AB, and CD is called the altitude of the triangle. Each triangle CAD or CDB is called a right-angled triangle. The side BC, or the side AC, opposite the right angle, is called the hypothenuse. The area or contents of a triangle is equal to half the product of its base by its altitude (Bk. IV., Prop. VI.). NOTE--All the references are to Davies' Legendre. Examples. 1. The base, AB, of a triangle is 50 yards, and the perpendicular, CD, 30 yards: what is the area? ANALYSIS.-We first multiply the base by the altitude, and the product is square yards, which we divide by 2 for the area. Ans. OPERATION. 50 30 2) 1500 750 sq. yards. 2. In a triangular field the base is 60 chains, and the perpendicular 12 chains: how much does it contain ? 3. There is a triangular field, of which the base is 45 rods, and the perpendicular 38 rods: what are its contents? 4. What are the contents of a triangle whose base is 75 chains, and perpendicular 36 chains? Rectangle and Parallelogram. 408. A RECTANGLE is a four-sided figure, or quadrilateral, like a square, in which the sides are perpendicular to each other, but the adjacent sides are not equal. 409. A PARALLELOGRAM is a quadrilateral which has its opposite sides equal and parallel, but its angles not right angles. The line DE, perpendicular to the base, is called the altitude. D The area of a square, rectangle, or parallelogram, is equal to the product of the base and altitude. Examples. 1. What is the area of a square ficld, of which the sides are each 66.16 chains? 2. What is the area of a square piece of land, of which the sides are 54 chains? 3. What is the area of a square piece of land, of which the sides are 75 rods each? 4. What are the contents of a rectangular field, the length of which is 80 rods, and the breadth 40 rods? 5. What are the contents of a field 80 rods square ? 6. What are the contents of a rectangular field, 30 chains long and 5 chains broad? 7. What are the contents of a field, 54 chains long and 18 rods broad? 8. The base of a parallelogram is 542 yards, and the perpendicular height 720 feet: what is the area? 9. The measure of a rectangular field is 24000 square feet, and its length is 200 feet: what is its breadth? Trapezoid. 410. A TRAPEZOID is a quadrilateral, ABCD, having two of its opposite sides, AB, DC, parallel. The perpendicular, EF, is called the altitude. D E The area of a trapezoid is equal to half the product of the sum of the two parallel sides by the altitude (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 parallel sides, and then multiply it by the altitude; after which we divide the product by 2 for the area. OPERATION. 643.02428.481071.50 = sum of parallel sides. Then, 1071.50 × 342.32=366795.88; and 366795.88 183397.94 = 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? Circle. 411. 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 center The curved line AEBD is called the circumference; the point C, the center; the line AB, passing through the center, diameter; and CB, a radius. The circumference, AEBD, is 3.1416 A D times as great as the diameter AB. 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. XVI.). 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. 3.1416 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 ? 412. 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? 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? 413. To find the area or contents of a circle. Rule.-Multiply the square of the radius by 3.1416 (Bk V., Prop XV.). Examples. 1. What is the area of a circle whose diameter is 12? 4. How many square yards in a circle whose diameter is 31 feet? 5. What is the area of a circle whose diameter ismile? 415. To find the surface of a sphere. Rule.-Multiply the square of the diameter by 3.1416 (Bk. VIII., Prop. X., Cor. 1.). Examples. 1. What is the surface of a sphere whose diameter is 6? 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. 416. A SOLID or VOLUME is a portion of space having three dimensions: length, breadth, and thickness. It is measured by a cube, called the cubic unit, or unit of volume. A CUBE is a volume having six equal faces, which are squares. If the sides of the cube be each one foot long, the figure is called 3 feet1 yard. 3 feet1 yard. a cubic foot. But when the sides of the cube are one yard, |