To two thirds of the product of the height of the segment by the chord add the cube of the height, divided by twice the chord. Ex. 1. Required the area of the segment B 2. Required the area of a segment whose height is 15 rods and whose chord is 24 rods. Ans. 1 acre 3 roods 30 rods 9.4 yards. 638. To find the area of a zone of a circle. From the area of the whole circle subtract the areas of the segments on the sides of the zone. Ex. 1. Required the area of a zone whose parallel sides are 23.25 and 20.8 feet, in a circle whose radius is 12 feet. Ans. 206+ sq. ft. 2. Required the area of a zone included between two chords of 16 feet each, the diameter of the circle being 20 feet. Ans. 224.7 sq. ft. 639. To find the area of a lune or crescent. Find the difference of the areas of the two segments formed by the arcs of the lune and its chord. Ex. 1. If the chord of two intersecting arcs is 72 feet, and the height of one of the segments is 30, and of the other 20 feet, what is the area of the crescent? Ans. 612 sq. ft. 640. To find the area of a circular ring. Multiply the sum of the diameters of the two circles by the difference of the diameters, and that product by .7854. Ex. 1. What is the area of the ring formed by two circles whose diameters are 10 and 20 yards? Ans. 235.62 sq. yd. 2. In the centre of a circular pond there is an island 128 yards in diameter; what is the area of the pond, provided the exact distance from any part of the outer side of the pond to the center of the island is 78 yards? Ans. 1 acre 1 rood 14 rods 17 yards 7.4 feet. 641. To find the side of a square that shall equal the area of a circle of a given diameter or circumference. Multiply the diameter of the circle by .886227. Or, Multiply the circumference of the circle by .282094. Ex. 1. I have a round field, 50 rods in diameter; what is the side of a square field that shall contain the same area? Ans. 44.31135+ rods. 2. I have a circular field 360 rods in circumference; what must be the side of a square field that shall contain the same area? Ans. 101.55+ rods. 3. John Smith had a farm which was 10,000 rods in circumference, which he sold at $71.75 per acre. He purchased another farm con taining the same quantity of land in the form of a square; required the length of one of its sides. Ans. 2820.94+ rods. 642. To find the diameter of a circle that shall contain the area of a given square. Multiply the side of the given square by 1.12838. Ex. 1. The side of a square is 44.31135 rods; required the diameter of a circular field containing the same area. 643. To find the side of the largest equilateral triangle that can be inscribed in a circle of a given diameter or circumference. Multiply the given diameter by .866025. Or, Ex. 1. There is a certain piece of round timber 30 inches in diameter; required the side of an equilateral triangular beam that may be hewn from it. Ans. 25.98 inches. 2. How large an equilateral triangle may be inscribed in a circle whose circumference is 5000 feet? Ans. 1378.320 feet. 3. Required the side of an equilateral triangular beam, that may be hewn from a round piece of timber 80 inches in circumference. Ans. 22.05 inches. 644. To find the side of the largest square that can be inscribed in a circle of a given diameter or circumference. Multiply the given diameter by .707106. Or, NOTE. To find the circumference of a circle required to exactly admit a square of a given side, divide the given side by .225079. Ex. 1. I have a piece of timber 30 inches in diameter; how large a square stick can be hewn from it? Ans. 21.21+ in. square. 2. Required the side of a square that may be inscribed in a circle 80 feet in diameter. Ans. 56.56848+ feet. 3. I have a circular field whose circumference is 5000 rods; what is the side of the largest square field that can be made in it? Ans. 1125.395+ rods. 4. How large a square stick may be hewn from a piece of round timber 100 inches in circumference? Ans. 22.5 inches square. 5. What must be the circumference of a tree that, when hewn, shall be 18 inches square? Ans. 79.97 inches. 6. I have a garden which is 20 rods square; required, in feet, the circumference of a circle that will enclose this garden. Ans. 1466.15+ feet. 645. To find the diameter of the three largest equal circles that can be inscribed in a circle of a given diameter. Divide the given diameter by 2.155. Ex. 1. Required the diameter of each of the largest three circles that can be inscribed in a circle 86.2 inches in diameter. Ans. 40 inches. ELLIPSE. E C G H B 646. An ellipse is a plane figure bounded by a curve, from any point of which the sum of the distances to two fixed points is equal to a given distance. The two fixed points are called the foci, as G,H in the ellipse AECBDFA. The major or transverse axis of an ellipse is its longest diameter, as A B. The minor or conjugate axis of an ellipse is its shortest diameter, as CD. The segment of an ellipse is a portion cut off from the ellipse, as F AE F. F D 647. To find the area of an ellipse, the two diameters being given. Multiply the two diameters together, and that product by .785398. Ex. 1. What is the area of an ellipse, whose two diameters are 24 and 18 inches? Ans. 339.2919 inches. 2. What is the area of an elliptical pond, whose longest diameter is 33 feet 5 inches, and whose shortest diameter 20 feet 3 inches? Ans. 59 sq. yd. 67 sq. in. MENSURATION OF SOLIDS. PRISMS AND CYLINDERS. 648. A prism is a figure whose ends or bases are any plane figures which are equal and similar, and parallel to each other, and whose sides are parallelograms. A triangular prism is one whose base is a triangle; as the figure A B. A square prism is one whose base is a square; a pentagonal prism, one whose base is a pentagon; and so on, according to the figure of the ends or bases. A parallelopiped is a prism whose ends or bases, as well as its sides, are parallelograms. 649. A cylinder is a round body of uniform diameter, and which has circular bases parallel to each other; as the figure CD. The perimeter of a prism or cylinder is the line that bounds its end or base; and the altitude or height is the distance between the ends or bases. The convex surface of a prism or cylinder is the entire surface, exclusive of the two ends or bases. A B C D 650. To find the surface of a prism, or of a cylinder. Multiply the perimeter of the given prism or cylinder by the height, and to the product add the area of the two ends. Ex. 1. Required the surface of a triangular prism, of which the dis tance between the ends is 13 feet, and the sides of the base 23, 34, and 19 inches. Ans. 85.22+ square feet. 2. Required the surface of a pentagonal prism, whose length is 14 feet, and each side of whose base is 33 inches. Ans. 218.52 sq. ft. 3. Required the surface of a cylinder 13 feet long, the circumference of whose base is 57 inches. Ans. 65.34 square feet. 4. How often must a cylinder, 5 feet 3 inches long, whose diameter is 21 inches, revolve, to roll an acre? Ans. 1509.18 times. 5. Required the wall-surface of a square room, whose sides are each 16 feet long and 10 feet high. Ans. 71 square yards. 651. To find the contents or volume of a prism or cylinder. Multiply the area of the base of the given prism or cylinder by the height. Ex. 1. What are the contents of a triangular prism, whose length is 12 feet, and each side of whose base is 24 feet? Ans. 32.47 cu. ft. 2. Required the volume of a triangular prism, whose length is 10 feet, and the three sides of whose triangular end or base are 5, 4, Ans. 60 cu. ft. and 3 feet. 3. How many cubic feet in a block of marble, whose length is 3 feet 2 inches, breadth 2 feet 8 inches, and depth 2 feet 6 inches? Ans. 21 cu. ft. 4. What is the volume of a cylinder, whose length is 9 feet, and the circumference of whose base is 6 feet? Ans. 25.78 cu. ft. PYRAMIDS AND CONES. 652. A pyramid is a solid having for its base some rectilineal figure, and for its sides triangles meeting in a common point called the vertex; as the figure A B. The slant height of a pyramid is a line drawn from the vertex to the middle of one of the sides of the base. B 653. A cone is a solid having a circle for its base, and tapering uniformly to a point called the vertex. The slant height of a cone is a line drawn from the vertex to the circumference of the base. 654. The altitude or height of a pyramid or of a cone is a line drawn from the vertex perpendicular to the plane of the base. 655. The frustum of a solid is the part that remains after cutting off the top by a plane parallel to the base; as the frustum of a cone C D. D 656. To find the surface of a pyramid or of a cone. C Multiply the perimeter or the circumference of the base by half of the slant height, and to the product add the area of the base. Ex. 1. Required the area of the surface of a square pyramid, whose base is 2 feet 8 inches square, and whose slant height is 3 feet 9 inches. 2. What is the convex surface of a cone, whose slant height is 20 feet, and the circumference of whose base is 9 feet? Ans. 90 feet. 657. To find the volume of a pyramid, or of a cone. Multiply the area of the base by one third of the altitude. Ex. 1. What is the solidity of a cone, whose height is 12 feet, and the diameter of whose base is 24 feet? Ans. 20.45+ feet. 2. What are the contents of a triangular pyramid, whose height is 14 feet 6 inches, and the sides of whose base are 5, 6, and 7 feet? Ans. 71.035 feet. 658. To find the surface of a frustum of a pyramid, or of a cone. Multiply the sum of the perimeters or of the circumferences of the two ends by half of the slant height; and to the product add the areas of the two ends. Ex. 1. Required the surface of a frustum of a pentagonal pyramid, whose slant height is 10 inches, and the sides of whose base are 3 and 5 inches. 2. What is the surface of the frustum of a cone, the diameters of the bases being 43 and 23 inches, and the slant height 9 feet? Ans. 90.72+ sq. ft. 659. To find the volume of a frustum of a pyramid, or of a cone. Multiply the areas of the two ends together, and extract the square root of the product. To this root add the two areas, and multiply their sum by one third of the altitude. Ex. 1. If the length of a frustum of a square pyramid be 18 feet 8 inches, the side of its greater base 27 inches, and that of its less 16 inches, what is the volume ? Ans. 61.228+ cu. ft. 2. What are the contents of a stick of timber, whose length is 40 feet, the diameter of the larger end being 24 inches, and of the smaller end 12 inches? Ans. 73 cu. ft., nearly. SPHERES, SPHEROIDS, &C. 660. A sphere is a solid, bounded by a curved surface, every part of which is equally distant from a point within, called the centre. The axis or diameter of a sphere is a line passing through the centre, and terminated by the surface; as the line A B. B The radius of a sphere is a line drawn from the centre to any part of the surface. |