26. Ask a carpenter to show you how he finds the pitch of a roof by using a rule and steel square; also, how he finds the length of the rafters required.

27. Find the diameter of a circle whose area is 314.16 sq. in. 28. What is the radius of a circle equal in area to a rectangle containing 45239.04 sq. ft.?

29. A baseball diamond is 90 ft. square. Edward stands 10 ft. from second base directly between it and third base. How far does he have to throw to reach the home plate?

=

3 is

Cubes and Cube Root. The cube of 2 (written 23) is 2 × 2 X2 8. The cube of 3 (written 33) is 3 x 3 x 3 = 27. the cube root of 27. What is the cube root of 8? The cube root of 8 is written VS.

Read the first two lines below carefully. Then supply the missing numbers in the lines following.

The cube of 4 is 64 and the 64 is 4.

The cube of 6 is 216 and the V216 is 6.

The cube of 9 is ... and the V... is...

Your knowledge of cubes and cube root can be applied to volumes; for instance, if the edge of a cube is 3 units, its volume is 27 cubic units. If you know that the volume of a cube is 27 cubic units, you can find its edge by finding the cube root of 27, which is 3.

What is the edge of a cube whose volume is 1000 cu. ft.? 729 cu. in.?

What is the edge of a cube which contains 27 cu. yd.?

Cube Root of a Perfect Cube. Just as we were able to find the square root of a perfect square by factoring, so we can find the cube root of a perfect cube by factoring. For example:

Find the cube root of these perfect cubes by factoring:

II. What is the edge of a cube whose volume is 4096 cu. ft.? 1728 cu. ft.? 1331 cu. yd.?

There is a general method of finding the cube root of any number, just as there is a general method of finding the square root, but it is not one of the subjects discussed in an elementary arithmetic Right Prism. The figure at the right represents a right prism. ABCDE and FGHKL, called bases, are parallel, which means that they cannot meet however far

K

they may be extended; they are also equal polygons. The other boundaries, called lateral faces,

are rectangles. AF, BG, and so on, are called E lateral edges. In a right prism any lateral edge

C

may be taken as the altitude of the prism. Name the lateral faces of this prism.

There are prisms which are not right prisms; these are called oblique prisms. Only right prisms will be treated in this book; they will be called merely prisms. The cube and the rectangular solid, which are classes of right prisms, have already been treated.

B

Meaning of Lateral Area of a Prism. The sum of the areas of the lateral faces of a prism is called the lateral area. Right Circular Cylinder. The figure at the right may be thought of as having been formed by the A revolution of rectangle ABCD about BC as an axis. The figure is a right circular cylinder. The surface formed by the side AD is the lateral surface and its area is the lateral area. The surfaces bounded by the circles formed by A and D are the bases. The axis BC is the altitude.

D

There are cylinders which are not right circular cylinders, but they will not be considered in this book. Right circular cylinders will be spoken of merely as cylinders.

Areas. The lateral surface of a prism or of a cylinder may be thought of as unrolled like a rectangular sheet of paper. How

does the perimeter of the base of the prism or the circumference of the base of the cylinder compare with the base of the rectangle? How does the altitude of the prism or the cylinder compare with the altitude of the rectangle? As you know how to find the area of a rectangle, how would you say the lateral area of a prism might be found? the lateral area of a cylinder?

1. The lateral area of a prism is equal to the product of its altitude and the perimeter of the base.

2. The lateral area of a cylinder is equal to the product of its altitude and the circumference of the base.

NOTE. See note, page 121.

Formulae. If H is the number of units in the altitude of a prism (see page 250) and P the number of units in its perimeter, show that its lateral area equals PX H. If S represents its lateral area, then:

S = P X H. (r)

If, in a cylinder, S represents its lateral area, H the number of units in its altitude, and P the number of units in its perimeter (circumference), show that:

S = P X H. (2)

If P represents the circumference of the base of the cylinder and R its radius, show that P = 2 πR. If 2 TR is substituted for P in equation (2), show that:

S = 2 πRH. (3)

In equation (1), what facts must be known concerning a prism before you can find its lateral area? What facts must be known concerning a cylinder before you can find its lateral area using equation (2)? using equation (3)?

AREAS OF PRISMS AND CYLINDERS

1. Find the lateral area of a prism each side of whose square base is 6 ft. and whose altitude is 3 ft.

2. Find the lateral area of a cylinder whose altitude is 6 in. and whose radius is 4 in.

3. Find the lateral area of a prism whose base is a square 4 ft. long, and whose altitude is 6 ft.

4. Find the lateral area of a prism whose base is a regular hexagon 2 ft. on a side, if the altitude of the prism is 8 ft.

5. Find the lateral area of a cylinder whose radius is 4 in. and whose altitude is 1 ft.

6. Find the number of square feet in the entire surface of a cube whose edge is 3 in.

Volumes. The rectangular solid (Figure 1), the prism (Figure 2), and the cylinder (Figure 3) have equal altitudes and the

FIG. 1.

FIG. 2.

FIG. 3.

same number of square units in their bases. How many square units in the base of Figure 1? Figure 2? Figure 3? In Figure 1 a layer 1 unit high contains how many cubic units? How many cubic units in each layer 1 unit high in Figures 2 and 3? How does the number of cubic units in 1 layer 1 unit high in Figure 1 compare with the number of cubic units 1 layer high in Figures 2 and 3?

As Figures 1, 2, and 3 have the same altitudes, how does the number of layers in the three figures compare? Then how do the volumes of the three figures compare? How would you say the number of cubic units in each of these figures, that is the volume of each, might be found?

The volume of a prism or of a cylinder is equal to the product of its base and altitude.