The Central Idea
Similarity allows one geometric figure to represent another at a different size while preserving shape.
Corresponding lengths change by the same multiplicative factor:
\[ \boxed{\text{new length}=k(\text{original length})}, \]where \(k\) is the scale factor.
If the same scale factor applies throughout a drawing or map, measurements made on the representation can be converted into real-world measurements, and real-world measurements can be converted back into representation measurements.
What Makes Figures Similar?
Two figures are similar when they have the same shape even if they have different sizes.
For corresponding parts:
- corresponding angles have equal measure;
- corresponding lengths are proportional.
If figure \(A\) is scaled to produce figure \(B\), then every corresponding length is multiplied by the same factor \(k\).
Scale Factor
The scale factor from an original figure to a new figure is
\[ \boxed{k=\frac{\text{new corresponding length}} {\text{original corresponding length}}}. \]If \(k>1\), the new figure is an enlargement.
If \(0
If \(k=1\), corresponding lengths are unchanged.
Example: Finding a Scale Factor
A rectangle with length \(6\) centimeters is enlarged so that the corresponding length becomes \(15\) centimeters.
The scale factor is
\[ k=\frac{15}{6}=2.5. \]Therefore,
\[ \boxed{k=2.5}. \]Every corresponding length in the enlarged rectangle must be \(2.5\) times the original length.
Corresponding Lengths
Suppose two similar figures have scale factor \(k\).
If an original length is \(L\), then its corresponding length is
\[ L'=kL. \]Conversely,
\[ \boxed{L=\frac{L'}{k}}. \]These two forms let us move in either direction between a representation and the object it represents.
Example: Finding a Missing Corresponding Length
A triangular drawing is enlarged by a scale factor of \(1.8\). One original side has length \(7\) centimeters.
The corresponding enlarged side is
\[ L'=1.8(7)=12.6. \]Thus,
\[ \boxed{L'=12.6\text{ cm}}. \]Ratios of Corresponding Lengths
Similarity can also be expressed through equal ratios.
If lengths \(a,b,c\) correspond to \(a',b',c'\), then
\[ \frac{a'}{a} = \frac{b'}{b} = \frac{c'}{c} = k. \]Equivalently,
\[ \boxed{\frac{a}{b}=\frac{a'}{b'}}. \]This proportional structure is what makes scale drawings and maps useful.
Example: Solving a Proportion
Two similar figures contain corresponding side lengths
\[ 4,\quad 10 \]in the first figure and
\[ 6,\quad x \]in the second.
Set up the proportion:
\[ \frac{6}{4}=\frac{x}{10}. \]Then
\[ 4x=60, \] \[ \boxed{x=15}. \]Correspondence Must Be Correct
Ratios work only when corresponding quantities are matched correctly.
If side \(a\) in one figure corresponds to side \(a'\) in another, and side \(b\) corresponds to \(b'\), then a valid proportion is
\[ \frac{a'}{a}=\frac{b'}{b}. \]Mixing a corresponding side with a noncorresponding side can produce a numerical answer that has no geometric meaning.
Scale Drawings
A scale drawing represents an object or region using a consistent ratio between drawing lengths and actual lengths.
Examples include:
- architectural floor plans;
- site plans;
- engineering drawings;
- mechanical diagrams;
- maps;
- models and enlarged technical illustrations.
A valid scale drawing preserves proportional dimensions even though the physical size of the drawing differs from the original.
Three Common Ways to State Scale
Scale is commonly expressed in three related forms.
Statement scale
For example,
\[ 1\text{ cm represents }5\text{ m}. \]Representative fraction or ratio
For example,
\[ 1:50{,}000. \]Graphic scale
A marked line or scale bar visually shows the relationship between map distance and real distance.
Each form communicates the same basic idea: drawing distance and actual distance are connected by a fixed proportional relationship.
Statement Scales
A statement such as
\[ 1\text{ cm}=4\text{ m} \]means that every \(1\) centimeter measured on the drawing represents \(4\) meters in the actual object.
If a wall measures \(3.5\) centimeters on the drawing, then its actual length is
\[ 3.5(4)=14. \]Therefore,
\[ \boxed{14\text{ m}}. \]Example: Actual Distance from a Drawing
A site plan uses the scale
\[ 1\text{ cm}=8\text{ m}. \]Two points are \(6.2\) centimeters apart on the plan.
The actual distance is
\[ 6.2(8)=49.6. \]Thus,
\[ \boxed{49.6\text{ m}}. \]Finding Drawing Length from Actual Length
Scale conversion also works in reverse.
If
\[ 1\text{ cm}=5\text{ m}, \]then an actual distance of \(42\) meters should appear on the drawing with length
\[ \frac{42}{5}=8.4. \]Therefore,
\[ \boxed{8.4\text{ cm}}. \]Ratio Scales Require Matching Units
A ratio scale such as
\[ 1:25{,}000 \]compares quantities measured in the same unit.
It can mean:
\[ 1\text{ cm on the map}=25{,}000\text{ cm in reality}, \]or
\[ 1\text{ in on the map}=25{,}000\text{ in in reality}. \]The ratio itself is dimensionless because the same units appear in the numerator and denominator.
Example: Interpreting a Ratio Scale
A map has scale
\[ 1:50{,}000. \]A distance of \(3\) centimeters on the map represents
\[ 3(50{,}000)=150{,}000\text{ cm}. \]Convert to meters:
\[ 150{,}000\text{ cm}=1{,}500\text{ m}. \]Convert to kilometers:
\[ 1{,}500\text{ m}=1.5\text{ km}. \]Therefore,
\[ \boxed{3\text{ cm on the map}=1.5\text{ km in reality}}. \]Large Scale and Small Scale
In mapping, the phrases large scale and small scale refer to the size of the representative fraction, not the physical size of the map sheet.
For example,
\[ 1:5{,}000 \]is a larger scale than
\[ 1:500{,}000. \]A larger-scale map generally covers a smaller geographic area with more visible detail. A smaller-scale map generally covers a larger geographic area with less detail.
Graphic Scales
A graphic scale, or scale bar, is a line divided into labeled real-world distances.
Its practical advantage is that it can remain useful when a map is enlarged or reduced together with the scale bar.
By contrast, a printed statement such as
\[ 1\text{ cm}=2\text{ km} \]becomes incorrect if the image is resized without updating the stated relationship.
Maps Are Spatial Models
A map is not the territory itself. It is a mathematical and visual model of selected spatial information.
A map may encode:
- position;
- distance;
- direction;
- boundaries;
- elevation;
- roads and paths;
- land use;
- physical or administrative regions.
The scale determines how measurements on the map correspond to measurements in the represented region.
Straight-Line Distance and Route Distance
A scale converts whichever path is measured.
A straight segment between two map points represents straight-line distance.
Measuring along a curved road or trail represents route distance.
These distances may differ substantially.
The scale conversion can be correct while the chosen path still answers the wrong question.
Example: Straight-Line Map Distance
A map uses
\[ 1\text{ cm}=12\text{ km}. \]Two cities are \(4.7\) centimeters apart in a straight line on the map.
Their represented straight-line distance is
\[ 4.7(12)=56.4. \]Therefore,
\[ \boxed{56.4\text{ km}}. \]This does not imply that the road distance between the cities is also \(56.4\) kilometers.
Curved Distances on Maps
Curved paths can be approximated by dividing them into short segments.
If measured map segments have lengths
\[ d_1,d_2,\ldots,d_n, \]then the approximate map-path length is
\[ d_{\text{map}}\approx d_1+d_2+\cdots+d_n. \]The map scale then converts this total to the represented real-world path length.
Shorter segments can generally follow a curved path more closely.
Scale Drawings and Coordinates
Coordinates can also be scaled.
A dilation centered at the origin with scale factor \(k\) sends
\[ (x,y) \]to
\[ \boxed{(kx,ky)}. \]For example, with \(k=3\),
\[ (2,-4)\longrightarrow(6,-12). \]Coordinate scaling makes the proportional structure of similarity explicit.
Scaling About a Point Other Than the Origin
If the center of scaling is
\[ C=(a,b), \]then a point
\[ P=(x,y) \]can be scaled by first measuring its displacement from \(C\), multiplying that displacement by \(k\), and then returning to the center:
\[ \boxed{ P'= \left( a+k(x-a), b+k(y-b) \right) }. \]This is the coordinate form of a dilation centered at \(C\).
Angles Under Uniform Scaling
Uniform scaling changes lengths but preserves angle measures.
If a triangle is enlarged by scale factor \(3\), each side becomes three times as long, but its corresponding angles remain unchanged.
That angle preservation is why the enlarged figure has the same shape as the original.
Perimeter Scaling
Perimeter is a one-dimensional measurement, so it scales by the same factor as length.
If
\[ P \]is the original perimeter and the scale factor is \(k\), then
\[ \boxed{P'=kP}. \]A figure enlarged by a factor of \(4\) has four times the perimeter.
Area Scaling
Area is two-dimensional, so it scales by the square of the linear scale factor.
If corresponding lengths are multiplied by \(k\), then
\[ \boxed{A'=k^2A}. \]A figure enlarged by a factor of \(3\) has
\[ 3^2=9 \]times the area.
Why Area Uses the Square of the Scale Factor
Consider a rectangle with dimensions
\[ L\times W. \]Its area is
\[ A=LW. \]After scaling by \(k\), the dimensions become
\[ kL\times kW. \]The new area is
\[ A'=(kL)(kW)=k^2LW. \]Therefore,
\[ \boxed{A'=k^2A}. \]Example: Area of a Scaled Figure
A floor-plan region has actual area
\[ 48\text{ m}^2. \]Suppose a second geometrically similar region has linear dimensions \(1.5\) times as large.
Its area is
\[ A'=(1.5)^2(48). \]Since
\[ (1.5)^2=2.25, \]we get
\[ A'=108. \]Therefore,
\[ \boxed{A'=108\text{ m}^2}. \]Volume Scaling
Volume is three-dimensional, so it scales by the cube of the linear scale factor.
If corresponding lengths are multiplied by \(k\), then
\[ \boxed{V'=k^3V}. \]An object enlarged by a factor of \(2\) has
\[ 2^3=8 \]times the volume.
Models and Prototypes
Physical models use the same scaling principles as drawings.
A model labeled
\[ 1:100 \]has linear dimensions one hundredth of the corresponding dimensions of the full-size object.
If a model wall is \(12\) centimeters long, the full-size wall is
\[ 12(100)=1{,}200\text{ cm}=12\text{ m}. \]Enlarged Scientific and Technical Images
Scale does not always mean reduction.
A small object can be enlarged so that details become visible.
If an actual feature is \(0.4\) millimeters long and an enlarged drawing shows it as \(20\) millimeters long, then
\[ k=\frac{20}{0.4}=50. \]The image is a
\[ \boxed{50\times} \]enlargement in linear dimension.
Scale and Resolution
A representation cannot show unlimited spatial detail.
As a larger real region is compressed into the same display area, small features may become too small to distinguish.
This creates a practical relationship between:
- scale;
- coverage;
- resolution;
- level of detail.
Choosing a useful scale therefore depends on the purpose of the drawing or map.
Scale and Measurement Precision
Suppose a drawing uses
\[ 1\text{ mm}=1\text{ m}. \]An uncertainty of only \(0.5\) millimeters on the drawing corresponds to
\[ 0.5\text{ m} \]in the represented object.
Small drawing-measurement errors can therefore represent much larger real-world uncertainties when the reduction is substantial.
Printed and Digital Resizing
A scale drawing can lose its original numerical scale if it is enlarged or reduced after creation.
Suppose a drawing originally uses
\[ 1\text{ cm}=10\text{ m}. \]If the entire drawing is printed at \(200\%\) size, each original \(1\)-centimeter drawing length becomes \(2\) centimeters on the new print.
The new effective scale is therefore
\[ \boxed{2\text{ cm}=10\text{ m}}, \]or equivalently,
\[ \boxed{1\text{ cm}=5\text{ m}}. \]Why a Graphic Scale Helps After Resizing
If a graphic scale is resized together with the map, its marked bar changes size at the same rate as the rest of the image.
The visual comparison can therefore remain valid.
A textual ratio or statement printed elsewhere does not automatically update when the image is resized.
Scale Does Not Guarantee Similarity
Multiplying horizontal distances by one factor and vertical distances by a different factor distorts shape.
For example,
\[ x'=2x,\qquad y'=3y \]is not a uniform scale transformation.
Circles generally become ellipses, and angles are not generally preserved.
A true similarity scaling uses the same factor in every spatial direction.
Maps and Projection Distortion
A flat map represents part of a curved Earth, so global or large-area maps cannot generally preserve every geometric property perfectly.
Depending on the map projection, some combination of:
- distance;
- area;
- shape;
- direction;
- scale
may vary across the map.
For small local regions, a stated scale may be an excellent approximation. For very large regions, map interpretation requires more attention to how the surface has been projected.
Local Scale Versus Global Scale
On some maps, scale is effectively constant over the region of interest.
On others, especially maps representing very large portions of Earth, scale may vary with position or direction.
This means a single ruler conversion should not automatically be assumed valid everywhere unless the map design supports that interpretation.
Reading Scale Before Measuring
Before measuring from a drawing or map, determine:
- what kind of scale is provided;
- whether the drawing has been resized;
- which units are being used;
- whether the scale applies uniformly;
- whether straight-line or path distance is required.
These checks prevent many common errors before calculation begins.
A Reliable Procedure for Scale Problems
- Identify the drawing or map scale.
- Write the scale as a clear ratio or conversion statement.
- Identify the representation measurement and the desired actual measurement, or vice versa.
- Convert units so the relationship is consistent.
- Set up a proportion or multiply by the appropriate scale factor.
- Solve for the unknown quantity.
- Attach the correct unit.
- Check whether enlargement or reduction moved the value in the expected direction.
Example: Full Scale-Drawing Workflow
A floor plan uses the scale
\[ 1\text{ cm}=2.5\text{ m}. \]A rectangular room measures \(3.6\) centimeters by \(2.8\) centimeters on the plan.
Convert each dimension:
\[ L=3.6(2.5)=9.0\text{ m}, \] \[ W=2.8(2.5)=7.0\text{ m}. \]Therefore, the actual room dimensions are
\[ \boxed{9.0\text{ m}\times7.0\text{ m}}. \]Its actual area is
\[ A=(9.0)(7.0)=63. \]Thus,
\[ \boxed{A=63\text{ m}^2}. \]Alternative Area Method Using the Scale Factor
The previous example can also be handled through area scaling.
Since
\[ 1\text{ cm}=2.5\text{ m}, \]one square centimeter on the plan corresponds to
\[ (2.5)^2=6.25\text{ m}^2. \]The plan area is
\[ (3.6)(2.8)=10.08\text{ cm}^2. \]Therefore,
\[ 10.08(6.25)=63\text{ m}^2. \]The same answer is recovered because area scales by the square of the linear factor.
Common Error: Reversing the Scale Factor
Suppose
\[ 1\text{ cm}=6\text{ m}. \]A \(4\)-centimeter drawing distance represents
\[ 4(6)=24\text{ m}, \]not
\[ \frac{4}{6}\text{ m}. \]Ask whether the conversion should enlarge or reduce the numerical value. Moving from a small drawing to the full-size object should usually increase the length.
Common Error: Mixing Corresponding Sides
If two similar rectangles have side pairs
\[ 4\leftrightarrow10 \]and
\[ 6\leftrightarrow15, \]then
\[ \frac{10}{4}=\frac{15}{6}=2.5. \]A ratio such as
\[ \frac{10}{6} \]compares noncorresponding sides and does not give the scale factor.
Common Error: Scaling Area Like Length
If all lengths double, area does not merely double.
With
\[ k=2, \]the area factor is
\[ k^2=4. \]Therefore,
\[ \boxed{A'=4A}. \]Common Error: Scaling Volume Like Area
If every linear dimension is tripled, the volume factor is
\[ 3^3=27, \]not \(3\) and not \(9\).
Linear, area, and volume measures respond differently to the same scale factor.
Common Error: Ignoring Resizing
A scanned, photocopied, projected, or digitally displayed plan may not retain its original printed dimensions.
A numeric scale printed on the source can therefore become invalid after resizing.
Verify the current scale before taking measurements from a reproduction.
Common Error: Measuring a Route as a Straight Segment
If a road curves around a hill or shoreline, its route distance is longer than the straight segment joining its endpoints.
A straight ruler measurement answers a straight-line distance question. It does not automatically answer a travel-distance question.
Common Error: Assuming Every Map Has Uniform Scale
Local engineering and site maps often behave approximately like uniform scale drawings over their intended region.
Large geographic maps may involve projection effects that make scale vary across the image.
The scale information supplied with the map should determine how distances are interpreted.
Practice 1: Scale Factor
A side of length \(8\) centimeters is enlarged to \(22\) centimeters. Find the scale factor.
Practice 2: Corresponding Length
Two figures are similar with scale factor \(1.6\) from the first to the second. A side in the first figure measures \(12\) centimeters. Find the corresponding side in the second.
Practice 3: Statement Scale
A drawing uses
\[ 1\text{ cm}=3.5\text{ m}. \]A wall measures \(6.4\) centimeters on the drawing. Find its actual length.
Practice 4: Drawing Length
A plan uses
\[ 1\text{ cm}=4\text{ m}. \]How long should a \(30\)-meter feature appear on the plan?
Practice 5: Ratio Scale
A map has scale
\[ 1:100{,}000. \]Two points are \(7.2\) centimeters apart on the map. Find the represented distance in kilometers.
Practice 6: Area Scaling
A similar figure is produced with linear scale factor \(3\). If the original area is \(14\text{ cm}^2\), find the new area.
Practice 7: Volume Scaling
A model is enlarged by a linear scale factor of \(2.5\). By what factor does its volume change?
Practice 8: Resized Drawing
A drawing originally uses
\[ 1\text{ cm}=8\text{ m}. \]The drawing is printed at \(50\%\) of its original size. What is the new effective scale in meters per centimeter?
Practice 9: Straight-Line or Route Distance?
A map shows a winding road between two towns. A student measures the direct segment between the town centers and converts it using the map scale.
What distance has the student estimated, and what additional work is needed to estimate the road distance?
Practice 10: Identify the Error
A figure is enlarged by a scale factor of \(4\). A student says its area becomes four times as large.
Explain the error and give the correct area factor.
Practice Answers
Practice 1
\[ k=\frac{22}{8}=2.75. \]Therefore,
\[ \boxed{k=2.75}. \]Practice 2
\[ L'=1.6(12)=19.2. \]Thus,
\[ \boxed{19.2\text{ cm}}. \]Practice 3
\[ 6.4(3.5)=22.4. \]Therefore,
\[ \boxed{22.4\text{ m}}. \]Practice 4
\[ \frac{30}{4}=7.5. \]Therefore,
\[ \boxed{7.5\text{ cm}}. \]Practice 5
The represented distance is
\[ 7.2(100{,}000)=720{,}000\text{ cm}. \]Since
\[ 100{,}000\text{ cm}=1\text{ km}, \]we obtain
\[ \boxed{7.2\text{ km}}. \]Practice 6
The area factor is
\[ 3^2=9. \]Therefore,
\[ A'=9(14)=126. \]Thus,
\[ \boxed{126\text{ cm}^2}. \]Practice 7
\[ 2.5^3=15.625. \]Therefore, the volume becomes
\[ \boxed{15.625} \]times as large.
Practice 8
Printing at \(50\%\) makes every drawing length half as large. A current \(1\)-centimeter segment corresponds to \(2\) centimeters on the original.
Since the original scale was
\[ 1\text{ cm}=8\text{ m}, \]the new scale is
\[ 1\text{ cm}=16\text{ m}. \]Therefore,
\[ \boxed{1\text{ cm}=16\text{ m}}. \]Practice 9
The student has estimated the straight-line distance between the towns.
To estimate road distance, the student must measure along the curved road, using short segments or another suitable path-measurement method, and then apply the map scale to that path length.
Practice 10
Area scales by the square of the linear scale factor:
\[ 4^2=16. \]Therefore, the new area is
\[ \boxed{16} \]times the original area.
What to Remember
- Similar figures preserve shape while corresponding lengths remain proportional.
- The scale factor is the ratio of corresponding new and original lengths.
- A scale drawing uses a consistent relationship between representation distance and actual distance.
- Statement scales, ratio scales, and graphic scales express the same basic proportional idea in different forms.
- Ratio scales require consistent units.
- Perimeter scales by \(k\), area by \(k^2\), and volume by \(k^3\).
- Uniform scaling preserves angles; nonuniform scaling generally distorts shape.
- Map distance depends on the path measured: straight-line distance and route distance are different quantities.
- Resizing a drawing can invalidate a printed numerical scale unless the scale is adjusted.
- Large geographic maps may include projection effects that complicate uniform scale assumptions.
The central relationship is:
\[ \boxed{\text{representation} \xleftrightarrow{\text{scale}} \text{actual spatial system}}. \]Where We Go Next
Scale tells us how distances in a representation correspond to distances in the world. Many real spatial problems also require us to describe direction and to locate points from measured angles and reference lines.
The next lesson, Navigation, Bearings, and Surveying, develops those methods for direction, position, triangulation, field measurement, and practical spatial location.