Same Shape, Different Size
A photograph can be enlarged without changing the shape of the objects in it. A building can be represented by a small architectural model. A map can represent hundreds of miles on a sheet of paper. A geometric figure can be enlarged or reduced while preserving its angles and overall form.
These situations all depend on scale.
Scale describes a proportional relationship between corresponding measurements. When every length is multiplied by the same factor, the resulting figure has the same shape but a different size.
This leads to one of the most important applications of proportion: similarity.
The Scale Factor
A scale factor tells us how much a figure, drawing, model, or measurement has been enlarged or reduced.
If an original length is \(L\) and the corresponding scaled length is \(L'\), then the scale factor \(k\) is
\[ k=\frac{L'}{L}. \]Equivalently,
\[ L'=kL. \]This is a direct proportional relationship.
If
\[ k=3, \]every corresponding length becomes three times as large.
If
\[ k=\frac12, \]every corresponding length becomes half as large.
Enlargements and Reductions
The value of the scale factor tells us whether a figure becomes larger or smaller.
- If \(k>1\), the image is an enlargement.
- If \(0<k<1\), the image is a reduction.
- If \(k=1\), the image has the same size as the original.
Suppose a segment 5 centimeters long is enlarged with scale factor 4.
\[ L'=4(5)=20. \]The new segment is 20 centimeters long.
If instead the scale factor is
\[ \frac35, \]then
\[ L'=\frac35(5)=3. \]The new segment is 3 centimeters long.
Dilations
In geometry, a transformation that enlarges or reduces a figure by a scale factor is called a dilation.
Under a dilation, all corresponding lengths are multiplied by the same scale factor.
Suppose a triangle has side lengths
\[ 3,\quad4,\quad5. \]A dilation with scale factor \(2\) produces side lengths
\[ 6,\quad8,\quad10. \]Each side is twice as long:
\[ \frac63 = \frac84 = \frac{10}{5} = 2. \]The size changed, but the proportional relationships among the sides did not.
What Makes Figures Similar?
Two geometric figures are similar when they have the same shape, even if they have different sizes.
For similar polygons, two conditions hold:
- Corresponding angles are equal.
- Corresponding side lengths are proportional.
Suppose two triangles have corresponding side lengths
\[ 3,\ 4,\ 5 \]and
\[ 9,\ 12,\ 15. \]Compare the corresponding sides:
\[ \frac93 = \frac{12}{4} = \frac{15}{5} = 3. \]The common scale factor is 3.
If the corresponding angles are also equal, the triangles are similar.
Corresponding Parts Must Match
When working with similar figures, the order in which measurements are compared matters.
If side \(AB\) corresponds to side \(DE\), and side \(BC\) corresponds to side \(EF\), then a valid proportion is
\[ \frac{AB}{DE} = \frac{BC}{EF}. \]The same orientation should be maintained throughout the proportion.
For example, if the smaller figure is placed in each numerator, then every numerator should represent a measurement from the smaller figure.
Mixing corresponding parts incorrectly can produce a mathematically valid equation that does not represent the geometry.
Finding a Missing Length
Suppose two triangles are similar. A side of length 6 in the first triangle corresponds to a side of length 15 in the second. Another side of length 10 in the first triangle corresponds to an unknown side \(x\).
Write a proportion using corresponding sides:
\[ \frac{6}{15} = \frac{10}{x}. \]Cross multiply:
\[ 6x=150. \]Therefore,
\[ x=25. \]The same result can be seen using the scale factor:
\[ k=\frac{15}{6}=2.5. \]Then
\[ x=2.5(10)=25. \]Scale Drawings
A scale drawing represents a real object using proportional measurements.
Suppose an architectural drawing uses the scale
\[ 1\text{ cm}:2\text{ m}. \]This means every 1 centimeter on the drawing represents 2 meters in the actual structure.
If a wall measures 7.5 centimeters on the drawing, its actual length is
\[ 7.5(2)=15. \]Therefore, the wall is
\[ 15\text{ m} \]long.
Use Compatible Units
Scale ratios compare lengths, so the units must be handled consistently.
Suppose a model car is 20 centimeters long and the real car is 4 meters long.
We should not immediately write
\[ \frac{20}{4} \]as the scale factor because the measurements use different units.
Convert 4 meters to centimeters:
\[ 4\text{ m}=400\text{ cm}. \]Now the model-to-real scale is
\[ \frac{20}{400} = \frac1{20}. \]The model therefore has a scale of
\[ 1:20. \]Every 1 unit of length on the model represents 20 of the same units on the real car.
Map Scales
Maps use scale to represent large real-world distances with much smaller measurements.
Suppose a map uses
\[ 1\text{ inch}:25\text{ miles}. \]If two cities are 3.6 inches apart on the map, the actual distance represented is
\[ 3.6(25)=90. \]The map therefore represents an actual distance of
\[ 90\text{ miles}. \]In this example, the map distance and actual distance are directly proportional.
Representative Fraction Scales
Some maps and technical drawings express scale as a ratio such as
\[ 1:50{,}000. \]This means one unit on the map represents 50,000 of the same unit in reality.
For example,
\[ 1\text{ cm on map} = 50{,}000\text{ cm in reality}. \]Convert:
\[ 50{,}000\text{ cm} = 500\text{ m} = 0.5\text{ km}. \]Therefore, on a \(1:50{,}000\) map,
\[ 1\text{ cm} = 0.5\text{ km}. \]How Perimeter Scales
If every length in a figure is multiplied by a scale factor \(k\), then the perimeter is also multiplied by \(k\).
Suppose a rectangle has side lengths
\[ 4\text{ cm} \qquad\text{and}\qquad 7\text{ cm}. \]Its perimeter is
\[ P=2(4)+2(7)=22\text{ cm}. \]Now enlarge the rectangle by a scale factor of 3. The new sides are
\[ 12\text{ cm} \qquad\text{and}\qquad 21\text{ cm}. \]The new perimeter is
\[ P'=2(12)+2(21)=66\text{ cm}. \]Notice that
\[ 66=3(22). \]Perimeter scales in the same way as length.
Area Does Not Scale the Same Way as Length
A common mistake is to assume that if lengths are multiplied by \(k\), areas are also multiplied by \(k\).
They are not.
Area involves two dimensions. If both dimensions are multiplied by \(k\), then
\[ A'=k^2A. \]The area scale factor is therefore
\[ k^2. \]Suppose a square has side length 4 centimeters. Its area is
\[ A=4^2=16\text{ cm}^2. \]Enlarge it by a length scale factor of 3. The new side length is
\[ 12\text{ cm}. \]The new area is
\[ A'=12^2=144\text{ cm}^2. \]Compare:
\[ \frac{144}{16}=9=3^2. \]The lengths tripled, but the area became nine times as large.
Why the Area Factor Is \(k^2\)
Consider a rectangle with length \(L\) and width \(W\).
Its area is
\[ A=LW. \]After scaling each length by \(k\),
\[ L'=kL \]and
\[ W'=kW. \]The new area is
\[ A'=(kL)(kW). \]Therefore,
\[ A'=k^2LW. \]Since
\[ A=LW, \]we obtain
\[ A'=k^2A. \]The exponent 2 appears because two independent dimensions are being scaled.
Volume Scales by \(k^3\)
Volume involves three dimensions. When all corresponding lengths are multiplied by \(k\), volume is multiplied by
\[ k^3. \]Thus,
\[ V'=k^3V. \]Suppose a cube has side length 2 centimeters.
Its volume is
\[ V=2^3=8\text{ cm}^3. \]Enlarge the cube by a scale factor of 4. Its new side length is
\[ 8\text{ cm}. \]Its new volume is
\[ V'=8^3=512\text{ cm}^3. \]Compare:
\[ \frac{512}{8}=64=4^3. \]The lengths became four times as large, but the volume became 64 times as large.
Length, Area, and Volume Scale Differently
If the length scale factor is \(k\), then:
| Measurement | Scale Factor |
|---|---|
| Length | \(k\) |
| Perimeter | \(k\) |
| Area | \(k^2\) |
| Volume | \(k^3\) |
This distinction is essential in geometry, modeling, engineering, design, and science.
Working Backward from an Area Scale
Sometimes the area ratio is known and the length scale factor must be found.
Suppose two similar figures have areas in the ratio
\[ 25:49. \]Since area scales by the square of the length factor,
\[ k^2=\frac{49}{25}. \]Taking the positive square root for ordinary geometric lengths,
\[ k=\frac75. \]Thus, corresponding lengths are in the ratio
\[ 5:7. \]Working Backward from a Volume Scale
A similar idea applies to volume.
Suppose two similar solids have volumes in the ratio
\[ 8:125. \]Since volume scales by the cube of the length factor,
\[ k^3=\frac{125}{8}. \]Take the cube root:
\[ k=\frac52. \]Corresponding lengths are therefore in the ratio
\[ 2:5. \]Why Similarity Is a Proportion Topic
Similarity may look like a geometric idea, but its numerical structure is fundamentally proportional.
If two similar figures have corresponding side lengths
\[ a,\ b,\ c \]and
\[ A,\ B,\ C, \]then there is a common scale factor \(k\) such that
\[ A=ka, \qquad B=kb, \qquad C=kc. \]Therefore,
\[ \frac{A}{a} = \frac{B}{b} = \frac{C}{c} = k. \]Similarity is therefore built on the same constant multiplicative relationship studied throughout proportion.
Similar Triangles
Similar triangles are especially important because they allow unknown lengths to be determined from proportional relationships.
Suppose triangles \(ABC\) and \(DEF\) are similar, with the correspondence
\[ A\leftrightarrow D, \qquad B\leftrightarrow E, \qquad C\leftrightarrow F. \]Then the corresponding sides satisfy
\[ \frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}. \]These equal ratios can be used to find missing side lengths whenever enough measurements are known.
Similar triangles later become important in geometry, trigonometry, surveying, indirect measurement, and many scientific applications.
Indirect Measurement
Similarity allows us to measure things that may be difficult or impossible to measure directly.
Imagine a 2-meter pole casting a 1.5-meter shadow at the same time that a tree casts a 9-meter shadow. If the sunlight creates similar triangles, the height \(h\) of the tree satisfies
\[ \frac{2}{1.5} = \frac{h}{9}. \]Cross multiply:
\[ 1.5h=18. \]Therefore,
\[ h=12. \]The tree is approximately
\[ 12\text{ meters} \]tall under the assumptions of the similar-triangle model.
Scale Models
Engineers, architects, designers, manufacturers, scientists, and hobbyists often use scale models.
Suppose a model airplane is built at a scale of
\[ 1:50. \]If the real airplane has a wingspan of 35 meters, then the model wingspan is
\[ \frac{35}{50}=0.7\text{ m}. \]Thus the model has a wingspan of
\[ 0.7\text{ m} \]or
\[ 70\text{ cm}. \]Every corresponding length on the model must follow the same \(1:50\) scale if the model is to remain geometrically similar to the original.
Photographs and Aspect Ratios
Proportional scaling is also important in photographs, screens, graphics, and digital media.
Suppose an image is
\[ 1200\times800 \]pixels.
Its width-to-height ratio is
\[ \frac{1200}{800} = \frac32. \]If the width is reduced to 600 pixels while preserving the same aspect ratio, the height must satisfy
\[ \frac{1200}{800} = \frac{600}{h}. \]The scale factor is
\[ \frac{600}{1200}=\frac12. \]Therefore,
\[ h=\frac12(800)=400. \]The resized image should be
\[ 600\times400. \]Changing only one dimension would distort the image because the proportional relationship would no longer be preserved.
Common Errors to Avoid
Comparing Noncorresponding Sides
Similarity proportions must compare matching parts of the figures.
Reversing Only One Ratio
If one ratio is written as smaller-to-larger, the other ratio should also be written smaller-to-larger.
Using Different Units without Converting
Before finding a numerical scale factor, convert measurements to compatible units.
Using \(k\) for Area
If the length factor is \(k\), the area factor is
\[ k^2, \]not \(k\).
Using \(k^2\) for Volume
Volume involves three dimensions, so its factor is
\[ k^3. \]Assuming Proportional Sides Alone Prove Polygon Similarity
For general polygons, corresponding angles must also match. Equal side ratios alone are not sufficient to establish similarity in every case.
Check Your Understanding
- A segment 8 centimeters long is enlarged by a scale factor of 2.5. What is its new length?
- A model uses a scale of \(1:40\). If a part is 6 centimeters long on the model, how long is the actual part in centimeters?
- Two similar triangles have corresponding sides 5 and 15. If another side of the smaller triangle is 8, what is the corresponding side of the larger triangle?
- A map uses the scale \[ 1\text{ cm}:12\text{ km}. \] Two locations are 7.5 centimeters apart on the map. What actual distance does this represent?
- A figure is enlarged with length scale factor 4. By what factor does its area increase?
- A solid is enlarged with length scale factor 3. By what factor does its volume increase?
- Two similar figures have areas in the ratio \(16:81\). What is the ratio of corresponding lengths?
- Two similar solids have volumes in the ratio \(27:216\). What is the ratio of corresponding lengths?
Show Answers
- \[ 8(2.5)=20. \] The new length is \[ 20\text{ cm}. \]
- \[ 6(40)=240. \] The actual part is \[ 240\text{ cm}. \]
- The scale factor is \[ \frac{15}{5}=3. \] Therefore, \[ 8(3)=24. \]
- \[ 7.5(12)=90. \] The actual distance is \[ 90\text{ km}. \]
- \[ 4^2=16. \] The area increases by a factor of 16.
- \[ 3^3=27. \] The volume increases by a factor of 27.
- Take square roots: \[ \sqrt{16}:\sqrt{81}=4:9. \]
- Simplify the volume ratio: \[ 27:216=1:8. \] Take cube roots: \[ 1:2. \]
Key Ideas
- A scale factor is the constant multiplier connecting corresponding lengths.
- A dilation multiplies every corresponding length by the same scale factor.
- Similar figures have equal corresponding angles and proportional corresponding side lengths.
- Scale drawings, maps, models, and resized images rely on proportional relationships.
- Measurements should be converted to compatible units before calculating a numerical scale factor.
- If the length scale factor is \(k\), perimeter also scales by \(k\).
- If the length scale factor is \(k\), area scales by \(k^2\).
- If the length scale factor is \(k\), volume scales by \(k^3\).
- Square roots recover length ratios from area ratios.
- Cube roots recover length ratios from volume ratios.
- Similarity is fundamentally a proportional relationship expressed through geometry.