32. What equals the volume of a pyramids of equal altitude, supposing the sum of their bases to be 30 sq. in. and the altitude 8"?

33. How can the area of the base of a pyramid be found if the number of units in its volume and the number of units in its altitude are given? How can the volume of a pyramid be found if the number of units in its altitude and the number of units in its base are given?

Regular Polyhedrons. 1. In what are the polyhedrons on page 200 alike?

2. Describe the greater radius of each. of the greater radii of each?

3. Describe the less radius of each. the less radii of each respectively?

What is true

What is true of

b

M

4. In the regular polyhedron M, a is a less radius and b a greater radius. Show the greater and less radii of this room. 5. Are all of the greater radii of the room equal? Are all of the less radii equal? Look carefully. Is it the

same distance from the center of the room to each of the four walls as to the ceiling and to the floor?

6. What is true of the greater and less radii respectively of a regular polyhedron ?

7. What is a regular polyhedron ?

A regular polyhedron is a solid whose greater and less radii are respectively equal.

8. In which solid is there the greatest difference in the length of the greater and less radii? In which are all the radii equal?

9. What things are respectively equal in a regular polygon? In a regular polyhedron ?

10. Into what equal solids can a regular polyhedron be separated?

1. Into how many equal pyramids can a cube be separated?

2. In this cube, find the edges and the base of each of the six pyramids into which the cube may be separated.

3. Find the edges and the base of each of the six pyramids into which this room may be separated.

4. What is the name of the altitude of the pyramids into which a regular polyhedron may be separated?

5. What is the number of sq. in. in the sum of the bases of the six equal pyramids into which a 12" cube may be separated? What is the altitude of the pyramids in a 12" cube? The number of units in its volume equals how many times the number of units in the base? By what must the number of units in the surface of a cube be multiplied to equal the number of units in the volume of the pyramids to which it is equal?

6. The number of units in the volume of any regular solid equals the product of what numbers?

7. If a sphere is a regular polyhedron, of what is it composed?

8. What equals the number of units in the volume of a sphere? Is it necessary to use the term less radius in giving a rule for finding the volume of a sphere? Why not?

9. Review the work on regular polygons.

Before arriving at an easy method of finding the volume of a sphere, one must know how to find its surface.

The altitude and the diameter of the base of the cylinder D are each equal to the diameter of the sphere 0. See page 204.

If a rectangle equal to the lateral surface of the cylinder D be placed about the sphere O, it could, by great skill in cutting and rearranging, be made to cover exactly the sphere 0.

1. Cut a rectangle equal to the lateral surface of a cylinder whose altitude and diameter are each 1 in. 2. Cut a rectangle equivalent to the surface of a sphere 1 in. in diameter.

3. Cut a rectangle equivalent to the surface of a sphere in. in diameter.

4. The surface of the spherein. in diameter equals what part of the surface of the sphere 1 in. in diameter?

5. Draw a rectangle 2 in. wide equivalent to the surface of a sphere 2 in. in diameter.

6. What is the ratio of the surface of a sphere 2 in. in diameter to the surface of a sphere 1 in. in diameter? to the surface of a sphere 4 in. in diameter ?

7. If a is the diameter of a sphere, what may be the dimensions of a rectangle equal to the surface of the sphere?

The altitude and the diameter of the cylinder are each equal to the diameter of the sphere and of the hemisphere. The length of the cord which covers the curved surface of the hemisphere equals the length of the cord which covers one-half of the lateral surface of the cylinder. Compare the lateral surface of the cylinder with the surface of the sphere. See page 204.

8. Observe spheres and cylinders and compare surfaces. 9. What are the dimensions of a rectangle equivalent to a circle 1 in. in diameter? 2 in. in diameter?

10. What are the dimensions of a rectangle equivalent to a great circle of a sphere 1 in. in diameter ?

11. What is the ratio of a great circle of a sphere to the surface of the sphere?

12. A great circle of the earth equals what part of its surface?

13. Discover the ratios of the surfaces of the solids in the following cut:

14. Draw a rectangle equivalent to the surface of a sphere 1 in. in diameter. Make rectangle 1 diameter wide.

15. Express in diameters the dimensions of a rectangle equivalent to the surface of any sphere. What is the ratio of this rectangle to the square of the diameter? 34 or 22 is the ratio of the surface of a sphere to what?