and that of an ellipse whose major diameter is 196 yds., and minor diameter 140 yds. ? Ex. LV. Miscellaneous Examples on the Trapezoid, Trapezium, Polygon, Circle, and Ellipse. (1) What is the area of a four-sided field, whose west side measures 900 lks., south side 975, east side 625, north side 500, and the diagonal from N.W. to S.E. 375 links? (2) The chord of an arc is 24 ft., and the chord of half the arc 15 ft.; find the radius of the circle. [See fig. Art. 113. A C=15 ft., and AD=12 ft. Hence the height CD= 9 ft. (Art. 72.) When these are known, the diameter may be found by Art. 115.] (3) The diameter of a circular enclosure is 196 yds.; find the circumference. (4) A B C D is a four-sided figure, the angles at A and D are right angles, and C D is parallel to A B; if C D=4.2 chs., A B 9.8 chs., and B C 7 chs., find the area, in square chains, of the whole figure. [Drop a perpendicular from C upon A B, then the figure will be seen to consist of a square whose side is 42 chs., and of a right-angled triangle whose hypotenuse is 7 chs., and one side 5'6 chs.] (5) How much ground does a circular walk 34 yds. wide take up, the diameter of the outer circle being 84 yds.? (6) What is the area of a trapezium whose diagonal is 6875 lks., and the two perpendiculars upon it 1125 and 675 lks.? (7) Find the area of a trapezoid whose two parallel sides are 1875 and 4375 lks., and perpendicular distance 625 lks. (8) The chord of an arc is 24 ft. and the height 9 ft.; find the length of the arc. [See note to Ex. XLVIII., Quest. 7.] (9) A mill-sail is 21 ft. long, and revolves regularly 9 times in a minute. Find the distance traversed by its extremity in 10 hours. (10) The span of a bridge, the form of which is an arc of a cirele, being 72 ft., and the height 12 ft.; find the radius. [See Art. 115. The span is the chord of the arc.] (11) Make a rough sketch and find the area of a field A B C D from the following measurements taken in links:A C=500, D E the perpendicular from D on A C=175, and B F the perpendicular from B on A C=225. (12) Find the area of an equilateral triangle whose side is 25 yds. (13) A circular bowling-green has a diameter of 743 yds. ; a walk round its outer edge is 10 ft. wide. How many square yards does the walk cover? (14) What is the area of an elliptical flower-bed 14 yds. long and 8 yds. broad? (15) The chord of an arc is 15 ft. and the height of the arc is 4 ft.; find the length of the arc. (16) The diameter of a circle is 252 yds.; find the circumference. (17) What is the area of a sector of a circle, the length of the arc being 30 ft. and the diameter 36 ft. ? [See Art. 120.] (18) The area of a circle is 41 ac. 9 sq. chs. 2650 sq. lks. ; find the diameter. (19) The diameters of two concentric circles are 38 chs. 50 lks. and 24 chs. 50 lks.; find, in ac. ro. po., the area of the included ring. (20) An octagonal summer-house, each of whose sides measures 4 ft., is floored with sixteen blocks of wood of equal size; find the surface of each block. (21) The axes of an elliptical piece of ground are 28 po. and 15 po. Find its value, at £100 per acre. (22) A bicycle is 31 ft. diameter, and turns exactly 168 times in going round a circular pond. Find the area of the pond. (23) What is the area, in square links, of a four-sided field whose west side measures 111 lks., south side 175 lks., east side 220 lks., north side 132 lks., and the diagonal from N.W. to S.E. 176 lks. ? (24) The area of a circle is 517 sq. yds. 5 sq. ft. 72 sq. in. ; find its radius. (25) The inner diameter of the base of a lighthouse is 21 ft. and the thickness of the wall 21 in.; find how many square feet of ground the base of the wall occupies. (26) What is the area of an oval whose greatest and least diameters are 32 yds. 2 ft. and 24 yds. ? (27) Required the area of a trapezoid whose parallel sides are 18 and 25 yds. respectively, and perpendicular distance 12 yds. (28) The radius of a circle is 45 yds. 1 ft. 6 in.; find the area of the semicircle. (29) What is the area of an equilateral triangle whose side is 2.6 yds.? (30) Find the expense of paving a circular court 35 yds. in diameter, at 1s. 6d. per sq. yd., leaving in the centre a lawn in the form of a regular hexagon, each side of which measures 10 yds. (31) Find the area of a regular pentagon whose side is 50 ft. (32) Find the area of an irregular polygon A B C D E, whose side A B 63, B C 55, C D = 132, D E 116, and E A = 84; the diagonal B E = 105, and the diagonal B D 143. = = = (33) What is the area of a sector whose radius is 63 ft., the angle which the arc subtends at the centre being 6° 40'? (34) The circumferences of two concentric circles are 154 and 264; what is the area of the included ring? (35) The perimeter of a regular heptagon is 84 yds.; find the area. (36) What is the diameter of a stone column whose circum. ference measures 16 ft. 6 in.? (37) Find the length of an arc of 1° 30′, the radius being 35 ft. (38) Find the value of a circular piece of ground whose diameter is 84 yds., at £60 10s. per acre. (39) What is the area of a trapezium whose sides A B, B C, CD, and D A are 55, 100, 75, and 20 chs. respectively, and the diagonal A C 65 chs.? (40) Required the radius of a circle which shall contain four times as much as another whose circumference is 176 ft. (41) An oblong grass-plot 120 ft. by 60 is to be levelled, at 6d. per square yard; and an oval lawn-tennis court 84 ft. by 36 is to be turfed within it, at 4d. per square yard. What will be the cost? (42) The radii of two concentric circles are 17 ft. and 241 ft.; what is the area of the included ring? (43) A semicircular-headed window is 3 ft. 6 in. wide, and its extreme height is 16 ft.; find the cost of glazing it, at 21s. a square yard. [Compare fig. to Ex. XXXVIII., Quest. 26. The radius of the semicircle is 1 ft. 9 in., the height of the rectangular portion will therefore be 16 ft.-1 ft. 9 in. 14 ft. 3 in. The area of the whole window will be the sum of the areas of a rectangle 14 ft. 3 in. by 3 ft. 6 in., and a semicircle whose diameter is 3 ft. 6 in.] (44) The radius of a circle is 63 ft., and the angle subtended by an arc at the centre is 22°; find the area of the sector. (45) Find the length of an arc of 7° 30′, the diameter of the circle being 168 ft. (46) In the middle of a circular court 28 ft. in diameter, is a well in the form of a regular hexagon, each side of which is (11) Make a rough sketch and find the area of a field A B C D from the following measurements taken in links :A C 500, D E the perpendicular from D on A C=175, and B F the perpendicular from B on A C=225. (12) Find the area of an equilateral triangle whose side is 25 yds. (13) A circular bowling-green has a diameter of 743 yds.; a walk round its outer edge is 10 ft. wide. How many square yards does the walk cover? (14) What is the area of an elliptical flower-bed 14 yds. long and 8 yds. broad? (15) The chord of an arc is 15 ft. and the height of the arc is 4 ft.; find the length of the arc. (16) The diameter of a circle is 252 yds.; find the circumference. (17) What is the area of a sector of a circle, the length of the arc being 30 ft. and the diameter 36 ft. ? [See Art. 120.] (18) The area of a circle is 41 ac. 9 sq. chs. 2650 sq. lks.; find the diameter. (19) The diameters of two concentric circles are 38 chs. 50 lks. and 24 chs. 50 lks.; find, in ac. ro. po., the area of the included ring. (20) An octagonal summer-house, each of whose sides measures 4 ft., is floored with sixteen blocks of wood of equal size; find the surface of each block. (21) The axes of an elliptical piece of ground are 28 po. and 15 po. Find its value, at £100 per acre. (22) A bicycle is 31 ft. diameter, and turns exactly 168 times in going round a circular pond. Find the area of the pond. (23) What is the area, in square links, of a four-sided field whose west side measures 111 lks., south side 175 lks., east side 220 lks., north side 132 lks., and the diagonal from N. W. to S.E. 176 lks. ? (24) The area of a circle is 517 sq. yds. 5 sq. ft. 72 sq. in.; find its radius. (25) The inner diameter of the base of a lighthouse is 21 ft. and the thickness of the wall 21 in.; find how many square feet of ground the base of the wall occupies. (26) What is the area of an oval whose greatest and least diameters are 32 yds. 2 ft. and 24 yds. ? (27) Required the area of a trapezoid whose parallel sides are 18 and 25 yds. respectively, and perpendicular distance 12 yds. (28) The radius of a circle is 45 yds. 1 ft. 6 in.; find the area of the semicircle. (29) What is the area of an equilateral triangle whose side is 2.6 yds.? (30) Find the expense of paving a circular court 35 yds. in diameter, at 1s. 6d. per sq. yd., leaving in the centre a lawn in the form of a regular hexagon, each side of which measures 10 yds. (31) Find the area of a regular pentagon whose side is 50 ft. (32) Find the area of an irregular polygon A B C D E, whose side A B = 63, B C = 55, C D = 132, D E = 116, and E A = 84; the diagonal B E = 105, and the diagonal B D = 143. == (33) What is the area of a sector whose radius is 63 ft., the angle which the arc subtends at the centre being 6° 40'? (34) The circumferences of two concentric circles are 154 and 264; what is the area of the included ring? (35) The perimeter of a regular heptagon is 84 yds.; find the area. (36) What is the diameter of a stone column whose circumference measures 16 ft. 6 in.? (37) Find the length of an arc of 1° 30′, the radius being 35 ft. (38) Find the value of a circular piece of ground whose diameter is 84 yds., at £60 10s. per acre. (39) What is the area of a trapezium whose sides A B, B C, CD, and D A are 55, 100, 75, and 20 chs. respectively, and the diagonal A C 65 chs.? (40) Required the radius of a circle which shall contain four times as much as another whose circumference is 176 ft. (41) An oblong grass-plot 120 ft. by 60 is to be levelled, at 6d. per square yard; and an oval lawn-tennis court 84 ft. by 36 is to be turfed within it, at 4d. per square yard. What will be the cost? (42) The radii of two concentric circles are 17 ft. and 241 ft.; what is the area of the included ring? (43) A semicircular-headed window is 3 ft. 6 in. wide, and its extreme height is 16 ft.; find the cost of glazing it, at 21s. a square yard. [Compare fig. to Ex. XXXVIII., Quest. 26. The radius of the semicircle is 1 ft. 9 in., the height of the rectangular portion will therefore be 16 ft.-1 ft. 9 in. 14 ft. 3 in. The area of the whole window will be the sum of the areas of a rectangle 14 ft. 3 in. by 3 ft. 6 in., and a semicircle whose diameter is 3 ft. 6 in.] (44) The radius of a circle is 63 ft., and the angle subtended by an arc at the centre is 221°; find the area of the sector. (45) Find the length of an arc of 7° 30', the diameter of the circle being 168 ft. (46) In the middle of a circular court 28 ft. in diameter, is a well in the form of a regular hexagon, each side of which is |