Proposition 20. Volume of a Sphere

230. Theorem. The volume of a sphere of radius r is

Proof. Let the sphere be inscribed in a cube (§ 227) whose edge is 2 r. Then the lines joining the center to the vertices of the cube are the edges of six pyramids of altitude r whose bases are the faces of the cube.

One such pyramid is shown in the figure.

The volume of each pyramid is a face of the cube multiplied by 3r (§ 124), and the volume of the six pyramids, or of the whole cube, is the area of the surface of the cube multiplied byr (Ax. 1).

Now let planes be tangent to the sphere at the points where the edges of the pyramids cut the sphere.

One such plane is shown in the figure.

We then have a circumscribed solid whose volume V', although greater than the volume V of the sphere, is nearer V than is the volume of the circumscribed cube.

Also the area A of the circumscribed solid is nearer the area S of the sphere than is the area of the cube.

Proceeding as before, let the center of the sphere be connected with the vertices of the new polyhedron. These connecting lines are the edges of pyramids of altitude r whose bases are the faces of the polyhedron.

Then the sum of the volumes of these pyramids is the area of the circumscribed polyhedron multiplied by r; that is, as before,

V'= A · } r.

If we continue to increase indefinitely the number of faces of the circumscribed polyhedron, we see that

Exercises. Volume of a Sphere

1. The volume of a sphere is the product of the area of its surface and one third of its radius.

2. The volume of a sphere of diameter d is πd3.

3. The volumes of two spheres are to each other as the cubes of their radii or as the cubes of their diameters.

4. If the radius of the earth is 4000 mi., and if the atmosphere extends 50 mi. above the surface of the earth, what is the volume of the atmosphere?

5. If a solid iron ball 4 in. in diameter weighs 9 lb., what is the weight of a spherical iron shell which is 2 in. thick and has an external diameter of 20 in.?

Exercises. Area and Volume of a Sphere

1. How many square feet of lead are needed to cover a hemispherical dome which is 66 ft. in circumference?

The student should always use 22 as the value of π, unless otherwise directed.

2. A hollow ball 8 ft. in diameter surmounts the dome of a church. Making no allowance for the support, how much will it cost to gild the surface of the ball at 10¢ per square inch?

3. Taking the circumference of the earth as 25,000 mi., find the area of the surface to the nearest million square miles.

Find the volumes of spheres whose radii are:

12. The diameter of a spherical basket ball is 10 in. Allowing 50 sq. in. for waste, how many square inches of leather are needed to cover it?

13. The distance across the top of a bowl in the shape of a portion of a sphere is 14 in. and the greatest depth is 7 in. Allowing 7 gal. to the cubic foot, how many pints of water does the bowl hold?

14. If the numbers expressing the area and the volume of a sphere are the same and the units of measure are the square inch and the cubic inch respectively, what is the diameter of the sphere?

15. The weights of two spheres are in the ratio 2:5 and the weights of 1 cu. in. of each of the substances of which they are composed are in the ratio 7:2. Find the ratio of the diameters.

IV. GENERAL REVIEW

Exercises. Polyhedrons

1. The lines drawn from each vertex of a tetrahedron to the point of intersection of the medians of the opposite face meet in a point P which divides each line so that the ratio of the shorter segment to the whole line is 1:4.

The point P is called the center of gravity of the tetrahedron.

2. In Ex. 1, the lines which join the midpoints of the opposite edges of the tetrahedron are each bisected by the center of gravity.

3. The plane which bisects a dihedral angle of a tetrahedron divides the opposite edge into segments proportional to the areas of the faces including the dihedral angle.

4. Show how to cut a cube by a plane so that the section shall be a regular hexagon.

5. If the face angles at the vertex of a triangular pyramid are all right angles, if the areas of the lateral faces are A, B, and C respectively, and if the area of the base is D, then A2+ B2+ C2= D2.

6. Show how to cut a tetrahedron by a plane so that the section shall be a parallelogram.

7. The altitude of a regular tetrahedron is equal to the sum of the four perpendiculars drawn from any point within the tetrahedron to the four faces.

8. Draw figures to show how to cut a cube so as to have a section of three sides; of four sides; of five sides; of as many more sides as possible.

9. The section of a regular octahedron made by a plane parallel to and midway between any pair of opposite faces is a regular hexagon.

Exercises. Formulas Relating to the Sphere

Deduce formulas for the following:

1. The area S of a zone of height on a sphere of radius r.

In the exercises upon this page, and in similar cases, the unit of area is the square of the unit of length, and the unit of volume is the cube of the unit of length. That is, if we think of r as feet, then Swill be square feet and V will be cubic feet.

2. The volume V of a sphere in terms of C, the circumference; in terms of S, the area of the sphere.

3. The radius r of a sphere in terms of V, the volume.

4. The area S of the zone, on a sphere of radius r, which is illuminated by an electric light at the height h above the surface.

5. The diameter d of a sphere in terms of S, the area of the sphere.

6. The altitude h of a zone of area S on a sphere of volume V.

7. The volume of the metal in a spherical iron shell of which the internal radius is r, and the thickness of the metal is t.

8. The weight of a spherical metal shell in which the inside radius is r, the thickness of the metal is t, and the weight of a cubic unit of the metal is w.

9. The diameter of the sphere upon which a zone of area S has an altitude h.

10. The area S' of a zone of altitude h upon a sphere of area S.

11. If R and r are the radii of the spheres circumscribed about and inscribed in a regular tetrahedron of edge 2 a, then R=3ra√6.