divide the product by 2, and the quotient will be the area (Bk. IV. Prop. VII). EXAMPLES. 1. Required the area or contents of the trapezoid ABCD, having given AB=643.02 feet, DC= 428.48 feet, and EF-342.32 feet. ANALYSIS.-We first find the sum of the sides, and then multiply it by the perpendicular height, after which we divide the product by 2, for the area. OPERATION. 643.02 +428.48 1071.50 = sum of parallel sides. Then, 1071.50×342.32=366795.88; = 183397.94 = and 366795.88 the area. 2. What is the area of a trapezoid, the parallel sides of which are 24.82 and 16.44 chains, and the perpendicular distance between them 10.30 chains? 3. Required the area of a trapezoid, whose parallel sides are 51 feet and 37 feet 6 inches, and the perpendicular distance between them 20 feet and 10 inches. 4. Required the area of a trapezoid, whose parallel sides are 41 and 24.5, and the perpendicular distance between them 21.5 yards. 5. What is the area of a trapezoid, whose parallel sides are 15 chains, and 24.5 chains, and the perpendicular height 30.80 chains? 6. What are the contents of a trapezoid, when the parallel sides are 40 and 64 chains, and the perpendicular distance between them 52 chains? 337. A circle is a portion of a plane bounded by a curved line, every point of which is equally distant from a certain point within, called the centre. The curved line AEBD is called the circumference; the point C the centre; the line AB passing through the centre a diameter; and CB the radius. The circumference AEBD is 3.1416 times as great as the diameter AB. A B Hence, if the diameter is 1, the circumference will be 3.1416. Therefore, if the diameter is known, the circumference is found by multiplying 3.1416 by the diameter (Bk. V. Prop. XIV). EXAMPLES. 1. The diameter of a circle is 8: what is the circumference? ANALYSIS. The circumference is found by simply multiplying 3.1416 by the di ameter. OPERATION. Ans. 25.1328 2. The diameter of a circle is 186: what is the circumference? 3. The diameter of a circle is 40: what is the circumference? 4. What is the circumference of a circle whose diameter is 57 ? 338. Since the circumference of a circle is 3.1416 times as great as the diameter, it follows, that if the circumference is known, we may find the diameter by dividing it by 3.1416. EXAMPLES. 1. What is the diameter of a circle whose circumference is 157.08? ANALYSIS.-We divide the circumference by 3.1416, the quotient 50 is the diameter. OPERATION. 3.1416)157.080(50 157.080 2. What is the diameter of a circle whose circumference is 23304.3888 ? 3. What is the diameter of a circle whose circumference is 13700? 337. What is a circle? What is the centre? What is the circumference? What is the diameter? What the radius? How many times greater is the circumference than the diameter? How do you find the circumference when the diameter is known? 338. How do you find the diameter when the circumference is known? 339. To find the area or contents of a circle. Multiply the square of the diameter by the decimal .7854 (Bk. V. Prop. XII. Cor. 2). EXAMPLES. 1. What is the area of a circle whose diameter is 12? ANALYSIS.-We first square the diameter, giving 144, which we then multiply by the decimal .7854: the product is the area of the circle. OPERATION. -2 12=144 144 x .7854 = 113.0976 Ans. 113.0976 2. What is the area of a circle whose diameter is 5? 3. What is the area of a circle whose diameter is 14? 4. How many square yards in a circle whose diameter is 3 feet? 340. A sphere is a figure terminated by a curved surface, all the points of which are equally distant from a certain point within, called the centre. The line AD, passing through its centre C, is called the diameter of the sphere, and AC its radius. 341. To find the surface of a sphere, Multiply the square of the diameter by 3.1416 (Bk. VIII. Prop. X. Cor.) EXAMPLES. 1. What is the surface of a sphere whose diameter is 6? ANALYSIS.-We simply multiply the number 3.1416 by the square of the diameter: the product is the surface. 339. How do you find the area of a circle? What is a radius? 2. What is the surface of a sphere whose diameter is 14? 3. Required the number of square inches in the surface of a sphere whose diameter is 3 feet or 36 inches. 4. Required the area of the surface of the earth, its mean diameter being 7918.7 miles. MENSURATION OF VOLUMES. 342. A SOLID or VOLUME is a figure having three dimensions: length, breadth, and thickness. It is measured by a cube called the cubic unit or unit of volume. 3 feet = 1 yard. A CUBE is a figure having six equal faces, which are squares. If the sides of the cube be each one foot long, the figure is called a cubic foot. But when the sides of the cube are one yard, as in the figure, it is called a cubic yard. The base of the cube, which is the face on which it stands, contains 3 x 3 3 feet = 1 yard. = 9 square feet. Therefore, 9 cubes, of one foot each, can be placed on the base. If the figure were one foot high it would contain 9 cubic feet; if it were 2 feet high it would contain two tiers of cubes, or 18 cubic feet; and if it were 3 feet high, it would contain three tiers, or 27 cubic feet. Hence, the contents of such a figure are equal to the product of its length, breadth, and height. 343. To find the contents of a sphere, Multiply the surface by the diameter, and divide the product by 6, the quotient will be the contents (Bk. VIII. Prop. XIV. Sch. 3). EXAMPLES. 1. What are the contents of a sphere whose diameter is 12? 342. What is a volume ? What is a cube? What is a cubic foot? What is a cubic yard? How many cubic feet in a cubic yard? What are the contents of a figure of three dimensions equal to ? 343. How do you find the contents of a sphere? ANALYSIS.-We first find the surface OPERATION. 12=144 by multiplying the square of the diam- multiply by 3.1416 solidity 6)5428.6848 = 904.7808 2. What are the contents of a sphere whose diameter is 8? 3. What are the contents of a sphere whose diameter is 16 inches? 4. What are the contents of the earth, its mean diameter being 7918.7 miles? 5. What are the contents of a sphere whose diameter is 12 feet? 344. A prism is a figure whose ends are equal plane figures and whose faces are parallelograms. The sum of the sides which bound the base is called the perimeter. of the base, and the sum of the parallelograms which bound the figure is called the convex surface. 345. To find the convex surface of a right prism. Multiply the perimeter of the base by the perpendicular height, and the product will be the convex surface (Bk. VII. Prop. I). EXAMPLES. 1. What is the convex surface of a prism whose base is bounded by five equal sides, each of which is 35 feet, the altitude being 52 feet? 2. What is the convex surface when there are eight equal sides, each 15 feet in length, and the altitude is 12 feet? 344. What is a prism? What is the perimeter of the base? What is the convex surface? 345. How do you find the convex surface of a prism? 346. How do you find the contents of a prism? |