18. Show that the convex surface of the frustum of a pyramid is equal to one-half the sum of the perimeters of the upper and the lower bases multiplied by the slant height.

19. Draw the pattern of a small shade for a candle. Make the upper opening 11⁄2 inches in diameter, the lower one 2 inches in diameter, and the slant height 2 inches.

20. How many square inches of tin will be required to make a pan, its upper base being 9 inches in diameter,

the lower base 6 inches in diameter, and the slant height 4 inches?

(Do not forget the bottom of the pan.)

1285. The frustum of a cone may be considered the frustum of a pyramid whose bases contain a very great number of sides. The convex surface of the frustum of a cone may, therefore, be found by multiplying the half sum of the circumference of the two bases by the slant height.

21. Find the convex surface of a frustum of a cone, the circumferences of the bases being 15 inches and 20 inches, respectively, and the slant height 10 inches.

22. How many square yards are there in the entire surface of a frustum of a cone, the radius of the upper base (r) being 3 yards, of the lower base (R) 5 yards, and the slant height 6 yards?

=

Tr2; of lower base =π π R2.

Surface of upper base Entire surface: =TX what? Multiply only once by 3.1416. 23. The diameter, AB, of the upper base of the frustum of a cone measures 6 feet, CD measures 8 feet, the slant height AC measures 9 feet. Find the slant height EA of the part cut from the cone C in making the frustum.

Let EA

=

x; ECx+9; AB=6; CD 8.

The triangles EAB and ECD are similar.

E

B

24. Find the convex surface of the whole cone, ECD, and the convex surface of the part cut off, EAB.

CG, CK, CD, CF, CA, and CI are radii; AD and FG are diameters.

G

1287. If a sphere be cut through at any part, the cut surface will be a circle. When the cutting plane passes through the center of the sphere, the circle is called a great circle; other circles are called small circles.

FXGC is a great circle; HYIB and JLEZ are small circles.

25. Find the length of an arc of 60° of a great circle of a sphere whose circumference is 25,000 miles.

26. Calling the arc AI in the preceding figure, 30°, the angle BCI will measure 30°. Calculate the radius BI of the small circle when the radius CI of the large circle is 4,000 miles. (IAH = arc of 60°; IH = chord of 60°.)

27. If I is 60° from G, a point on the equator, find the length of the circumference of the small circle HYI, assuming the circumference of a great circle to be 25,000 miles.

28. What is the ratio between the length of a degree on the small circle HYI, and the length of a degree of a great circle?

29. Calculate the radius of a small circle formed by passing a plane parallel to GCXF through a point on GA 45 degrees from G.

1288. Surface of Sphere.

We have seen (Art. 1151) that it may be experimentally shown that the surface of a sphere is equal to the

surface of four of its great circles.

Calling the radius of the sphere R, its surface is 4π R2.

30. Find the surface of a sphere whose diameter is 6 inches.

31. How many square inches are there

in the convex surface of a hemisphere whose radius is 3 inches? What is the area of the great circle that forms the base of the hemisphere?

Find the entire surface of the hemisphere.

32. Is there any difference between the convex surface of a sphere and its entire surface? Why?

VOLUMES.

1289. Prisms and Cylinders.

1. How many one-inch cubes will cover the base of a box 4 inches by 3 inches? If the box is 2 inches deep, how many oneinch cubes will it contain? How many cubic

inches are there in the volume of a right prism whose base is a rectangle measuring 4 inches by 3 inches, and whose altitude is 6 inches?

2. If the above hollow prism were divided into two equal parts by a thin partition extending from a vertical edge to one diagonally opposite, how many cubic inches of sand would each part contain?

3. How many cubic inches are there in the volume of a prism whose base is a right-angled triangle 3 by 4 by 5 inches, and whose altitude is 6 inches?

4. Find the volume of a triangular prism, the area of the base being 6 square inches, and the altitude 6 inches.

Find the volume of a triangular prism, each side of whose base measures 6 inches, its altitude being 8 inches.

5. What are the solid contents of a pentagonal prism formed by fastening together three triangular prisms whose bases contain, respectively, 12, 16, and 18 square inches, the altitude of each being 15 inches?

6. If a very great number of triangular prisms of the same height are united so as to form a cylinder whose base contains 12.5664 square inches, and whose altitude measures 5 inches, what are the solid contents?

1290. Pyramids and Cones.

With a center at C, and a convenient radius, describe an arc AB. Mark off four equal portions v, w, x, and y; and draw the equal chords. Cutting out CAvwxy, with an additional narrow

strip along Cy for gumming, and creasing along the lines Cv, Cw, Cx, and Cy, we can fold the paper into a square pyramid.

Measure its altitude, and make a square prism of equal altitude and with an equal base.

Filling the pyramid with sand, and pouring the sand into the prism, it will be found

A

v

B

บ

that the latter will contain the contents of the former three times; that is, the volume of a square pyramid is one-third that of a square prism having an equal base and an equal altitude.

The same ratio will be found true in the case of a triangular, or any other pyramid, as compared with the corresponding prism, and of the cone as compared with a cylinder.

1291. The volume of a pyramid or of a cone is equal, therefore, to the area of the base multiplied by one-third of the altitude.

1292. Frustums of Pyramids and Cones.

7. Find the volume of a square pyramid whose altitude is 12 inches, one side of the base measuring 6 inches.

Find the volume of a square pyramid whose altitude is 6 inches, one side of the base measuring 3 inches.

3

3

6

6

8. Find the volume of the frustum of a square pyramid whose altitude is 6 inches, one side of the upper base 3 inches, and one side of the lower base 6 inches.

E

9. A square pyramid whose altitude measures 18 inches, and each side of whose base measures 15 inches, is divided into two parts by a plane, FGHI, parallel to the base, the distance, EJ, of the plane from the vertex, E, being 6 inches.

The ratio between the edge, EB, of the whole pyramid and the edge, A EG, of the part cut off will be equal

to that between EJ and EK; that

B

F

JH

G

K

C

is, 6:18 1:3. The same will be the ratio between BC and GH, and the latter will be one-third

of 15 inches long, or 5 inches.

Find the volume of the large pyramid and that of the small pyramid.

10. Each side of the upper base of the frustum of a square pyramid measures 5 inches; each side of the lower base measures 15 inches; the perpendicular distance between them measures 12 inches. Find the solid contents.

Find the convex surface of the above frustum. Find its entire surface.

NOTE.What is the slant height?