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THEOREMS AND PROBLEMS

1. Derive the following formulas for finding the volume, V, of a hollow circular cylinder of length h and cross sectional dimensions as given in the figure:

(1) V=Th(R+r) (R―r).

(2) V=Tht(D+d).

(3) V=Tht(d+t).

R

(4) V=Tht(D−t).

2. Construct a plane through a point and tangent to

a given right circular cylinder.

k

3. If the radius of one cylinder is equal to the altitude of a second, and the radius of the second is equal to the altitude of the first, what is the ratio of their volumes?

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4. The intersection of two planes each tangent to a circular cylinder is parallel to the elements of the cylinder.

5. Prove that the volume of a right circular cylinder is equal to its lateral area times one-half the radius of its base.

6. The volume of two right circular cylinders are equal. Write a proportion between their lateral areas and their radii.

7. If a straight line has more than two points common to the curved surface of a right circular cylinder, the line is an element of the surface.

CHAPTER VIII

PYRAMIDS AND CONES

693. A moving straight line that always contains a fixed point, and always intersects a given straight line, generates a plane. Why?

§ 542

694. Pyramidal Surface. A moving straight line that always contains a fixed point, and always intersects a broken line not coplanar with it, generates a pyramidal surface.

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695. Conical Surface. A moving straight line that always contains a fixed point, and always intersects a plane curved line not coplanar with it, generates a conical surface.

696. The fixed point is called the vertex of the pyramidal, or conical, surface.

697. The two parts of the pyramidal, or conical, surface on opposite sides of the vertex are called nappes.

The words generatrix, directrix, element, and closed surface have the same significance here as in §§ 618, 619.

The directrix is not necessarily closed, but in this text only those that are closed are considered.

The word "section" has the same significance as before (§ 620), however, the cutting plane must not pass through the vertex.

698. A pyramid is the solid formed by cutting all the elements of one nappe of a

closed pyramidal surface by a plane.

699. A cone is the solid form by cutting all the elements of one nappe of a

closed conical surface by a plane.

The meanings of the words: base, lateral face, lateral edge, base edge, lateral area, total area, are apparent from the definitions of §§ 623, 624. 700. The altitude of a pyramid, or cone, is the perpendicular distance from the vertex to the base.

701. Pyramids Classified According to the Number of Lateral Faces. Pyramids, like prisms, are classified as triangular, quadrangular, pentagonal, etc., according as their bases are triangles, quadrilaterals, pentagons, etc.

702. Regular Pyramids. A pyramid whose base is a regular polygon, and whose altitude meets the center of its base, is called a regular pyramid.

The altitude of a regular pyramid is called its axis.

The altitude of one of the lateral faces of a regular pyramid is called the slant height.

In the figure, OV is the altitude, BV a lateral edge, and NV the slant height.

V

E

o!

D

B

C

REGULAR PYRAMID

703. Circular Cones. A cone whose base is a circle is called a circular cone.

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The radius of a circular cone is the radius of its base. Since a right circular cone may be generated by revolving a right triangle about one of its sides as an axis it is often called a cone of revolution.

The length of an element of a right circular cone is its slant height.

705. Prove the following facts concerning cones and pyramids:

(1) The lateral faces of a regular pyramid are congruent isosceles triangles.

(2) The lateral edges of a regular pyramid are equal.

(3) The elements of a right circular cone are equal.

(4) The axis of a right circular cone coincides with its altitude. (5) A straight line drawn from the vertex of a cone to any point in the perimeter of its base is an element.

(6) The section of a circular cone made by a plane containing an element is a triangle.

(7) The section of a pyramid made by a plane through its vertex is a triangle.

EXERCISES

1. Find an element of a right circular cone whose altitude is 15 and radius 7.

2. Find the altitude of a right circular cone whose slant height is s and radius r.

3. Find the altitude of a regular pyramid each face and the base being an equilateral triangle 10 in. on a side.

4. Into how many parts do the nappes of a conical surface divide space?

5. Why is it stated in the definition of a pyramidal surface that the vertex and directrix must not be coplanar?

6. The altitude of a right circular cone is 10 in. and the radius of the base is 6 in. Find the area of a section made by a plane passing through the vertex and 3 in. from the center of the base. In the figure find the area of VAB.

A

E

D

B

706. Theorem. The lateral edges and the altitude of a pyramid, or the elements and the altitude of a cone, are divided proportionally by a plane parallel to its base.

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Pass a plane through the vertex parallel to the base and apply § 568.

707. Theorem. The section of a pyramid made by a plane parallel to the base is similar to the base.

Given pyramid V-AD, cut by plane P
parallel to base AD, forming the section FI.
To prove polygon FI polygon AD.
Proof. AFVG ▲AVB, ▲GVH~^BVC,

V

PFI

GH

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1. The lateral edges of a pyramid are respectively 15 in., 12 in., and 16 in. Find the parts of each made by a plane that is parallel to the base and which divides the altitude into parts that are in the ratio of 1:3.

2. The area of the base of a pyramid is 125 sq. in. Find the area of a section 8 in. from the base and parallel to it, if two corresponding edges of the base and the section are respectively 10 in. and 8 in.

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