12. How many square feet in the roof of a house 50 ft. long and the rafters on each side of the roof 25 ft. long? 50 x 50 = 2500 sq. ft. 13. How many shingles, each shingle covering 6 in. by 4, will it take to roof the house? 6 2500 sq. ft. x XAA 24 = 15000 shingles. 14. How many square yards of plastering in a house 20 ft. front, 60 ft. deep and 36 ft. high; three stories in height ? and how many shingles to roof it, each covering 8 in. by 6, and the entire length of the roof 64 ft., the width 20 ft. ? 1st Ans. 1040 sq. yd. plastering. 2d Ans. 3840 shingles. 15. What is the area of a triangle whose base is 24 in. and the perpendicular distance from the vertical angle to the base 9 in. ? 12 x = 108 inches. MEASUREMENT OF CIRCLES. DEF.—A Circle is a plane figure bounded by a curved line called the circumference, every point of which is equally distant from a point within, called the centre. PROBLEMS. 1. To find the circumference of a circle, having given the diameter. Multiply the diameter by 3.1416 (Geometry). 2. To find the diameter, having given the circumference. Divide the circumference by 3.1416. 3. To find the area of a circle. Multiply the circumference by one-fourth the diameter. Let D = diameter, then D x 3.1416 = circumference, D and D x 8.X4X6 X = D2 x .7854; 4 .7854 or, multiply the square of the diameter by .7854. EXAMPLES. 1. Find the circumference of a circle whose diameter is 2 ft. Ans. 6.2832 ft. 2. Find the diameter of a circle whose circumference is 6.2832 ft. Ans. 2 ft. 3. Find the area of a circle whose diameter is 2 ft. a 2 x 2 x .7854 = 3.1416 sq. ft. PROB. 4.- Find the space between two concentric circles. Find the area of both circles, and the difference of the areas will be the area of the space. Ex. 4. Two circles have the same centre; the larger one has a diameter of 4 feet and the smaller one of 2 ft. What is the area of the space between them ? 4 X 4 = 16 12 x .7854 = 9.4248 sq. ft. CUBIC MEASURE, DEFINITIONS. 1. A Right Prism is a solid which has equal and parallel polygons for its bases, and its edges are perpendicular to the bases. 2. If the bases are squares and each face square and equal to a base, the figure is called a Cube. 3. When the bases are circles, the figure becomes a Cylinder. 4. Figures with a polygon for a base and tapering to a point are called Pyramids. 5. If the upper part is cut off by a plane parallel to the base, the lower part is called the Frustum of a Pyramid. 6. When the base is a circle and the figure tapers to a point, it is called a Cone. 17. If the upper part is cut off by a plane parallel to the base, the lower part is called the Frustum of a Cone. PROBLEMS. a To find the volume of a prism or cylinder. REM.—The lateral surface of a prism or cylinder is the product of the altitude and the perimeter of the base. EXAMPLES. 1. What is the volume of a prism whose base contains 6 sq. ft. and altitude 5 feet? 6 sq. ft. x 5 ft. = 30 cu. ft. 2. What is the volume of a cube whose edges are each 3 feet? 3x3 x3 = 27 cu. ft. 3. What is the volume of a cylinder whose base contains 12 sq. ft. and its altitude is 5 ft. 12 sq. ft. x 5 ft. = 60 cu. ft. PROBLEM. To find the volume of a pyramid or cone. EXAMPLES. 1. What is the volume of a pyramid whose base contains 15 sq. ft. and altitude 9 ft. ? 15 x3 = 45 cu. ft. 2. The volume of a pyramid is 90 cu. ft. and the altituds 9 ft. ; what is the area of basa ? Ans. 30 sq. ft. 3. What is the volume of a cone whose base contains 36 sq. ft. and its altitude is 18 ft. ? Ans. 216 cu. ft. PROBLEM. To find the volume of the frustum of a pyramid or cone. The volume of the frustum is equivalent to three pyramids or cones, one having the lower base for its base, the second having the upper base for its base, each hav ing for its altitude the altitude of the frustum, and the volume of the third pyramid or cone is a mean proportional between the other two. Therefore, extract the square root of the product of the areas of the lower and upper bases, and add together this root and the areas of the two bases and multiply their sum by one-third the altitude of the frustum, and this product will be the volume of the frustum. EXAMPLES. 1. The areas of the lower and upper bases of the frustum of a pyramid are 16 sq. ft. and 9 sq. ft., and the altitude is 12 ft.; what is the volume of the frustum. 16x9 = V144 = 12 16 37 x 4 = 148 cu. ft. DEFINITIONS. 1. A Sphere is a solid with a curved surface, every part of which is equally distant from a point within called the centre. 2. The Axis or Diameter of a sphere is a straight line passing through the centre and terminated at both ends by the surface. 3. A Radius is one-half the diameter, or a straight line drawn from the centre to any point of the surface. |