MENSURATION. 405. MENSURATION is the art of measuring, and embraces all the methods of determining the contents of geometrical figures. It is divided into two parts, the Mensuration of Surfaces, and the Mensuration of Volumes. MENSURATION OF SURFACES. 1 foot. 406. Surfaces have length and breadth. They are measured by means of a square, which is called the unit of surface. A SQUARE is the space included between four equal lines, drawn perpendicular to each other. Each line is called a side of the square. each side be one foot, the figure is called a square foot. If The number of small squares that is contained in any large square, is always equal to the product of two of the sides of the large square. As in the figure, 3 × 3 = 9 square feet. The number of square inches contained in a square foot is equal to 12 × 12 = 144. 1 foot. If the sides of a square be each four feet, the square will contain sixteen square feet. For, in the large square there are sixteen small squares, the sides of which are each one foot. Therefore, the square whose side is four feet, contains sixteen square feet. Triangle. 407. A TRIANGLE is a figure bounded by three straight lines. Thus, ACB is a triangle. The lines BA, AC, BC, are called sides; and the corners, B, A, and C, are called angles. The side AB is the base. When a line like CD is drawn, making the angle CDA equal to the angle CDB, A then CD is said to be at right angles to 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 per pendicular, CD, 30 yards: what is 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 E 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 field, of which the sides are each 66.16 chains? 2. What is the area of a square piece of land, of which the ides 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 C 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 fect, 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. 643.02 +428.481071.50 = sum of parallel sides. Then, 1071.50 × 342.32366795.88; and 366795.88 = 183397.94 the area. 2 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, a diameter; and CB, a radius. The circumference, AEBD, is 3.1416 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. 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? |