11. Find the total volume of three square pyramids, the altitude of each being 12 inches, and the areas of their bases being 25 sq. in., 225 sq. in., and 75 sq. in., respectively.

√75

15

5

12. Find the number of cubic feet in a block of stone whose shape is that of a frustum of a square pyramid 4 feet high, each side of the upper base measuring 3 feet, and each side of the lower base 5 feet.

1293. The volume of the frustum of a pyramid is equal to the sum of the volumes of three pyramids, each having an altitude equal to that of the frustum; the base of one of them being equal in area to that of the lower base of the frustum, the base of a second being equal in area to that of the upper base of the frustum, and the base of a third being a mean proportional between the area of the other two.

Base of first = 3 x 3 sq. ft.; of second, 5×5 sq. ft.; of third, √9 × 25 sq. ft. 15 sq. ft.

NOTE. The mean proportional between two numbers is equal to the square root of their product.

13. Find the volume of the frustum of a square pyramid, its upper base containing 64 square inches, and its lower base 196 square inches, its altitude being 18 inches.

1294. Note that the mean proportional between 64 and 196 is 8 × 14, or 112. Since each is multiplied by one-third of the altitude, the operation is shortened by adding together the three areas, 64, 196, and 112, and multiplying their sum by one-third of 18.

Calling the altitude A, the side of the large square S, of small square s, the volume V, we have

1295. The volume of the frustum of a cone is found in the same way as that of the frustum of a pyramid.

altitude (area upper base + area lower base + area mean proportional).

Calling the radius of the upper base r, and that of the lower base R, the area of the upper base will be r2, of the lower base R2, of mean proportional πr R.

V=} A(πr2 + πR2 + πîR)

Since (or 3.1416) is a common factor, we can save time by first adding r2, R2, and rR, and then multiplying by π.

V = Aπ(r2 + R2 + rR).

3

14. The diameters of the bases of the frustum of a cone measure 8 and 15 inches, respectively; the altitude is 18 inches. Find the volume.

15. How many cubic inches of water will a pan hold, whose lower base is 12 inches in diameter, whose upper base is 16 inches in diameter, and whose depth is 6 inches? How many gallons?

1296. The pupils should make a frustum of a square pyramid of convenient size, and the three corresponding pyramids, as given in the rule. Fill the latter with sand, and pour the contents of all three into the frustum.

To make the frustum, draw two concentric circles. Lay off equal arcs, AB, BD, DE, EF Draw the chords and radii from the extremities of each chord. Draw the chords ab, bd, de, and ef. Cut out, after constructing a square for either the upper or the lower base, and taking care to provide flaps for pasting.

α

C

B

G

H

To get a mean proportional between ab and AB for one side of the base of the third pyramid, lay off a line IJ equal in length to ab + AB. On this line construct a semicircle.

Make IK equal to AB, and at Kerect a perpendic-
ular KM. KM is a mean proportional between A
ab and AB.

1297. Oblique Prisms.

M

Bab

KJ

G

We have seen that a rectangle, ABCD, and a parallelogram, EFGH, are equal in area when the bases, AB and EF, and the altitudes, AD and HX, are equal, each to each. This can be shown by cutting both out of paper, and by shifting the tri

angle HEX to the right side of the parallelogram.

1298. In a somewhat similar way, we can show that an

oblique prism is equal to a

right prism that has an equal base and an equal altitude.

Make from a potato or a tur

nip an oblique prism having

rectangular bases, and change it to a right prism of the same height by cutting and shifting a portion.

1299. The volume of any prism (or cylinder) is found by multiplying the area of the base by the altitude.

1300. In the same way it can be shown that the volume of any pyramid or cone is equal to the product of the area of the base by one-third the altitude.

1301. The Sphere.

A sphere may be considered as made up of a great number of pyramids whose bases together make the surface of the sphere,

and whose vertices all meet at the center of the sphere, making their altitudes each equal to the radius of the sphere.

The volume of a sphere is equal, therefore, to its surface X radius.

16. Find the volume of a sphere whose radius is 9 inches.

17. What is the volume of a sphere whose diameter is 9 inches?

Find the volume of a cone whose altitude is 9 inches, diameter of base 9 inches.

How does the volume of the cone compare with the volume of the sphere?

How does the volume of the sphere compare with the volume of a cylinder 9 inches in diameter and 9 inches high?

Make a paper

1302. Take a clay sphere of a convenient size. cylinder that will exactly contain it, the height of the cylinder being equal to the diameter of the sphere. Make a hollow cone of the same diameter and altitude.

Place the sphere in the cylinder, carefully fill the cone with water, and pour it into the cylinder, which should then be filled to the top, showing that the volume of the cylinder is equal to that of the sphere and the cone together.