Find the areas of circles whose circumferences are as follows:

17. Find the perimeter and area of a square circumscribed about a circle whose diameter is 8 ft.

18. Find the difference between the cir- 4 cumference of a circle and the perimeter of a circumscribed square whose side is 12 ft.

19. Find the difference between the area of a circle and the area of a circumscribed

square whose length is 10 ft.

A Circumscribed
Square

20. A square courtyard, as large as possible, is made within a circle whose diameter is 66 ft., and the remainder of the circle is sodded. How many

square feet are sodded? Answer correct

to the nearest .1 ft.

SUGGESTION.

The diameter of the circle is

the diagonal of the square.

21. A cow is fastened by a rope 20 yards long to a stake in the center of

a square lot containing 1600 sq. yd. Over how much land can she graze?

22. Find the area of the largest square that can be cut from a circular board whose diameter is 20 in.

MENSURATION OF SOLIDS

General Definitions

464. A solid is any bounded portion of space; as, a cube.

465. The volume of a solid is the number of cubic units it contains.

Right Prisms and Cylinders of Revolution

Definitions

466. A right prism is a solid bounded by two equal and parallel polygons, called bases, and by rectangles called

lateral faces.

467. The sum of the areas of the lateral faces is called the lateral area.

The perpendicular line joining the bases is the altitude; as, mn.

n

K

REMARK. For treatment of the cube and rectangular solid, which are classes of right prisms, F see pp. 192-194. For partial treatment of the cylinder of revolution, see pp. 232-233. In this book right prisms and cylinders of revolution only are treated; they will be called merely prisms and cylinders.

Areas

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

paper, the perimeter of the base of the prism or the circumference of the base of the cylinder being the base of the rectangle, and the altitude of the prism or the cylinder, the altitude of the rectangle.

The area of a rectangle being the product of the number of units in the altitude and the number of units in the base, it follows that:

469. I. The lateral area of a prism is equal to the product of the number of units in the altitude and the number of units in the perimeter of the base.

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

Volumes

470. The rectangular solid (Fig. 1), the prism (Fig. 2), and the cylinder (Fig. 3) have equal altitudes and the same number of square units in their bases.

Fig. 1.

Fig. 2.

Fig. 3.

In Fig. 1 it is seen that a layer 1 unit high contains as many cubic units as there are square units in the base; hence for each layer one unit high in Figs. 2 and 3 there are as many cubic units as there are square units in the base. In each figure the number of cubic units is equal to the number of cubic units in a layer multiplied by the number of layers; it follows that:

471. The volume of a prism or of a cylinder is equal to the product of the number of units in the altitude and the number of square units in the base.

472. If S denotes the lateral area, V the volume, H the number of units in the altitude, P the number of units in the perimeter of the base, and B the number of square units in the base of a prism, (2) V=Bx H.

(1) S=Px H,

If S denotes the lateral area, V the volume, H the number of units in the altitude, R the number of units in the radius of the base of a cylinder,

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

=

SUGGESTION. In a prism, SPx H. [§ 472, 1]

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

SUGGESTION. In a cylinder, S=2 TRH. [$ 472, 3]

.. S= 2 × 3.1416 × 4 × 6.

3. Find the volume of a prism the area of whose base is 64.96 sq. ft. and whose altitude is 8 ft.

SUGGESTION. In a prism, V=B × H. [§ 472, 2] .. V= 64.96 x 8.

4. Find the volume of a cylinder whose altitude is 5 in, and whose radius is 6 in.

SUGGESTION. In a cylinder, V=TR2 H. [§ 472, 4]

.. V3.1416 x 62 x 5.

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

6. 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.

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

8. Find the volume of a prism whose base is a square 8 ft. on a side, and whose altitude is 18 in.

9. What is the volume of a cylinder whose radius is 6 in., and whose altitude is 1 ft. 6 in. ?

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

II. Find, correct to the nearest gallon, the capacity of a tank in the form of a cylinder, the

radius of whose base is 3 ft. and whose altitude is 6 ft.

12. This figure represents a cylindrical silo 20 ft. deep and 10 ft. in diameter. How many cubic feet will it hold?

13. What is one side of the base of a square prism whose volume is 150 cu. ft. and whose altitude is 6 ft.?

14. Find the weight of a granite pillar of uniform size, whose base is a circle with a radius of 12 in. and whose height is 14 ft., if 1 cu. ft. of granite weighs 165ğ lb.