1st remainder =29.7-15.4=14.3 = 9.7 .. Area√29.7 × 14.3 × 9.7 × 5.7=145.84 square yards. 403. To find the area of a trapezium: Multiply either of the diagonals by the sum of the verticals drawn to it from the opposite angles, and halve the product. Or divide the trapezium into two triangles, and proceed as for triangles. Ex. 1. Let a diagonal be 65 yards, and the verticals on it drawn from the opposite angles 28 and 32 yards; what is the area of the trapezium? Area = 65 ×(28+32), = 1966 square yards. 404. To find the area of a trapezoid, multiply the sum of the parallel sides by the vertical distance between them, and take half the product. Ex 1. Given the parallel sides of a trapezoid, equal 24 and 16 feet, and its breadth, or vertical distance equals 12 feet. Find the area. (24+16)×12 Area 2 405. The area of a polygon, or of any irregular figure, is found by dividing it into triangles, or trapeziums, or into both; the areas of these are determined and their sum is the area of the polygon. 406. To find the area of a circle, multiply the circumference by the diameter, and take one-fourth of the product. Or, multiply the square of the radius by 3.14159. Or, multiply the square of the diameter by .785398. 407. To find the area of a sector, multiply the subtending arc by half the radius. Or, find the area of the whole circle, multiply it by the number of degrees in the arc, and this product, divided by 360°, gives the area. 408. To find the area of an ellipse, multiply the product of the two diameters by .785398. Ex. 1. What is the area of a circle, the circumference of which is 15 feet, and diameter 4.77 feet? Ex. 2. Find the area of a sector, the radius of which is 50 and the arc 56° 30'. Ex. 3. The diameters of an ellipse are 18 feet and 8 feet; the area is required. Area 18x8x.785398-113.0973 square feet. = 409. EXERCISES. 1. Find the area of a rectangular garden, the length of which is 100 yards, and breadth 84 yards. 2. If the side of a square table be 4 feet 8 inches, what is its area? 3. How many acres does a triangular field contain, the base of which is 1440 yards, and its altitude 960 yards? 4. How many paving stones, the surface of each of which is 9 inches by 5, will be required to pave a street 450 yards long and 8 yards broad? 5. What is the side of a square, the area of which is the same as that of a triangle, and the sides of which are 72.60 and 56 feet? 6. Find the area of a circle, the diameter of which is 5.6 yards; and the side of a square of equivalent area. 7. The circumference of a circular pond is 720 yards; find its area. 8. It is required to find the radius of a circle, of the same area as two other circles, the radii of which are, respectively, 5 and 7.6 yards. 9. A square is covered with half-crowns, the diameter of which is 1.3 inches, and there are 12 on each side. Find the vacant space between the coins. 10. The interior diameter of a building is 56 feet, and the thickness of the wall 1 foot 4 inches. Find the surface of the ground upon which the wall stands. 11. The areas of an elliptic ceiling are 32 feet 8 inches, and 20 feet 9 inches. What is its area? 12. Required, the area of the sector of a circle, the arc and radius of which are each 7.4 feet. MENSURATION OF SOLIDS. 410. Geometrical bodies are either terminated by plane faces or by curved faces; solid bodies alone maintain their forms, liquid and gaseous bodies take the form of the vessels which contain them. Solid bodies bounded by plane surfaces are called polyhedron, they are; the prism, whose bases are equal and parellel polygons, and the lateral faces parallelograms. A prism is said to be trigonal, tetragonal, pentagonal, &c., if its bases be triangles, tetragons, pentagons, &c, The expressions three-sided, four-sided, five-sided prism, &c., are sometimes employed The altitude of a prism is the distance between its parallel bases. A prism the bases of which are parallelograms is called parallelepiped; when the lateral faces are vertical to the bases, we have instances of right-parallelepipeds, such as a closed box, a brick, &c. The cube is an example of a right parallelepiped, its six-faces, are equal squares. A pyramid has for its base a polygon, and for its lateral faces triangles, the vertices of which meet in a point called the vertex of the pyramid. Pyramids are trigonal, tetragonal, pentagonal, &c., if their base be a triangle, a tetragon, a pentagon, &c. The altitude of a pyramid is the vertical drawn from the vertex to the base. The cylinder is a prism having circles for its bases, such as a garden roller, a round pillar, &c. The cone is a pyramid having a circular base, such as a sugar loaf. In the right cone the axis is vertical to the centre of the base, in all other cases the cone is oblique. A frustum of a pyramid or a cone, is what remains when the top has been taken away, the solid is then said to have been truncated A sphere is a solid, bounded in every direction by a curved surface, which is everywhere at the same distance from a certain point within it called the centre. SUPERFICIAL MEASURE OF SOLIDS. 411. The lateral surface of a right prism, is equal to the product of the perimeter of its base by the altitude. The lateral surface of a regular pyramid, is formed by multiplying the perimeter of its base by the altitude of the lateral faces and taking half of the product. The convex surface of a cylinder, is obtained by multiplying the circumference of the base by the altitude. The convex surface of a cone, is equal to the product of the circumference of the base and half the slant height. The convex surface of a truncated cone, is equal to the sum of the circumferences of both bases, mutliplied by half the slant altitude. The surface of a sphere is equal to the convex surface of the circumscribing cylinder, or to the product of the square of the diameter by 3.1415926. The surface of a spherical segment, is equal to the convex surface of the corresponding portion of the circumscribing cylinder. MEASURE OF CONTENTS OF SOLIDS. 412. The dimensions of a solid are the lines representing its length, breadth, and thickness. The unit of bodies is a cube the length, breadth, and thickness of which are each equal to the lineal unit. The solidity of a prism, or cylinder, is equal to the product of the base by the altitude. The contents of a right parallelepiped is found by multiplying its three dimensions, or the three edges terminating at the same The solidity of a pyramid, or cone, is equal to the third part of the area of the base multiplied by the altitude. The solidity of a frustum of a pyramid, or of a cone, is found by adding the area of both bases to four times the area of the mean section parallel to the bases; multiply this sum by the altitude, and take one-sixth of the result. The content of a sphere is equal to the cube of its diameter multiplied by .5236, or is equal to two-thirds of the solidity of the circumscribing cylinder. 413. EXAMPLES OF SUPERFICIAL MEASURE. Ex. 1. How many square feet of wood are required to make a box, the length of which is 4 feet, depth 3 feet 6 inches, and breadth 1 foot 6 inches? Here lateral faces=(8+3) 34=38 square feet. square feet. .. wood required=384+12=50 square feet. Ex. 2. What quantity of canvass is required for an octagonal tent, each side being 8 feet, and the slant height 10 feet? Here surface: = 8 × 8 × 10 =320 square feet, or 35 square yards. Ex. 3. A cylindrical iron chimney, 5 feet in diameter and 20 feet high, is made with sheet-iron, 8 feet 6 inches long and 1 yard broad. How many sheets were required ? Surface of chimney=5 × 3.1416×20=314.16 square feet. Surface of each sheet=84×3=25.5 square feet. Ex. 4. If the diameter of the base of a right cone be .58 yards, and the distance of the vertex to any point of the circumference of the base .92 yard. Find the entire surface of the Area of base = .582 ×.785398=.2642 square yard. .. whole surface=.8382+.2642=1.1 square yards nearly. Ex. 5. A pail has the shape of a frustum of a cone; the radii of its bases are 1.25 feet and .75 foot, the height of its side is .5 yard. Find its convex surface. |