249. A circle may be regarded as made up of a large number of triangles, the sum of whose bases forms the circumference of the circle, and whose altitude is the radius of the circle. Hence the RULE. I. To find the area of a circle, multiply the circumference by half the radius. Area Circumference X Radius. Since Circumference = 2 X Radius X 7, Area = 2 × Radius X X Radius, Area Radius', or 7 R2. = or, Hence, II. To find the area of a circle, multiply the square of the radius by 3.1416. WRITTEN EXERCISES. Find the area of a circle whose 1. Radius is 10 ft. 2. Diameter is 40 rd. 3. Circumference is 18 ch. 4. Radius is 3 ft. 6 in. 5. A horse is tied to a stake by a rope 18 ft. long; over how many square yards can he graze? 6. A has a garden that is 40 rods square, and B has a circular garden 40 rods in diameter; what is the difference in area? 7. F has a circular fish-pond 30 yards in diameter: he makes a circular gravel walk around it 6 ft. wide; what is the area of the walk? 8. There is a circular farm whose diameter is 30 chains; what is it worth, at $55 an acre? 250. A square is inscribed in a circle when each of its angles is in the circumference. Since A B C is a right triangle, But A B WRITTEN EXERCISES. 1. Find the side of the largest square that can be cut out of a circular board 16 in. in diameter. = or, Hence the A B2 + BC2 BC. Hence 2 A B2 = A C2, or 256. 128, = = = = D A C. A 11.361+ in., Ans. RULE. To find the side of the largest square that can be cut from a circle, divide the square of the diameter by 2 and extract the square root. 2. Find the side of a square that can be cut from a circle 80 inches in diameter. 3. Find the side of a square field that can be made from a circular field 40 rd. in radius. 4. How large a square can be cut out of a circular yard 100 ft. in circumference? 5. Find the difference between the area of a circular field 60 rd. in circumference and that of the largest square field that can be cut from it. 6. The radius of a circle is 1.5 inches; what is the side of the inscribed square? SOLIDS. 251. A Solid is that which has length, breadth, and thickness. The Volume of a solid is the number of times that it contains another solid regarded as the unit of measure. 252. The Prism. A Prism is a solid whose ends are equal polygons and whose sides are parallelograms. The bases are the equal polygons, and the parallelograms form the convex surface. Prisms are named from their bases. A Rectangular Parallelopiped, or Square Prism, is a prism whose six faces are all rectangles. A Cube is a rectangular parallelopiped whose six faces are squares. all A Right Prism is a prism whose edges are perpendicular to its bases: the prisms in the cut above are right prisms. RULE. The convex surface of a right prism is equal to the perimeter of the base multiplied by the altitude. WRITTEN EXERCISES. 1. Find the convex surface of a triangular prism the three sides of whose base are 6, 8, and 10 in., respectively, and altitude 40 in. 2. Find the convex surface of a square prism the sides of whose base are each 10 in. and altitude 50 in. 3. Find the convex surface of a pentangular prism, each of the sides of the base being 8 ft. and the altitude 12 ft. 4. Find the surface of a cube whose edge is 6 inches. 5. Find the entire surface of a triangular prism the sides of whose base are 8, 10, and 12 in., respectively, and altitude 20 in. NOTE. To find the entire surface, add the area of the bases. 6. Find the entire surface of a triangular prism the sides of whose base are each 20 ft. and altitude 40 ft. 253. Let the figure ABCDEF represent a rectangular parallelopiped, or square prism, whose length is 4 in., width 3 in., and altitude 9 in.; what is its volume? If we divide the length, width, and height into inches and pass planes through these points of division, the prism will be divided into equal cubes. There will be 12 cubic inches in one layer; and since there are 9 such layers, the volume will equal 9 times 12 cubic inches, or 108 cubic inches. Hence the 9 B E D RULE. The volume of a rectangular prism is equal to the area of the base multiplied by the altitude. A rectangular prism may be divided into two equal triangular prisms, each having the altitude of the rectangular prism and a base equal to half the base of the rectangular prism. Hence the RULE. The volume of a triangular prism is equal to the area of the base multiplied by the altitude. Any prism may be divided into triangular prisms, each of which is equal to the area of the base multiplied by the common altitude, and their sum, which is the volume of the prism, is equal to the sum of their bases or the base of the prism multiplied by the altitude. Hence the general RULE. The volume of any prism is equal to the area of the base multiplied by the altitude. WRITTEN EXERCISES. 1. Find the volume of a square prism whose altitude is 40 ft. and the sides of the base 4 ft. 2. What is the volume of a rectangular prism whose altitude is 60 ft. and the sides of the base 4 ft. and 8 ft., respectively? 3. Find the volume of a triangular prism whose altitude is 100 ft. and the sides of the base each 20 ft. 4. Find the volume of a triangular prism whose altitude is 40 ft. and the base a right triangle, with the sides about the right angle 8 ft. and 12 ft., respectively. 5. The water in a stream flows at the rate of 3 miles an hour; if the stream is 4 ft. wide and 3 ft. deep, how many cubic yards of water pass a given point in 24 hours? |