APPENDIX. MENSURATION. $196. A triangle is a figure bounded by three straight lines. Thus, BAC, is a triangle. The three lines BA, AC, BC, are called sides and the three corners, B, A, and C, are called angles. side BC is called the base. The When a line like AD is drawn making the angle ADB equal to the angle ADC, then AD is said to be perpendicular to BC, and AD is called the altitude of the triangle. Each triangle BAD or DAC is called a right angled triangle. The side BA or the side AC, opposite the right angle, is called the hypothenuse. The area or content of a triangle is equal to half the product of its base by its altitude. EXAMPLES. 1. The base of a triangle is 40 yards and the perpendicular 20 yards: what is the area? 2. In a triangular field the base is 40 chains and the perpendicular 15 chains: how much does it contain? (see § 64.) Ans. 30 acres 3. There is a triangular field of which the base is 35 rods and the perpendicular 26 rods: what is its content? Ans. 2A. 3R. 15P. 4. What is the area of a square field of which the sides are each 33,08 chains? Ans. 109A. 1R. 28P+. 5. What is the area of a square piece of land of which the sides are 27 chains? 6. What is the area of a square the sides are 25 rods each? Ans. piece of land of which Ans. 3A. 3R. 25P. } § 197. A rectangle is a four-sided figure like a square, in which the sides are perpendicular to each other, but the adjacent sides are not equal. The area or content of a rectangle is equal to the length multiplied by the breadth. EXAMPLES. 1. What is the content of a rectangular field the length of which is 40 rods and the breath 20 rods? Ans. 5 acres. 2. What is the content of a field 40 rods square? Ans. 10 acres. 3. What is the content of a rectangular field 15 chains long and 5 chains broad. Ans. 4. What is the content of a field 25 chains long by 20 chains broad? Ans. 50 acres. 5. What is the content of a field 27 chains long and 9 rods broad. Ans. 6A. OR. 12P. § 198. A circle is a portion of a plane bounded by a curved line, every part 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. 1 A The circumference AEBD is 3,1416 times greater than the diameter AB. Hence, if the diameter is 1, the circumference will be 3,1416. Hence, also, if the diameter is known, the circumference is found by multiplying 3,1416 by the diameter. EXAMPLES. 1. The diameter of a circle is 4, what is the circum ference? The circumference is found by simply multiylying 3,1416 by the diameter. OPERATION. 3,1416 4 Ans. 12,5664. 2. The diameter of a circle is 93, what is the circumference? Ans. 3. The diameter of a circle is 20, what is the circumference? Ans. 62,832. § 199. Since the circumference of a circle is 3,1416 times greater than 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 78,54. We divide the circumfer ence by 3,1416, the quotient 25 is the diameter. OPERATION. 3,1416)78,5400(25 62832 157080 157080 2. What is the diameter of a circle whose circumference is 11652,1944 ? Ans. 37,09. 3. What is the diameter of a circle whose circumference is 6850? Ans. 2180,41+. § 200. The area or content of a circle is found by multiplying the square of the diameter by the decimal,7854 EXAMPLES. OPERATION. 62=36 ,7854x36=28,2744 1. What is the area of a circle whose diameter is 6? We first square the diameter, giving 36, which we then multiply by the decimal,7854: the product is the area of the circle. Ans. 28,2744 2. What is the area of a circle whose diameter is 10? Ans. 78,54. 3. What is the area of a circle whose diameter is 7? Ans. 4. How many square yards in a circle whose diameter is 3 feet. Ans. 1,069016+. § 201. The surface of a sphere is formed by multiplying the square of the diameter by the decimal 3,1416. EXAMPLES. 1. What is the surface of a sphere whose diameter is 12? We simply multiply the decimal 3,1416 by the square of the diameter: the product is the surface. OPERATION. 3,1416 122=144 Ans. 452,3904 2. What is the surface of a sphere whose diameter is 7? Ans. 153,9384. 3. Required the number of square inches in the surface of a sphere whose diameter is 2 feet or 24 inches? Ans. 4. Required the area of the surface of the earth, its mean diameter being 7918,7 miles? " Ans. 196996571,722104 sq. miles. § 202. To find the solidity of a sphere-Multiply the surface by the diameter and divide the product by 6—the quotient will be the solidity. EXAMPLES. 1. What is the solidity of a sphere whose diameter is 12? 2. What is the solidity of a sphere whose diameter is 4 ? Ans. 33,5104. 3. What is the solidity of the earth, its mean diameter being 7918,7 miles? Ans. 259992792079,860+. § 203. To find the solid content of a prism-Multiply the area of the base by the altitude, and the product will be the content. EXAMPLES. 1. What is the content of a square prism, each side of the square which forms the base being 15, and the altitude of the prism 20 feet? We first find the area of the square which forms the base, and then multiply by the altitude. OPERATION. 152=225 20 Ans. 4500 2. What is the solid content of a cube each side of which is 24 inches? Ans. 13824 solid in. 3. How many cubic feet in a block of marble of which the length is 3 feet 2 inches, breadth 2 feet 8 inches, and height or thickness 2 feet 6 inches? Ans. 21 solid ft. 4. How many gallons of water, ale measure, will a cistern contain, whose 'dimensions are the same as in the last example? (See $ 67, NOTE.) Ans. 1291 gal. 5. Required the solidity of a triangular prism whose height is 10 feet, and area of the base 350? Ans. 3500. § 204. To find the convex surface of a cylinderMultiply the circumference of its base by the altitude. EXAMPLES. 1. What is the convex surface of a cylinder, the diameter of whose base is 20 and the altitude 50? We first multiply the diameter by 3,1416 which gives the circumference of the base. Then multiplying by the altitude, we obtain the convex surface. OPERATION. 3,1416 62,8320 50 Ans. 3141,6000 |