Use scale factors in two and three dimensions

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Lesson progressPractice problems 0/3

Difficulty: Intermediate
Estimated time: 27 minutes
Domains: Geometry and Trigonometry
Techniques: Scale-factorArea-scalingVolume-scalingRecover-scale-factor

What you’ll learn

Spot a scaling question about similar figures or solids.
Use the linear scale factor $k$ for lengths and perimeters.
Explain why area and surface area use $k^2$ but volume uses $k^3$ .
Handle a fractional scale factor when a figure shrinks.
Find $k$ from an area ratio with a square root, or from a volume ratio with a cube root.

Prerequisites

Why this matters on the SAT

Scale the measurement, not only the picture

Shrink a floor to a model where every length is $\frac15$ as long, and its area doesn’t shrink by $\frac15$ . SAT questions often give you the scale factor for one kind of measurement, like length, and ask about another, like area or volume. Here’s a typical one.

Solution to the example

Going from the actual floor to the model, every length is multiplied by $\frac15$ . Area spreads in two directions, so that $\frac15$ applies twice:

\left(\frac15\right)^2=\frac1{25}.

Now apply the area factor to the actual area:

450\left(\frac1{25}\right)=18.

The answer is A. Choice B is $450\times\frac15=90$ , which shrinks the floor in only one direction. Choices C and D are bigger than $450$ , and a smaller model can’t have a bigger floor.

Both steps here are quick by hand. When a power or root isn’t one you know, you set the problem up the same way and let Desmos do the arithmetic.

SAT example

The shapes are similar, but you need to turn the length factor you’re given into an area factor. Figure not drawn to scale.

An architect makes a scale model of a ballroom floor. Each length in the model is $\frac15$ of its corresponding length in the actual floor. The actual floor has area $450$ square meters. Which choice gives the area, in square meters, of the model floor?

A
$18$
B
$90$
C
$2{,}250$
D
$11{,}250$

Recognize geometric scaling

Scaling works when two objects are geometrically similar: they have the same shape, and every pair of matching lengths is multiplied by the same number. If an original length is $L$ , the matching scaled length is

kL.

That number $k$ is the linear scale factor. It takes you from the original object to the scaled one.

If $k>1$ , the object gets bigger.
If $0<k<1$ , the object gets smaller.
Going the other way, from the scaled object back to the original, the factor is $\frac1k$ .

You don’t have to prove that the shapes are similar. When the question says the figures or solids are similar, or that every length is multiplied by the same factor, that’s all you need.

Some questions only look like this. Converting units, like feet to inches, isn’t this kind of scaling, and neither is a cost that grows with the number of tickets. And if you need one missing side of a pair of similar triangles, a single proportion does the job; that has its own lesson.

Check your understanding:

A rectangular photo is stretched to twice its width, but its height stays the same. Is there one scale factor that takes the original rectangle to the new one?

Count the directions being scaled

The power of k counts how many length directions the measurement multiplies together.

Why $k$ , then $k^2$ , then $k^3$ ? Each length direction in a measurement brings its own factor of $k$ . So count the directions, and you have the power.

Length and perimeter use $k$

A length runs in one direction, so it picks up one factor of $k$ :

L\longrightarrow kL.

Perimeter is side lengths added up. If a triangle has sides $a$ , $b$ , and $c$ , its scaled perimeter is

ka+kb+kc=k(a+b+c).

Each side brings one $k$ , so the whole perimeter is multiplied by $k$ .

Area and surface area use $k^2$

A rectangle’s area multiplies two length directions, so it picks up two factors of $k$ :

\begin{aligned} A_{\text{scaled}} &=(kL)(kW)\\[1.4em] &=k^2LW\\[1.4em] &=k^2A_{\text{original}}. \end{aligned}

Triangles, circles, and other similar flat shapes work the same way. Surface area uses $k^2$ too, because each face of a solid is an area, which is two-dimensional. Every face grows by $k^2$ , so their total does too.

Volume uses $k^3$

A rectangular prism’s volume multiplies three length directions:

\begin{aligned} V_{\text{scaled}} &=(kL)(kW)(kH)\\[1.4em] &=k^3LWH\\[1.4em] &=k^3V_{\text{original}}. \end{aligned}

Other similar solids follow the same pattern, even with a different formula. A sphere has $V=\frac43\pi r^3$ . Replace $r$ with $kr$ , and you get

\frac43\pi(kr)^3 =k^3\left(\frac43\pi r^3\right).

The formula looks different, but three scaled directions still give $k^3$ .

Check your understanding:

A solid is enlarged with linear scale factor $4$ . By what factors do its surface area and volume change?

Common mistake:

It’s tempting to use $k$ for every measurement. Before you calculate, label what the question asks for as linear, square, or cubic, then use $k$ , $k^2$ , or $k^3$ . The units back you up: plain units like centimeters mean one factor, square units mean two, and cubic units mean three.

Work backward from area or volume

Sometimes a question runs the other way. It gives you the area or volume ratio and asks for the linear scale factor. So undo the power that made the ratio: an area ratio came from squaring $k$ , and a volume ratio came from cubing it.

\frac{A_{\text{scaled}}}{A_{\text{original}}}=R,

then $k^2=R$ , so

k=\sqrt{R}.

\frac{V_{\text{scaled}}}{V_{\text{original}}}=R,

then $k^3=R$ , so

k=\sqrt[3]{R}.

Take the positive root, because a scale factor compares lengths, and lengths are positive. And keep the ratio the right way up: scaled over original gives you the factor from the original to the scaled object.

Check your understanding:

The area of figure B is $\frac49$ the area of similar figure A. What is the linear scale factor from A to B?

Example: Recover a length factor from volume

Worked example

The prisms are only a sketch. Find the linear factor

k

from the two volumes instead of measuring the drawing.

Display cases A and B are geometrically similar. Case A has a volume of $96$ cubic centimeters, and case B has a volume of $324$ cubic centimeters. The width of case A is $8$ centimeters. What is the width, in centimeters, of case B?

Step 1

Choose the direction

We want the factor from case A to case B, so put B on top:

\frac{V_B}{V_A} =\frac{324}{96}.

The bigger volume is on top, so the $k$ we find should be greater than $1$ .

Step 2

Simplify the volume ratio

Divide the top and bottom by $12$ :

\frac{324}{96} =\frac{27}{8}.

Volume is cubic, so

k^3=\frac{27}{8}.

Step 3

Undo the cube

Take the positive cube root. Both $27$ and $8$ are perfect cubes, so this one works by hand:

\begin{aligned} k &=\sqrt[3]{\frac{27}{8}}\\[1.4em] &=\frac{\sqrt[3]{27}}{\sqrt[3]{8}}\\[1.4em] &=\frac32. \end{aligned}

Check it by cubing:

\left(\frac32\right)^3=\frac{27}{8},

which matches the volume ratio. And $\frac32$ is greater than $1$ , just as step 1 predicted.

Step 4

Scale the width

Width is a length, so it uses $k$ , not $k^3$ :

\text{width of B} =8\left(\frac32\right) =12.

Case B is $\boxed{12}$ centimeters wide.

Check your understanding:

For the same two cases, by what factor is the surface area of case B greater than the surface area of case A?

Common mistake:

It’s easy to use the volume ratio $\frac{27}{8}$ as the length factor. But if every length grew by $\frac{27}{8}$ , the volume would grow by $\left(\frac{27}{8}\right)^3$ , far more than $\frac{27}{8}$ . Take the positive cube root first, then use that factor on a length.

Choose the power before the tool

Here’s the whole routine, in order:

Pick a direction and put the scaled object over the original, like $\frac{V_B}{V_A}$ for A to B.
Name what the question asks for: a length or perimeter is linear, an area or surface area is square, and a volume is cubic.
That tells you the factor you need: $k$ , $k^2$ , or $k^3$ .
If you’re given an area or volume ratio, take the square root of an area ratio or the cube root of a volume ratio to get $k$ .
Work familiar numbers like $\left(\frac32\right)^2$ or $\sqrt[3]{27}$ in your head, and type unfamiliar ones into Desmos as one complete expression.
Check the size of your answer: a reduction should come out smaller, and an enlargement bigger.

Desmos can’t tell whether a question wants perimeter, surface area, or volume, and that choice sets the power. Once you’ve made it, typing an unfamiliar power or root straight into Desmos is faster and safer than working it out by hand.

For example, say the scaled volume is $3.7$ times the original volume and the question asks about surface area. The volume ratio gives you $k$ , and surface area then squares it. $3.7$ isn’t a cube you’ll know, so enter

3.7^(1/3)

to get $k$ , or go straight to the surface-area factor with

3.7^(2/3).

Desmos gives $k\approx1.547$ and a surface-area factor of about $2.392$ . The calculator did the arithmetic, but you chose the powers: volume to length, then length to area.

Try it yourself:

In each practice problem, write linear, square, or cubic next to what the question asks for before you calculate. Then write the factor you need, $k$ , $k^2$ , or $k^3$ , before you put in any numbers.

Practice problems

The first two work by hand. The third has a cube root you won’t know offhand, so plan it first, then let Desmos do the arithmetic.

Scale a banner’s area

Practice problem

Two rectangular banners are similar. Each length of banner B is $\frac32$ times the corresponding length of banner A. If banner A has area $80$ square feet, which choice gives the area, in square feet, of banner B?

Calculator loads as you approach

Pick the area factor first. Use the calculator only for arithmetic.

Reverse a model’s surface area

Practice problem

A model cube is geometrically similar to a full-size cube. The model’s edge length is $\frac25$ of the full-size cube’s edge length. The model has surface area $60$ square centimeters. What is the surface area, in square centimeters, of the full-size cube?

Your answer

Calculator loads as you approach

Keep track of which cube is smaller before you undo the area factor.

Transfer volume information to surface area

Practice problem

Spheres P and Q are geometrically similar. The volume of sphere Q is $\frac{2197}{512}$ times the volume of sphere P. The surface area of sphere P is $128\pi$ square units. Which choice gives the surface area, in square units, of sphere Q?

Calculator loads as you approach

Plan it first: cube root, then square. Then enter the whole calculation.

Finish the lesson

3 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

Similar objects share one linear scale factor $k$ for every pair of matching lengths.
Lengths and perimeters scale by $k$ , because they use one length direction.
Areas and surface areas scale by $k^2$ , because they use two.
Volumes scale by $k^3$ , because they use three.
A fractional $k$ means a reduction, and it still gets squared or cubed.
To find $k$ , take the positive square root of an area ratio or the positive cube root of a volume ratio.
Keep the ratio scaled over original, and let the units check your power.
Choose $k$ , $k^2$ , or $k^3$ by thinking it through. Then do familiar powers and roots by hand, and give unfamiliar ones to Desmos.

Next lesson

Use line and angle relationships

Use vertical angles, linear pairs, parallel lines, and transversals to find unknown angle measures.

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Practice

Practice this lesson

103 SAT questions use what this lesson teaches. Practice a few in a study session at the difficulty you choose.

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What you’ll learn

Prerequisites

Scale the measurement, not only the picture

Recognize geometric scaling

Count the directions being scaled

Length and perimeter use kkk

Area and surface area use k2k^2k2

Volume uses k3k^3k3

Work backward from area or volume

Example: Recover a length factor from volume

Choose the direction

Simplify the volume ratio

Undo the cube

Scale the width

Choose the power before the tool

Practice problems

Scale a banner’s area

Reverse a model’s surface area

Transfer volume information to surface area

3 practice examples left

Quick recap

Use line and angle relationships

Practice this lesson

Length and perimeter use $k$

Area and surface area use $k^2$

Volume uses $k^3$