Besides being able to find the volume of solids, we must also be able to find the area of their surfaces, for it is only in this way that we can tell how much lumber is required to construct a packing case, how much cardboard is needed to make a shoe box, etc. In the prism shown, the surface around is a rectangle 6" long by 4" high, containing 24 sq. in. The two bases are rectangles 2" long 1" wide, each containing 2 sq. in. or a total of 4 sq. in. The entire surface therefore contains 24 sq. in. + 4 sq. in., or 28 sq. in. Exercise 41-Written. Make drawings of the surfaces of the following rectangular prisms (to scale of 1" to 1", or " to 1") in the same way as is shown in Figure I, and find the area of the entire surface of each: 7. How many square inches of cardboard are needed to make a box for an umbrella, the size of the box being 4" wide, 4" deep, 4′ high, if no allowances are made? 8. How many square feet of zinc are needed to line a case 2' deep, 3' wide, 4' long? 9. How many square yards of paper are needed to line a case 4' X 4' X 4'? 10. How many board feet of 1" lumber are needed to make a box with a cover, the outside dimensions being 20" x 30" x 40"? 11. Find the area of the entire surface of a prism 2" X 6" X 8". 12. Find the area of the entire surface of a cube 81" each way. 13. Find the area of the entire surface of a cube having 5" edges. 14. Find the area of the entire surface of a prism 4" X 4" X 6". 15. Find the area of the entire surface of a cigar box 2" X 6" X 8". 16. Find the area of the four walls, floor, and ceiling of a room 14' X 22' X 10'. Make a cylinder out of heavy paper. A "cylinder" is a solid bounded by a uniformly curved side and two parallel circular ends or bases of equal size. Pipes, tin cans, and round pencils usually have the shape of cylinders. To find the area of the entire surface of a cylinder, we must find the sum of the areas of the curved side and the two circular bases. You already know how to find the areas of the two circular bases; therefore, the only new point for you to learn is how to find the area of the curved side. Can you tell how? (Use the terms of the cylinder.) In the drawing here shown, we find the circumference of the base to be 11"; therefore, the curved side is 11" long and 6" wide, its area being 11 sq. in. X 6 = 66 sq. in.; the area of each base is of 11 sq. in. X (31⁄2 ÷ 2) = 95 sq. in. or 194 sq. in. in both bases, or 66 sq. in. 191 sq. in. 85 sq. in. in the area of the entire surface. Exercise 42-Written. = 1. Cut a paper into an oblong 4" X 8", and form a hollow cylinder 8" long; what is the circumference of this cylinder? Watch the bases carefully. 2. What is the area of the curved side of this cylinder? 3. What is the diameter of one of the bases of this cylinder? 4. What is the area of each of the two bases? 5. What is the total area of the curved side and the two bases? 6. A section of rain pipe is 6" in diameter and 36" long; what are the dimensions of a sheet of galvanized iron of the correct size to make this pipe, allowing 1" for the seam? 7. What is the area of this sheet of iron? 8. A cylindrical smoke-stack is 12' high and 1' in diameter; what is the area of its curved side? How many square feet of metal were needed to make this stack, if the metal overlaps 3" at the seam? 9. What is the area of the curved side of a pencil 7′′ long and 3" in diameter? 10. A mailing tube is 20" long and 3" in diameter; what is the area of its curved side? How many such tubes could be cut from a sheet of cardboard 20" by 37.7", allowing nothing for the seam? Exercise 43—Oral. 1. A rectangular prism has how many sides? How many bases? How many surfaces? 2. How do we find the area of the entire surface of a rectangular prism? 3. Name the dimensions of each of the surfaces of a prism 2" X 4" × 8". 4. How many sides has a cylinder? bases? How How many 5. What is the shape of the side of a cylinder when the curve is straightened? 6. What is the shape of each of the bases of a cylinder? 7. Take a cylinder and tell how we find the dimensions of the curved side. 8. How do we find the area of the curved side of a cylinder? 9. How do we find the area of each of the two bases of a cylinder? 10. Take a cylinder and tell the class all you can about it. Take 2 minutes. 11. Take a prism and tell the class all you can about it. Take 2 minutes. 12. A rectangular prism has how many edges? 13. The dimensions of a rectangular prism are 5" X 10" X 15"; how many of its edges are 5" long? How many of its edges are 10" long? How many of its edges are 15" long? 14. A cylinder has how many edges? 15. The diameter of a cylinder is 10"; what is the length of each of its edges? |