Geometry (B): READ: Similar Solids Reading

READ: Similar Solids Reading

Similar Solids

Learning Objectives

Find the volumes of solids with bases of equal areas.

Introduction

You’ve learned formulas for calculating the volume of different types of solids—prisms, pyramids, cylinders, and spheres. In most cases, the formulas provided had special conditions. For example, the formula for the volume of a cylinder was specific for a right cylinder.

Now the question arises: What happens when you consider the volume of two cylinders that have an equal base but one cylinder is non-right—that is, oblique. Does an oblique cylinder have the same volume as a right cylinder if the two share bases of the same area?

Parts of a Solid

Given, two cylinders with the same height and radius. One cylinder is a right cylinder, the other is oblique. To see if the volume of the oblique cylinder is equivalent to the volume of the right cylinder, first observe the two solids.

Since they both have the same circular radius, they both have congruent bases with area:

$A = \pi r^2.$

Now cut the right cylinder into a series of $n$ cross-section disks each with height $1$ and radius $r$ .

It should be clear from the diagram that the total volume of the $n$ disks is equal to the volume of the original cylinder.

Now start with the same set of disks. Shift each disk over to the right. The volume of the shifted disks must be exactly the same as the unshifted disks, since both figures are made out of the same disks.

It follows that the volume of the oblique figure is equal to the volume of the original right cylinder.

In other words, if the radius and height of each figure are congruent:

$V \text{(right cylinder)} &= V \text{(oblique cylinder)}\\ V \text{(any cylinder)} &= \frac{1}{3} Bh\\ V \text{(any cylinder)} &= \frac{1}{3} \pi r^2 h$

The principle shown above was developed in the seventeenth century by Italian mathematician Francisco Cavalieri. It is known as Cavalieri’s Principle. (Liu Hui also discovered the same principle in third-century China, but was not given credit for it until recently.) The principle is valid for any solid studied in this chapter.

Volume of a Solid Postulate (Cavalieri’s Principle):

The volumes of two objects are equal if the areas of their corresponding cross-sections are in all cases equal. Two cross-sections correspond if they are intersections of the figure with planes equidistant from a chosen base plane.

Example 1

Prove (informally) that the two circular cones with the same radius and height are equal in volume.

As before, we can break down the right circular cone into disks.

Now shift the disks over.

You can see that the shifted-over figure, since it uses the very same disks as the straight figure, must have the same volume. In fact, you can shift the disks any way you like. Since you are always using the same set of disks, the volume is the same. $\blacklozenge$

Keep in mind that Cavalieri’s Principle will work for any two solids as long as their bases are equal in area (not necessarily congruent) and their cross sections change in the same way.

Example 2

A rectangular pyramid and a circular cone have the same height, and base area. Are their volumes congruent?

Yes. Even though the two figures are different, both can be computed by using the following formula:

$V \text{(cone)} = \frac{1}{3} B_{circle} h \;\text{and} \ V \text{(pyramid)} = \frac{1}{3} B_{rectangle} h$

Since

$B_{circle} = B_{rectangle}$

Then

$V \text{(cone)} = V \text{(rectangle)}$

Similar or Not Similar

Two solids of the same type with equal ratios of corresponding linear measures (such as heights or radii) are called similar solids.

To be similar, figures need to have corresponding linear measures that are in proportion to one another. If these linear measures are not in proportion, the figures are not similar.

Example 1

Are these two figures similar?

If the figures are similar, all ratios for corresponding measures must be the same.

The ratios are:

$\text{width} = \frac{6}{9} = \frac{2}{3}\\ \text{height} = \frac{14}{21} = \frac{2}{3}\\ \text{depth} = \frac{8}{12} = \frac{2}{3}$

Since the three ratios are equal, you can conclude that the figures are similar.

Example 2

Cone A has height $20$ and radius $5$ . Cone B has height $18$ and radius $6$ . Are the two cones similar?

If the figures are similar all ratios for corresponding measures must be the same.

The ratios are:

$\text{height} = \frac{20}{16} = \frac{5}{4}\\ \text{radius} = \frac{18}{6} = \frac{3}{1}$

Since the ratios are different, the two figures are not similar.

Compare Surface Areas and Volumes of Similar Figures

When you compare similar two-dimensional figures, area changes as a function of the square of the ratio of

For example, take a look at the areas of these two similar figures.

The ratio between corresponding sides is:

$\frac{\text{length}\ (A)} {\text{length}\ (B)} = \frac{12} {6} = \frac{2} {1}$

The ratio between the areas of the two figures is the square of the ratio of the linear measurement:

$\frac{\text{area}(A)} {\text{area}(B)} = \frac{12 \cdot 8} {6 \cdot 4} = \frac{96} {24} = \frac{4} {1} = \frac{2^2} {1}$

This relationship holds for solid figures as well. The ratio of the areas of two similar figures is equal to the square of the ratio between the corresponding linear sides.

Example 3

Find the ratio of the surface area between the two similar figures C and D.

Since the two figures are similar, you can use the ratio between any two corresponding measurements to find the answer. Here, only the radius has been supplied, so:

$\frac{\text{radius}\ (C)} {\text{radius}\ (D)} = \frac{6} {4} = \frac{3} {2}$

The ratio between the areas of the two figures is the square of the ratio of the linear measurements:

$\frac{\text{area}\ (C)} {\text{area}\ (D)} = \left( \frac{3} {2} \right )^2 = \frac{9} {4}$

Example 4

If the surface area of the small cylinder in the problem above is 80π, what is the surface area of the larger cylinder?

From above we, know that:

$\frac{\text{area}\ (C)} {\text{area}\ (D)} = \left( \frac{3} {2} \right)^2 = \frac{9} {4}$

So the surface area can be found by setting up equal ratios

$\frac{9} {4} = \frac{n} {80\pi}$

Solve for $n$ .

$n = 180\pi$

The ratio of the volumes of two similar figures is equal to the cube of the ratio between the corresponding linear sides.

Example 5

Find the ratio of the volume between the two similar figures C and D.

As with surface area, since the two figures are similar you can use the height, depth, or width of the figures to find the linear ratio. In this example we will use the widths of the two figures.

$\frac{w(\text{small})} {w(\text{large})} = \frac{15} {20} = \frac{3} {4}$

The ratio between the volumes of the two figures is the cube of the ratio of the linear measurements:

$\frac{\text{volume}\ (C)} {\text{volume}\ (D)} = \left( \frac{3} {4}\right )^3 = \frac{27} {64}$

Does this cube relationship agree with the actual measurements? Compute the volume of each figure.

$\frac{\text{volume (small)}} {\text{volume (large)}} = \frac{5 \times 9 \times 15} {6 2/3 \times 12 \times 20} = \frac{535} {1600} = \frac{27} {64}$

As you can see, the ratio holds. We can summarize the information in this lesson in the following postulate.

Similar Solids Postulate:

If two solid figures, A and B are similar and the ratio of their linear measurements is $\frac{a}{b}$ , then the ratio of their surface areas is:

$\frac{ \text{surface area A}} { \text{surface area B}} = \left( \frac{a}{b} \right)^2$

The ratio of their volumes is:

$\frac{\text{volume A}}{\text{volume B}} = \left( \frac{a}{b} \right)^3$

Scale Factors and Models

The ratio of the linear measurements between two similar figures is called the scaling factor. For example, we can find the scaling factor for cylinders $E$ and $F$ by finding the ratio of any two corresponding measurements.

Using the heights, we find a scaling factor of:

$\frac{h (small)}{h (large)} = \frac{8}{16} = \frac{1}{2}.$

You can use a scaling factor to make a model.

Example 6

Doug is making a model of the Statue of Liberty. The real statue has a height of $111\;\mathrm{feet}$ and a nose that is $4.5\;\mathrm{feet}$ in length. Doug’s model statue has a height of $3\;\mathrm{feet}$ . How long should the nose on Doug’s model be?

First find the scaling factor.

$\frac{height (model)}{height (statue)} = \frac{3}{111} = \frac{1}{37} = 0.027$

To find the length of the nose, simply multiply the height of the model’s nose by the scaling factor.

$\text{nose(model)} &= \text{nose(statue)} \cdot \text{(scaling factor)}\\ &= 4.5 \cdot 0.027\\ &= 0.122 \ \text{feet}$

In inches, the quantity would be:

$\text{nose(model)} &= 0.122 \ \text{feet} \cdot 12 \ \text{inches/feet}\\ &= 1.46 \ \text{inches}$

Example 7

An architect makes a scale model of a building shaped like a rectangular prism. The model measures $1.4\;\mathrm{ft}$ in height, $0.6\;\mathrm{inches}$ in width, and $0.2\;\mathrm{inches}$ in depth. The real building will be $420\;\mathrm{feet}$ tall. How wide will the real building be?

First find the scaling factor.

$\frac{height (real)} {height (model)} = \frac{420} {1.4} = \frac{300} {1} = 300$

To find the width, simply multiply the width of the model by the scaling factor.

$\text{width(real)} &= \text{width(model)} \cdot \text{(scaling factor)}\\ &= 0.6 \cdot 300\\ &= 180 \ \text{feet}$

Review Questions

How does the volume of a cube change if the sides of a cube are multiplied by 4? Explain.
In a cone if the radius and height are doubled what happens to the volume? Explain.
In a rectangular solid, is the sides are doubled what happens to the volume? Explain.
Two spheres have radii of $5$ and $9$ . What is the ratio of their volumes?
The ratio of the volumes of two similar pyramids is $8:27$ . What is the ratio of their total surface areas?
1. Are all spheres similar?
2. Are all cylinders similar?
3. Are all cubes similar? Explain your answers to each of these.
The ratio of the volumes of two tetrahedron is $1000:1$ . The smaller tetrahedron has a side of length $6\;\mathrm{centimeters}$ . What is the side length of the larger tetrahedron?

Refer to these two similar cylinders in problems 8 – 10:

What is the similarity ratio of cylinder A to cylinder B?
What is the ratio of surface area of cylinder A to cylinder B?
What is the ratio of the volume of cylinder B to cylinder A?

Review Answers

The volume will be $64\;\mathrm{times}$ greater. $\;\mathrm{Volume} = {s}^3 \;\mathrm{New \ volume}= (4s)^3$
Volume will be $8\;\mathrm{times}$ greater.
The volume will be $8\;\mathrm{times}$ greater $(2w)(2l)(2h) = 8\;\mathrm{wlh} = 8$ (volume of first rectangular solid)
$5^3/9^3$
$4/9$
All spheres and all cubes are similar since each has only one linear measure. All cylinders are not similar. They can only be similar if the ratio of the radii $=$ the ratio of the heights.
$60 \;\mathrm{cm}$
$20/5 = 4/1$
$16/1$
$1/4^3$

Last modified: Wednesday, 7 July 2010, 1:36 PM

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