117. Now it is evident that the area of the sector A OBC bears the same ratio to the area of the whole circle that the number of degrees in the angle A O B bears to 360°. (Compare Art. 108.) 118. To find the area of a sector of a circle, when the area of the circle and the angle of the sector are given. RULE. AS 360 is to the number of degrees in the angle of the sector, so is the area of the circle to the area of the sector. [When the radius or diameter is given, the area of the circle must first be found by Art. 102.] Ex. The radius of a circle is 7 in., and the angle of the sector is 45°; what is the area of the sector? Here, area of circle = 154 sq. in. (Art. 102) 360° 45° 154:: area of sector. 45 × 154 = 19.25. 360 Thus, the area of the sector is 19.25 sq. in. Ex. L. (1) The area of a circle is 243, the angle of the sector is 40°; find the area of the sector. (2) The diameter of a circle is 70 ft., the angle which the arc subtends at the centre is 36°; required the area of the sector. (3) The radius of a circle is 42 yds., the angle subtended by the arc at the centre is 30°; what is the area of the sector? (4) The radius of a circle is 11 ft. 8 in., the angle of the sector is 18°; required the area of the sector. (5) Find the area of a sector whose radius is 28 ft., and whose arc contains 135°. (6) The area of a sector being 125 sq. ft., and of the whole circle 400 sq. ft.; find the angle of the sector. [As, Area of circle: area of sector:: 360°: angle of sector (Compare Art. 110).] (7) The area of a sector is 231 sq. ft., and the diameter of the circle is 168 ft.; required the number of degrees in the angle of the sector. (8) The area of a sector is 77 sq. ft., and the radius of the circle is 28 ft.; find the number of degrees in the angle of the sector. (9) The area of a sector is 93.555 sq. ft., the angle of the sector is 120°; find the radius of the circle. [As, Angle of sector: 360°:: arca of sector: area of circle. When the area of the circle is known, the radius may be found by Art. 103.] (10) The area of a sector is 616 sq. ft., the angle of the sector is 10°; what is the radius? 119. Since the area of a circle is equal to the product of half the diameter and half the circumference (Art. 101), therefore the area of a sector, which is a portion of a circle, may be found by multiplying the arc by the radius and taking half the product. 120. To find the area of a sector of a circle when the radius and the length of the arc are given. RULE. Multiply the length of the arc by the radius, and divide the product by 2. Ex. The radius of a circle is 15 in., and the length of an arc of a sector is 18 in.; what is its area? Area of sector=x 18 x 15-135 sq. in. Ex. LI. (1) The length of an arc of a sector is 1 ft. 6 in., and the radius is 3 ft. 6 in.; required its area. (2) The radius of a sector is 25 ft., and the length of the arc 8 ft.; what is its area? (3) The radius of a circle is 6 ft., and the arc of the sector is equal to the radius; find the area of the sector. (4) The radius of a circle is 2 ft. 6 in., and the arc of the sector is equal to the radius; find the area of the sector. THE SEGMENT OF A CIRCLE. 121. A Segment of a circle is the figure bounded by a chord and the arc it cuts off. Thus ABC is a segment of the circle ABCE, being bounded by the A chord A C and the arc ABC. O A and OO are the radii, and BD is the height of the segment. 122. To find the area of a segment of a circle when the chord of the arc and its height are given. B D E G RULE. To two-thirds of the product of the chord and height, add the cube of the height divided by twice the chord, and the sum will give the area of the segment. Ex. If the chord be 72 and height 6, what is the area of the segment? 6 = × 72 × Area of segment 288 144 (1) Required the area of the segment whose height is 3 and chord 18. (2) The chord is 20 in., and the height is 4 in.; find the area of the segment. (3) The chord is 12 in., and the height is 3 in.; required the area of the segment. (4) The chord is 21 and the height 6; find the area of the segment. (5) Required the area of a segment of a circle the chord of which is 30 and height 10. (6) The chord of an arc is 36 in., and height 6 in.; find the area of the segment. THE CIRCULAR RING. 123. When two circles are described from the same centre, but with different radii, they are called concentric circles, and the space included between their circumferences is called a circular ring. Thus the circle A E B is de- A scribed from the centre O, at the distance O A, and the circle CFD from the same centre at the distance O C. E C D 124. The area of the circular ring is evidently the difference between the area of the larger circle A EB and of the smaller circle CF D. 125. To find the area of a circular ring. RULE. Subtract the area of the smaller circle from the area of the larger circle, and the remainder will be the area of the ring. Ex. What is the area of a circular ring when the radii of the two circles are 7 ft. and 14 ft. respectively? Area of outer circle=11x28x28=616 sq. ft. (Art. 102). Area of circular ring=616 154-462 sq. ft. Ex. LIII. (1) The diameter of the inner circle of a ring is 203 ft., and of the outer circle 217 ft.; find the area of the ring. (2) What is the area of a circular ring whose internal and external diameters are 63 ft. and 77 ft. respectively? (3) A gravel walk 7 yds. wide runs round the outside of a circular plot of land whose diameter is 42 yds.; find the area of the walk. [To find the external diameter, twice the width of the walk must be added to the internal diameter. Compare Art. 18.] (4) A circular court 84 yds. in diameter has a circular grassplot 70 yds. in diameter in the centre; what will it cost to pave the remainder (which forms a road round), at 44d. per sq. yd.? (5) Find, in acres, etc., the area of a circular ring whose inner diameter is 14 ch., and outer diameter 17 ch. 50 ĺks. (6) The circumference of an amphitheatre is 132 yds., and of the arena 66 yds. How many spectators can be accommodated if each occupies 4 sq. ft.? (7) A circular race-course whose diameter is 44 yds. contains a carriage-drive 3 yds. wide running round the inside of it 1 yd. from the edge. What will it cost to turf the gravel walk, at 4d. per sq. yd. ? [External diameter of walk=42 yds., internal diameter=35 yds.] (8) Find the cost of covering with flagstones, at 4s. 6d. per sq. yd., a path 2 ft. 4 in. wide round the outside of a circular flower-bed whose diameter is 15 ft. 2 in. (9) A mason is to put a stone curb to a circular well, at 1s. 6d. per sq. ft. The breadth of the curb is to be 21 in., and the diameter of the well 3 ft.; what will be the expense ? THE ELLIPSE. 126. The Ellipse, or Oval, is the figure described by the earth and all the other planets in their journey round the sun. ABCD is an ellipse, AB is the transverse diameter or major axis, and C D the conjugate C 3 127. To find the area of an ellipse. RULE. Multiply the product of the two diameters by 11. Ex. What is the area of an ellipse of which the major diameter is 35 yds., and the minor diameter 14 yds.? Area of ellipse 1×35×14=385 sq. yds. Ex. LIV. Find the areas of the ellipses whose major and minor axes are respectively : (1) 231 yds. and 70 yds. (3) 64 ft. 2 in. and 6 ft. 5 in. (2) 196 yds. and 42 yds. (4) 3 yds. 2 ft. 1 in. and 1yd. 2 ft. 10 in. (5) The diameters of an elliptical piece of ground are 85 ch. 75 lks. and 24 chs. 50 lks.; find its area in ac. ro. po. (6) In a rectangular court 49 yds. long and 40 yds. broad, there is an elliptical grass-plot, whose transverse diameter is 42 yds. and conjugate diameter 36 yds. How much of the court is not grass? (7) A rectangular grass-plot 56 ft. by 30 ft. has an oval flower-bed in its centre touching each side. Find the area of the grass surface in the four corners. [Compare fig. to Ex. XLI. Quest. 32. The length and breadth of the rectangle will be the transverse and conjugate diameters of the inscribed ellipse.] (8) A carriage drive 7ft. wide runs round an elliptical lawn whose major axis is 35 ft. and minor axis 24 ft.; find the expense of gravelling it, at 24d. per sq. yd. |