Gibilisco S. Everyday Math Demystified: A Self-Teaching Guide

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The interior of the cone consists of the set of all points within the cone. The

surface might or might not be included in the deﬁnition of the interior.

A cylinder has a circular or elliptical base, and a circular or elliptical top

that is congruent to the base and that lies in a plane parallel to the base. The

cylinder itself consists of the union of the following sets of points:

The base circle or ellipse.

All points inside the base circle or ellipse and that lie in its plane.

The top circle or ellipse.

All points inside the top circle or ellipse and that lie in its plane.

All line segments connecting corresponding points on the base circle

or ellipse and top circle or ellipse (not including their interiors).

The interior of the cylinder consists of the set of all points within the cylinder.

The surface might or might not be included in the deﬁnition of the interior.

These are general deﬁnitions, and they encompass a great variety of

objects! Here, we’ll look only at cones and cylinders whose bases are circles.

THE RIGHT CIRCULAR CONE

A right circular cone has a base that is a circle, and an apex point on a

line perpendicular to the base and passing through its center. This type

of cone, when the height is about twice the diameter of the base, has

a ‘‘dunce cap’’ shape. Dry sand, when poured into an enormous pile on a

ﬂat surface, acquires a shape that is roughly that of a right circular cone.

An example of this type of object is shown in Fig. 10-12.

Fig. 10-12. A right circular cone.

CHAPTER 10 Geometry in Space 241

SURFACE AREA OF RIGHT CIRCULAR CONE

Imagine a right circular cone as shown in Fig. 10-12. Let P be the apex of

the cone, and let Q be the center of the base. Let r be the radius of the

base, let h be the height of the cone (the length of line segment PQ), and

let s be the slant height of the cone as measured from any point on the

edge of the base to the apex P. The surface area S

of the cone, including

the base, is given by either of the following formulas:

¼ pr

þ prs

¼ pr

þ prðr

þ h

1=2

The surface area S

of the cone, not including the base, is called the lateral

surface area and is given by either of the following:

¼ prs

¼ prðr

þ h

1=2

VOLUME OF RIGHT CIRCULAR CONE

Imagine a right circular cone as deﬁned above and in Fig. 10-12. The

volume, V, of the interior of the ﬁgure is given by:

V ¼ pr

h=3

THE SLANT CIRCULAR CONE

A slant circular cone has a base that is a circle, just as does the right circular

cone. But the apex is such that a line, perpendicular to the plane containing

the base and running from the apex through the plane containing the base,

does not pass through the center of the base. Such a cone has a ‘‘blown-

over’’ or ‘‘cantilevered’’ appearance (Fig. 10-13).

VOLUME OF SLANT CIRCULAR CONE

Imagine a cone whose base is a circle. Let P be the apex of the cone, and let Q

be a point in the plane X containing the base such that line segment PQ is

perpendicular to X (Fig. 10-13). Let h be the height of the cone (the length

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242

of line segment PQ). Let r be the radius of the base. Then the volume, V ,of

the corresponding cone is given by:

V ¼ pr

h=3

This is the same as the formula for the volume of a right circular cone.

THE RIGHT CIRCULAR CYLINDER

A right circular cylinder has a circular base and a circular top. The base and

the top lie in parallel planes. The center of the base and the center of the top

lie along a line that is perpendicular to both the plane containing the base and

the plane containing the top. A general example is shown in Fig. 10-14.

SURFACE AREA OF RIGHT CIRCULAR CYLINDER

Imagine a right circular cylinder where P is the center of the top and Q is

the center of the base (Fig. 10-14). Let r be the radius of the cylinder, and

Fig. 10-13. A slant circular cone.

Fig. 10-14. A right circular cylinder.

CHAPTER 10 Geometry in Space 243

let h be the height (the length of line segment PQ). Then the surface area

of the cylinder, including the base, is given by:

¼ 2prh þ 2pr

¼ 2prðh þ rÞ

The lateral surface area S

of the cylinder (not including the base) is given by:

¼ 2prh

VOLUME OF RIGHT CIRCULAR CYLINDER

Imagine a right circular cylinder as deﬁned above and shown in Fig. 10-14.

The volume, V, of the cylinder is given by:

V ¼ pr

THE SLANT CIRCULAR CYLINDER

A slant circular cylinder has a circular base and a circular top. The base

and the top lie in parallel planes. The center of the base and the center of

the top lie along a line that is not perpendicular to the planes that contain

them (Fig. 10-15).

VOLUME OF SLANT CIRCULAR CYLINDER

Imagine a slant circular cylinder as deﬁned above and in Fig. 10-15. The

volume, V, of the corresponding solid is given by:

V ¼ pr

Fig. 10-15. A slant circular cylinder.

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244

THE SPHERE

Consider a speciﬁc point P in 3D space. The surface of a sphere S consists

of the set of all points at a speciﬁc distance or radius r from point P. The

interior of sphere S, including the surface, consists of the set of all points

whose distance from point P is less than or equal to r. The interior of sphere

S, not including the surface, consists of the set of all points whose distance

from P is less than r.

SURFACE AREA OF SPHERE

Imagine a sphere S having radius r as shown in Fig. 10-16. The surface

area, A, of the sphere is given by:

A ¼ 4pr

VOLUME OF SPHERE

Imagine a sphere S as deﬁned above and in Fig. 10-16. The volume, V, of the

solid enclosed by the sphere is given by:

V ¼ 4pr

This volume applies to the interior of sphere S, either including the surface

or not including it, because the surface has zero volume.

Fig. 10-16. A sphere.

CHAPTER 10 Geometry in Space 245

PROBLEM 10-4

A cylindrical water tower is exactly 30 m high and exactly 10 m in radius.

How many liters of water can it hold, assuming the entire interior can be

ﬁlled with water? Round the answer oﬀ to the nearest liter.

SOLUTION 10-4

Use the formula for the volume of a right circular cylinder to ﬁnd the volume

in cubic meters:

V ¼ pr

Plug in the numbers. Let r ¼ 10, h ¼ 30, and p ¼ 3.14159:

V ¼ 3:14159  10

 30

¼ 3:14159  100  30

¼ 9424:77 m

There are 1000 liters per cubic meter. This means that the amount of water

the tower can hold, in liters, is equal to 9424.77  1000, or 9,424,770.

PROBLEM 10-5

A circus tent is shaped like a right circular cone. Its diameter is 50 m and the

height at the center is 20 m. How much canvas is in the tent? Express the

answer to the nearest square meter.

SOLUTION 10-5

Use the formula for the lateral surface area, S, of the right circular cone:

S ¼ prðr

þ h

1=2

We know that the diameter is 50 m, so the radius is 25 m. Therefore, r ¼ 25.

We also know that h ¼ 20. Let p ¼ 3.14159. Then:

S ¼ 3:14159  25 ð25

þ 20

1=2

¼ 3:14159  25 ð625 þ 400Þ

1=2

¼ 3:14159  25  1025

1=2

¼ 3:14159  25  32:0156

¼ 2514:4972201 m

There are 2514 m

of canvas, rounded oﬀ to the nearest square meter.

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246

PROBLEM 10-6

Suppose a football ﬁeld is to be covered by an inﬂatable dome that takes the

shape of a half-sphere. If the radius of the dome is 100 m, what is the volume

of air enclosed by the dome, in cubic meters? Find the result to the nearest

1000 m

SOLUTION 10-6

First, we must ﬁnd the volume V of a sphere whose radius is 100 m, and then

divide the result by 2. Let p ¼ 3.14159. Using the formula with r ¼ 100 gives

this result:

V ¼ð4  3:14159  100

Þ=3

¼ð4  3:14159  1 ó000ó000Þ=3

¼ 4ó188ó786:667 m

Thus V/2 ¼ 4,188,786.667/2 ¼ 2,094,393.333 m

. Rounding oﬀ to the nearest

1000 m

, we obtain the ﬁgure of 2,094,000 m

as the volume of air enclosed

by the dome.

Quiz

Refer to the text in this chapter if necessary. A good score is eight correct.

Answers are in the back of the book.

1. What is the approximate surface area of the sides (not including

the base or top) of a water tower that is shaped like a right circular

cylinder, that has a radius of exactly 5 meters, and that is exactly 20

meters tall?

(a) 1571 m

(b) 628.3 m

(d) It cannot be determined without more information.

2. If the height of a silo that is shaped like a right circular cylinder is cut

in half while its radius remains the same, what happens to its surface

area (not including the base or the top)?

(a) It decreases to 1/2 of its previous value.

(b) It decreases to 1/4 of its previous value.

(d) It decreases to 1/16 of its previous value.

CHAPTER 10 Geometry in Space 247

3. If the height of a silo that is shaped like a right circular cylinder is

cut in half while its radius remains the same, what happens to its

volume?

(a) It decreases to 1/2 of its previous value.

(b) It decreases to 1/4 of its previous value.

(d) It decreases to 1/16 of its previous value.

4. What is the approximate volume, in cubic centimeters (cm

), of a ball

that is exactly 20 cm in diameter? (Remember that the diameter of a

sphere is twice the radius.)

(a) 62.83 cm

(b) 1257 cm

(d) It cannot be determined without more information.

5. Which of the following statements is not true in Euclidean 3D space?

(a) If two straight lines are parallel, then they are skew.

(b) If two straight lines are parallel, then they lie in the same plane.

plane.

(d) If two straight lines are parallel, then they do not intersect.

6. Which of the following statements is always true in Euclidean 3D

space?

(a) Four points determine a unique ﬂat plane.

(b) Three points can have more than one ﬂat plane in common.

(d) None of the statements (a), (b), or (c) is always true in Euclidean

3D space.

7. If the radius of a sphere increases by a factor of 10, the volume of the

sphere increases by a factor of

(a) 100

(b) (400/3)p

(d) (4000/3)p

8. The boundary of a closed plane region

(a) adds nothing to its area

(b) increases its volume

(d) is always a straight line

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248

9. A cube is a special form of

(a) rhombus

(b) pyramid

(d) parallelepiped

10. If the radius of a sphere increases by a factor of 9, the ratio of the

volume of the sphere to its surface area increases by a factor of

(a) 3

(b) 9

(d) 81

CHAPTER 10 Geometry in Space 249

CHAPTER

Graphing It

We’ve already done some work with simple two-dimensional (2D) graphs.

In this chapter, we’ll take a closer look at the two most common coordinate

systems on which 2D graphs are plotted. They are the Cartesian plane (named

after the French mathematician Rene

Descartes who invented it in the 1600s)

and the polar coordinate plane.

The Cartesian Plane

The Cartesian plane, also called the rectangular coordinate plane or rectangu-

lar coordinates, is deﬁned by two number lines that intersect at a right angle.

Figure 11-1 illustrates a basic set of rectangular coordinates. Both number

lines have equal increments. This means that on either axis, points corre-

sponding to consecutive integers are the same distance apart, no matter

where on the axis we look. The two number lines intersect at their zero

points. The horizontal (right-and-left) axis is the x axis; the vertical

(up-and-down) axis is the y axis.

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