A cylinder is a solid figure described by the revolution of a rectangle about one of its sides, which remains fixed.

Thus, if the rectangle ABCD revolves about the side AB, it describes the cylinder represented by the figure. AB is said to be the axis of the cylinder, and its height is the length of the axis.

The whole surface of the cylinder consists of the curved surface, described by CD, and the two circular ends, described by AD and BC. Either end, on which the cylinder may be supposed to rest, may be called its base.

B

D

It will be seen from these figures that if the surface of a cylinder were rolled out flat it would make a rectangle, as

ABCD, the length of which would be the circumference of the cylinder, and the breadth the height. To illustrate this, form a cylinder with a rectangular sheet of paper.

RULE 1: To find the surface area of a cylinder: To twice the area of the base add the area of the rectangle formed by the circumference and the height.

RULE 2: To find the volume of a cylinder: Multiply the area of the base by the height.

ILLUSTRATIVE EXERCISES

1. Find the surface area of a cylinder 14 in. in diameter and 20 in. high.

Circumference 14 x 22+7=44 in.

=

Area of curved surface = 44 × 20 = 880 sq. in.

Area of base = 7 x 7 × 22 ÷ 7 =

154 sq. in.

Complete area = (154 + 154 +880) sq. in.

= 1188 sq. in.

2. Find the volume in cubic inches of a cylinder 14 in. in diameter and 10 in. high.

Area of base = 7 × 7 × 22 ÷ 7 =154 sq. in.

Volume (10 × 154) cu. in.

=

1. Find the curved surface of a cylinder the diameter of which is 14 in. and height 7 in.

2. The diameter of a cylinder is 3 ft. 6 in. and the height 3 ft. 4 in. Find the entire area.

3. The radius of the base of a cylinder is 5 in., and its curved surface is 440 sq. in. Find its height.

4. The diameter of a cylindrical granite column is 21 in. and its height is 16 ft. Find the cost of polishing its curved surfaces at 36 a square foot.

5. How many square yards are covered in 90 revolutions of a cylindrical roller whose length is 4 ft. 6 in. and whose diameter is 3 ft. 6 in. ?

6. Find the volume of a cylinder of which the radius of the base is 7 in. and the height 8 in.

7. Find (in pounds) the weight of a solid iron cylinder 1 ft. 9 in. long, the diameter of the base being 1 ft. 9 in.; supposing that one cubic foot of iron weighs 486 lb.

8. The volume of a cylinder is 308 cu. in. and the radius 31 in. Find the height.

9. The volume of a cylinder 14 in. long is equal to that of a cube having an edge of 11 in. Find the radius of the cylinder.

10. A circular well is 15 ft. deep and its diameter is 3 ft. 6 in. How many cubic yards of earth were taken out in digging it?

Lesson No. 12. Pyramids

A pyramid is a solid the base of which is any plane figure, and whose sides are triangles which meet in a point called the vertex of the pyramid.

D

B

The accompanying figure represents a pyramid, with base

ABCD, and vertex E.

The base may be a triangle, a square, a rectangle, an octagon, or any regular or irregular plane figure contained by straight lines.

A pyramid is said to be right when a perpendicular dropped from the vertex on the base meets the base at its central point; that is, the center of its inscribed or circumscribed circle, if the base is a regular figure, or the intersection of its diagonals if the base is a rectangle.

The sum of the triangular faces is called the slant surface of the pyramid.

RULE 1: The slant surface of a right pyramid equals onehalf the perimeter of the base multiplied by the slant height.

NOTE. The whole surface equals the slant surface plus the area of the base.

RULE 2: The volume of a right pyramid equals one-third the area of the base multiplied by the perpendicular height. NOTE. In the exercises below all pyramids are supposed to be right pyramids.

EXERCISES

1. Find the slant surface of a pyramid 21 in. in perpendicular height, standing on a square base whose side is 40 in.

2. Find the volume in cubic inches of a pyramid whose perpendicular height is 21 in., standing on a square base whose side is 40 in.

3. A right pyramid, 3 ft. in perpendicular height, stands on a square base whose side is 8 ft. Find the area of one of the triangular surfaces.

4. Find the whole surface, including the base, of a right pyramid of which the perpendicular height is 2 ft., and the base a square on a side of 20 in.

5. Find the volume of a pyramid the base of which is a square on a side of 5 in., and the perpendicular height is

6. The base of a pyramid is a rectangle measuring 8 in. by 4 in., and the height is 1 ft. Find the volume.

7. Find the height of a pyramid in which the volume is 84 cu. in., and the base a square on a side of 6 in.

8. A pyramid, standing on a square base, is 15 in. in height. If the volume is 320 cu. in., find the side of the base.

9. The base of a pyramid is a right-angled triangle. The sides containing the right angle are 2 ft. and 5 ft.; the perpendicular height is 4 ft. Find the volume.

10. The height of a pyramid standing on a rectangular base is 2 ft. 8 in., and its volume is 5 cu. ft. If the length of the base is 3 ft. 8 in., find its breadth.

Lesson No. 13. Cones

A right circular cone is a solid described by the revolution of a right-angled triangle about one of its sides (one containing the right angle) which remains fixed.