7. The lateral surface of the frustum of a cone measures 565.488 sq. in.; the diameter of its upper base is 10"; its slant height is 12"; what is the diameter of its lower base?

8. The lateral surface of the frustum of a cone

measures 311.0184 sq. ft.; the radius of its upper base is 4 ft.; the radius of its lower base is 5 ft.; what is its slant height?

9. The upper base of the frustum of a pyramid is a 10"; its lower base is a 22"; its altitude is 8"; what is its slant height?

10. The upper base of the frustum of a cone is 6" in diameter; the lower base is 60" in diameter; its slant height is 45"; what is its altitude?

A "sphere" is a round solid bounded by a uniformly curved surface, every point of which is equally distant from a point within, called the center.

The circle made by cutting a sphere into two equal parts or hemispheres, by a plane passing through the center, is called a "great circle" of the sphere. Name a great circle of the earth.

The distance from surface to surface through the center is the diameter of the sphere, and is the same as the diameter of the great circle.

The distance from the center to any point on the surface is the radius of the sphere, and is the same as the radius of the great circle.

The circumference of a sphere is the same as the circumference of its great circle. The circumference is the longest curved line around the sphere.

By placing a tack in the center of the curved surface of one of the hemispheres, and another tack in the center of the flat surface of the other hemisphere, and winding cord carefully around each to cover the two surfaces, you will find that exactly twice as much cord is required to cover the curved surface as is required to cover the flat surface; hence, the area of the curved surface of the entire sphere equals four times the area of its great circle. The area of the great circle equals TR2; therefore, the area of the curved surface equals 4πR2. You can also prove this by cutting several circles out of paper; make the circles the same diameter as the diameter of a wooden ball (you can cut an old croquet ball in half); tear one circle at a time into small pieces and paste to cover the ball. How many circles are needed?

If a sphere were cut into small pyramids as is shown in the illustration, each of the pyramids would have an altitude equal to the radius of the sphere, and the combined area of the bases of the small pyramids would be approximately the same as the area of the surface of the sphere. Since we find the volume of a pyramid

by multiplying the area of its base by the altitude, we can find the volume of a sphere by multiplying the area of its curved surface by its radius. Therefore,

R 3

4

since 4R2= area, X 4TR2 or R3,

3

Exercise 49-Oral.

Choose one of your classmates to answer.

His row

asks him any of these questions. After losing or answering successfully five of these, another row takes it up, etc.

1. Name several things you know which are spheres in shape.

2. When a sphere is cut into two equal parts by a plane passing through the center, what is each of the two parts called?

3. What name is given to the circular plane surface made by cutting a sphere into two equal parts? 4. How is the diameter of a sphere measured? How the radius? How the circumference?

5. With what measurement of the great circle of a sphere does the diameter of the sphere correspond? The radius? The circumference?

6. Does it require more cord to cover the curved surface or the flat surface of a hemisphere? How many times as much? How many times as much is required to cover the curved surfaces of both hemispheres as is required to cover the flat surface of one hemisphere?

7. What is the ratio of the area of the curved surface of a sphere to the area of the flat surface of one of its hemispheres?

8. How do we find the area of any circle? How do we find the area of the great circle of a sphere?

How do we find the area of the curved surface of a sphere?

9. If a sphere is cut into small pyramids as is shown in the illustration, what dimension of the sphere would correspond to the altitude of each of the pyramids?

10. Which dimension of the sphere would correspond to the combined area of the bases of all of the small pyramids?

11. How do we find the volume of any pyramid? 12. If we know the radius of a sphere and the area of its curved surface, how can we find the volume? 13. Since R2 = area of any circle, state in the form of an equation the formula for finding the area of the curved surface of a sphere.

14. What change do you make in your equation to show volume?

15. Given the radius of a sphere, state the shortest way of finding its volume.

16. Take a ball and tell all the rules you know about it.

Exercise 50-Written.

1. The radius of a sphere is 10"; what is the area of its great circle? What is the area of its curved surface?

2. The diameter of a sphere is 14 ft.; what is the area of its curved surface?

3. What is the area of the cover of a baseball 21" in diameter?

4. The diameter of the earth is approximately 8,000 miles; what is the area of the earth's surface? 5. The outside of the dome of an astronomical observatory is in the form of a hemisphere 50 ft. in diameter; at 25¢ per square yard, what would be the cost of painting the outside of this dome? 6. The area of the surface of the moon is about 12,566,400 sq. mi.; what is the area of its great circle? What is the length of its radius? What is the length of its diameter?

7. What is the volume of an orange whose area is 50.25 sq. in. and whose radius is 2"?

8. What is the volume of the largest sphere that can be turned from a 6" cube of wood?

9. Circular discs of flat sheet metal are often pressed by machinery into the form of hemispheres. The area of the circular disc must equal the area of the finished hemisphere. What is the diameter of the disc required to make a hemisphere whose diameter is 12"?

10. What would be the diameter of the sphere which could be made by joining the two hemispheres pressed out of two brass discs each 10" in diameter? What would be the volume of this sphere.

Exercise 51-Oral Review.

1. How many degrees of arc are there in a circle? In a semicircle? In a quadrant? From the equator to the North Pole? From the North Pole to the South Pole?