Proposition 17. Bases of a Cylinder

134. Theorem. The bases of a cylinder are congruent.

Given the cylinder c with bases b and b'.

Proof. Let X, Y, Z be any three points on the perimeter of b, and let XX', YY', ZZ' be elements.

Place b on b' so that X lies on X', and Y on Y'; then Z, which is any third point on the perimeter of b, has one and only one corresponding point on the perimeter of b'. Hence the same is true of every point on the perimeter of b; that is, b can be made to coincide with b'.

.. b is congruent to b'.

$ 7,1

Exercises. Cylinders

1. Every section of a cylinder made by a plane passing through two elements is a parallelogram.

2. Every section of a right cylinder made by a plane passing through two elements is a rectangle.

3. Any two sections of a cylinder made by parallel planes which cut all the elements are congruent.

4. Any section of a cylinder parallel to the base is congruent to the base.

5. The straight line joining the centers of the bases of a circular cylinder passes through the centers of all sections of the cylinder parallel to the bases.

6. If a rectangle revolves about one of its sides, it forms a right circular cylinder.

7. If two right circular cylinders have equal bases and equal altitudes, they are congruent.

8. The locus of points at a given distance from a given line is a cylindric surface.

The given line is called the axis of the surface.

9. The center of any section of a circular cylinder parallel to the base is on the axis.

10. If parallel planes cut all the elements of a cylindric surface, the sections thus formed are congruent.

11. If a section of an oblique cylinder is made by a plane parallel to an element, is the resulting figure a parallelogram? Can it be a rectangle? Give the proofs.

12. From the center of the upper base of a right circular cylinder 4 in. high lines are drawn to the perimeter of the lower base. If the diameters of the bases are 6 in., what is the length of each line?

135. Tangent Plane. A plane which contains an element of a cylinder, but does not cut the surface, is called a tangent plane.

From this definition it is evident that

A plane passing through a tangent to the base of a circular cylinder and the element through the point of contact is tangent to the cylinder.

If a plane is tangent to a circular cylinder, its intersection with the plane of the base is tangent to the base.

136. Inscribed Prism. A prism whose lateral edges are elements of a cylinder and whose bases are inscribed in the bases of the cylinder is called an inscribed prism.

The first figure above shows an inscribed prism. The cylinder is said to be circumscribed about the prism.

137. Circumscribed Prism. A prism whose lateral faces are tangent to the lateral surface of a cylinder and whose bases are circumscribed about the bases of the cylinder is called a circumscribed prism.

The second figure above shows a circumscribed prism. The cylinder is said to be inscribed in the prism.

138. Transverse Section. A section of a cylinder made by a plane that cuts all the elements is called a transverse section of the cylinder.

If the plane is perpendicular to the elements, the section is called a right section.

139. Cylinder as a Limit. From the principles outlined in §§ 28 and 29, and from the nature of inscribed and circumscribed prisms, the properties of the cylinder stated below may be assumed without proof.

If a prism whose base is a regular polygon is inscribed in or circumscribed about a circular cylinder, and if the number of sides of the prism is indefinitely increased,

1. The volume of the cylinder is the limit of the volume of the prism.

2. The lateral area of the cylinder is the limit of the lateral area of the prism.

3. The perimeter of any transverse section of the cylinder is the limit of the perimeter of the corresponding section of the prism.

As we increase the number of sides of the inscribed or circumscribed prism whose base is a regular polygon, the perimeter of the base approaches the circle as its limit (§ 29, 1). This brings the lateral surface of each prism nearer and nearer the cylindric surface.

Proposition 18. Lateral Area of a Cylinder

140. Theorem. The lateral area of a circular cylinder is the product of an element and the perimeter of a right section.

Given c, a circular cylinder; S, the lateral area; e, an element; and p, the perimeter of a rt. section.

Proof. Let a prism whose base is a regular polygon be inscribed in c, and let L be the lateral area and p' the perimeter of a rt. section.

Then

L= ep'.

§ 93

If the number of lateral faces is indefinitely increased,

141. Corollary. In a cylinder of revolution of lateral area

S, total area T, altitude h, and radius r,

S=2πrh, and T=2 πr (h+r).