Scaling Lengths, Areas and Volumes, SAT math lesson

A photo printed twice as wide and twice as tall needs four times the paper, not twice. A model car built at twice the size needs eight times the plastic. Lengths, areas and volumes respond to the same stretch in different ways, and SAT questions test exactly that difference. Once you see why, the rule is quick to use in both directions.

See also Similar Triangles and Changing Dimensions in Foundations for Calculus.

Key terms

Scale factor

Definition A scaling with scale factor $k > 0$ multiplies every distance in a figure by the same number $k$ . If two points are $d$ units apart in the original figure, their images are $kd$ units apart in the new figure. A factor $k > 1$ enlarges the figure, $0 < k < 1$ shrinks it, and $k = 1$ keeps every length.

In plain words The scale factor is the number you multiply every length by. All lengths, not just some of them.

Example A triangle has sides $5$ , $12$ and $13$ . Scaling it by $k = 3$ gives sides $15$ , $36$ and $39$ , and its perimeter grows from $30$ to $90$ , also by the factor $3$ . Scaling it by $k = \frac{1}{2}$ gives sides $2.5$ , $6$ and $6.5$ .

Similar figures

Definition Two figures are similar when one can be turned into the other by a scaling with some factor $k > 0$ , possibly followed by a rotation, reflection or translation. For similar figures, every length in the second figure divided by the matching length in the first equals the same number $k$ ,

\frac{\text{new length}}{\text{original length}} = k.

This holds for sides, perimeters, heights, diagonals, radii and every other one-dimensional measure. Matching angles are equal. All the power rules below assume the two figures are similar, so every dimension is scaled by the same $k$ .

In plain words Similar figures have the same shape and possibly a different size. One ratio of lengths describes the whole size change.

Example Two similar rectangles have widths $4$ and $10$ , so the scale factor from the small one to the large one is

k = \frac{10}{4} = 2.5.

If the small rectangle has length $6$ , the large one has length $2.5 \cdot 6 = 15$ . Any circle is similar to any other circle, and a circle of radius $3$ becomes a circle of radius $7.5$ under the same factor.

Square scale factor for area

Definition If a figure is scaled by the factor $k > 0$ , every area in it is multiplied by $k^2$ ,

\frac{\text{new area}}{\text{original area}} = k^2.

The reason is that area is counted in unit squares. Each small square of side $s$ becomes a square of side $ks$ with area $k^2 s^2$ . A figure made of such squares, or approximated by them as closely as you like for curved shapes, has its area multiplied by $k^2$ . The same holds for surface area of a solid.

In plain words Double every length and each little square becomes a $2$ by $2$ block of four squares. Area follows the square of the scale factor.

Example A triangle with area $12$ square inches is scaled by $k = 3$ . Its new area is

3^2 \cdot 12 = 108 \text{ square inches,}

not $36$ . You can see it directly with a $1$ by $1$ square scaled by $3$ . It becomes a $3$ by $3$ square, which holds nine of the original squares.

Cubic scale factor for volume

Definition If a solid is scaled by the factor $k > 0$ , its volume is multiplied by $k^3$ ,

\frac{\text{new volume}}{\text{original volume}} = k^3.

Each small cube of edge $s$ becomes a cube of edge $ks$ with volume $k^3 s^3$ , and the solid is built from, or approximated by, such cubes.

In plain words Length is one dimension, area two, volume three. The exponent on $k$ matches the number of dimensions.

Example A cylinder has radius $2$ centimeters, height $5$ centimeters and volume $\pi \cdot 2^2 \cdot 5 = 20\pi$ cubic centimeters. A similar cylinder with $k = 2$ has radius $4$ and height $10$ , and its volume is

\pi \cdot 4^2 \cdot 10 = 160\pi = 2^3 \cdot 20\pi.

The volume grew by the factor $8$ , while the curved surface area grew by $4$ and the height by $2$ .

Measure	Multiplied by
Length, perimeter, radius	$k$
Area, surface area	$k^2$
Volume	$k^3$

Recovering a length ratio

Definition For similar figures with area ratio $A$ and volume ratio $V$ (new over original), the scale factor is

k = \sqrt{A} \quad \text{and} \quad k = \sqrt[3]{V}.

These work because $k > 0$ , so each positive number has exactly one positive square root and one real cube root. An area ratio is not a length ratio. A volume ratio is not one either.

In plain words Work backward by undoing the power. Take the square root of an area ratio, or the cube root of a volume ratio, to get back to lengths.

Example Two similar cones have volumes $54$ and $16$ cubic feet. The volume ratio from the large cone to the small one is

\frac{16}{54} = \frac{8}{27} = \left(\frac{2}{3}\right)^3,

so $k = \frac{2}{3}$ . If the large cone is $9$ feet tall, the small one is $\frac{2}{3} \cdot 9 = 6$ feet tall. The ratio of their surface areas is $\left(\frac{2}{3}\right)^2 = \frac{4}{9}$ .

Nonuniform dimension changes

Definition If different dimensions of a figure are multiplied by different factors, the figure is not similar to the original and the rules $k^2$ and $k^3$ do not apply. For a rectangle with length multiplied by $a > 0$ and width multiplied by $b > 0$ , the area is multiplied by $ab$ . For a box with its three edges multiplied by $a$ , $b$ and $c$ , the volume is multiplied by $abc$ .

In plain words When only some dimensions change, multiply the factors of the dimensions that actually change. Do not square or cube anything automatically.

Example A rectangle’s length grows by $20\%$ and its width shrinks by $20\%$ . The area factor is

1.2 \cdot 0.8 = 0.96,

so the area drops by $4\%$ , even though the two percentages look like they cancel. For a cylinder whose radius doubles while the height stays the same, the volume factor is $2^2 \cdot 1 = 4$ , because the radius appears squared in $V = \pi r^2 h$ and the height does not change.

Units and unit conversion

Definition Lengths are measured in linear units such as meters, areas in square units such as square meters, and volumes in cubic units such as cubic meters. Converting units does not change the object, but a length conversion factor gets squared for area and cubed for volume. Since $1 \text{ m} = 100 \text{ cm}$ ,

\begin{aligned} 1 \text{ m}^2 &= 100^2 \text{ cm}^2 = 10{,}000 \text{ cm}^2, \\[0.4em] 1 \text{ m}^3 &= 100^3 \text{ cm}^3 = 1{,}000{,}000 \text{ cm}^3. \end{aligned}

Scaling a shape makes a different, bigger or smaller object. Converting units describes the same object in other units. A map or blueprint scale combines both, since it pairs a drawing length with a real length.

In plain words The same power rule shows up when you change units, because a square meter is a square that is $100$ centimeters on each side. Keep track of whether a question changes the object or only the units.

Example A map uses $1$ centimeter for $5$ kilometers. A park covers $3$ square centimeters on the map. Each square centimeter on the map is a square $5$ kilometers on a side in reality, which is $25$ square kilometers, so the park covers

3 \cdot 25 = 75 \text{ square kilometers,}

not $15$ . The drawing and the park are similar figures. The length scale is $5$ kilometers per centimeter, so the area scale is $5^2 = 25$ square kilometers per square centimeter.

Check yourself

Two similar cylinders have volumes $54\pi$ and $16\pi$ cubic centimeters. The larger cylinder has radius $6$ centimeters. What is the radius of the smaller cylinder, in centimeters? (A) $\frac{16}{9}$ (B) $\frac{8}{3}$ (C) $4$ (D) $9$

Show the answer

The correct choice is (C). The volume ratio from large to small is

\frac{16\pi}{54\pi} = \frac{8}{27},

whose cube root is $\frac{2}{3}$ . So the smaller radius is $\frac{2}{3} \cdot 6 = 4$ . Choice (A) multiplies the radius by the volume ratio itself, and (B) uses $\frac{4}{9}$ , the area ratio. Choice (D) uses the scale factor upside down and gives a radius larger than $6$ .

The length of a rectangle is increased by $20\%$ and its width is decreased by $20\%$ . By what percent does the area of the rectangle decrease?

Show the answer

The answer is $4$ . The area is multiplied by $1.2 \cdot 0.8 = 0.96$ , which is a $4\%$ decrease. The figure is not similar to the original, so no single scale factor is squared here. Entering $0$ would assume the two changes cancel, and they do not.

A cube has surface area $54$ square inches. A second cube has surface area $216$ square inches. What is the volume of the second cube, in cubic inches? (A) $108$ (B) $216$ (C) $432$ (D) $1{,}728$

Show the answer

The correct choice is (B). The surface area ratio is $\frac{216}{54} = 4$ , so $k = \sqrt{4} = 2$ . The first cube has $6s^2 = 54$ , so $s = 3$ and its volume is $27$ . The second volume is $2^3 \cdot 27 = 216$ , which matches a cube of edge $6$ . Choice (A) multiplies $27$ by the area ratio $4$ , and (C) multiplies the surface area $54$ by $8$ , mixing area with volume. Choice (D) treats the area ratio $4$ as the scale factor and cubes it.

Scaling Lengths, Areas and Volumes

Theory, part 113

Practice, part 114

Key terms

Scale factor

Similar figures

Square scale factor for area

Cubic scale factor for volume

Recovering a length ratio

Nonuniform dimension changes

Units and unit conversion

Check yourself

How this lesson connects

Builds on