CHAPTER X MENSURATION Mensuration is the art of computing lengths, areas, and volumes of geometrical magnitudes by arithmetical rules. A length is a definite portion of a line in terms of its unit of measure; an area, that of a surface; a volume, that of a solid. The unit of measure of a surface is a square; of a volume, a cube. LENGTHS. Sides of a right-angled triangle. I. TO FIND HYPOTENUSE WHEN BASE AND PERPENDICULAR ARE GIVEN. RULE. Add the squares of base and perpendicular and extract square root of the sum. II. TO FIND BASE OR PERPENDICULAR WHEN THE HYPOTENUSE AND EITHER SIDE ARE GIVEN. RULE. From the square of hypotenuse subtract the square of the given side and extract the square root of the remainder. PROBLEMS. 1. Base equals 4.8 ft.; perpendicular, 3.6 ft.; what is hypotenuse? Ans. 6 ft. 2. Base equals 15 yds.; perpendicular, 9 yds. 1 ft.; what is hypotenuse? Ans. 17 yds. 2 ft. 3. Hypotenuse equals 67.43 yds.; perpendicular, 52.6; what is the base? Ans. 42.17+. 4. Hypotenuse equals 52.32 ft.; base, 32.11; what is perpendicular? Ans. 41.30+. Lines of a Circle. The ratio of the circumference of a circle to the diameter is 3.1415926+, which ratio is represented by the Greek letter π, called pi. I. TO FIND THE CIRCUMFERENCE WHEN THE DIAMETER IS GIVEN. RULE.-Multiply the diameter by 3.1416. II. TO FIND THE DIAMETER WHEN THE CIRCUMFERENCE IS GIVEN. RULE.-Divide the circumference by 3.1416. PROBLEMS. 1. Diameter equals 3.2 in.; what is the circumference? Ans. 10.05 in. 2. Diameter equals 1 ft. 3 in.; what is the circumference? Ans. 3 ft. 11.12 in. 3. Circumference equals 63.9 ft.; what is the diameter? Ans. 20.34 ft. 4. Circumference equals 6 yd. 1 ft. 4 in.; what is the diameter in feet? Ans. 6.154 ft. AREAS. I. TO FIND AREA WHEN BASE AND ALTITUDE ARE GIVEN. RULE.-Multiply the base by half the altitude. II. TO FIND AREA WHEN THREE SIDES ARE GIVEN. RULE. From half the sum of the three sides subtract each side separately; multiply the half sum and three remainders together, and extract the square root of the product. PROBLEMS. 1. Base equals 4.75 ft.; altitude, 3.5 ft.; what is area? Ans. 8.312 sq. ft. 2. Base equals 2 ft. 7 in.; altitude, 2 ft. 5 in.; what is area? Ans. 3.12 sq. ft. 3. Sides equal 3 ft., 4 ft., and 5 ft.; what is area? Ans. 6 sq. ft. 4 Sides equal 1 ft. 10 in., 2 ft., and 3 ft. 2 in.; what is area? Ans. 1 sq. ft. 101.9 sq. in. Parallelogram. TO FIND AREA WHEN BASE AND ALTITUDE ARE GIVEN. PROBLEMS. 1. Base equals 9 ft. 4 in.; altitude, 2 ft. 5 in.; what is area? Ans. 22.5 sq. ft. 2 Base equals 2 yds. 2.25 ft.; altitude 5 ft. 9 in.; what is area in square feet? Ans. 47.4375 sq. ft. Trapezoid. TO FIND AREA WHEN PARALLEL SIDES AND PERPENDICULAR DISTANCE BETWEEN THEM ARE GIVEN. RULE- Multiply half the sum of parallel sides by the perpendicular distance. PROBLEMS. 1. Parallel sides equal 2.25 ft. and 2.75 ft.; perp. dist., 2.5 ft.; what is area? Ans. 6.25 sq. ft. 2. Parallel sides equal 3 ft. 5 in., and 2 ft. 7 in.; perp. dist. 2 ft. 8 in.; what is area? Ans. 8 sq. ft. Circle. I. TO FIND AREA WHEN DIAMETER AND CIRCUMFERENCE ARE GIVEN. RULE.-Multiply the diameter by one-fourth of the circum ference. II. TO FIND AREA WHEN THE DIAMETER IS GIVEN. III. TO FIND AREA WHEN THE RADIUS IS GIVEN. PROBLEMS. 1. Diameter equals 221 ft., circumference 69.9 ft.; what is area? Ans. 388.82 sq. ft. 2. Diameter equals 2 ft. 5 in., circumference 7 ft. 7 in.; what is area? Ans. 4 sq. ft. 83.75 sq. in. 3. Diameter equals 3.2 inches; what is the area? Ans. 8.042 sq. in. 4. Diameter equals 1 ft. 3 in.; what is the area? Ans. 1.227 sq. ft. in. 5 Radius equals 5 in.; what is the area? Ans. 78.54 sq. 6. Radius equals 2.25 in.; what is the area? Ans. 15.9 sq. in. Convex Surface, Right Prism, or Cylinder. TO FIND AREA WHEN PERIMETER OF BASE AND ALTITUDE ARE GIVEN. RULE.-Multiply perimeter of base by the altitude. PROBLEMS. 1. The sides of base of a right prism equal 51, 6, 8, 9, and 10 in., the altitude 11 in.; what is area of convex surface? Ans. 3 sq. ft. 12.375 sq. in. 2. The diameter of a right cylinder equals 1 ft. 2 in., the altitude 1 ft. 9 in.; what is area of convex surface? Ans. 6 sq. ft. 92.6 sq. in. Convex Surface, Right Pyramid, or Cone. TO FIND AREA WHEN PERIMETER OF BASE AND SLANT HEIGHT ARE GIVEN. RULE.-Multiply sum of perimeter of bases by one-half the slant height. 1. PROBLEMS. Each side of a triangular pyramid equals 3 ft. 6 in., the slant height 18 ft.; what is area of convex surface? Ans. 94.5 sq. ft. 2. The diameter of the base of a conical tent equals 16 ft., the altitude 14 ft.; how many square yards of canvas in the tent? Ans. 45.016 sq. yds. Convex Surface, Frustum, Right Pyramid, or Cone. TO FIND AREA WHEN PERIMETERS OF BASES AND SLANT HEIGHT ARE GIVEN. RULE.-Multiply sum of perimeters of bases by one-half the slant height. PROBLEMS. Each side of lower base of a frustum of a quadrangular pyramid equals 10 in., of upper base 9 in., the slant height 20 in.; what is area of convex surface? Ans. 5 sq. ft. 40 sq. in. 2. The circumferences of the bases of a frustum of a right cone are 22 in. and 15.71 in., the slant height 26 in.; what is area of convex surface? Ans. 3 sq. ft. 58.23 sq. in. |