The Central Idea
Suppose a small rectangle measures
\[ 4\text{ cm}\times6\text{ cm}. \]A larger rectangle measures
\[ 10\text{ cm}\times15\text{ cm}. \]Compare corresponding sides:
\[ \frac{10}{4}=2.5, \] \[ \frac{15}{6}=2.5. \]Every corresponding length was multiplied by the same number.
That common multiplier is the scale factor.
The rectangles have the same shape but different sizes, so they are similar.
What Does Similar Mean?
Two figures are similar when they have the same shape, even if they do not have the same size.
For polygons, similarity means:
- corresponding angles are equal;
- corresponding side lengths are proportional.
Similarity preserves geometric structure while allowing uniform enlargement or reduction.
What Is a Scale Factor?
A scale factor is the multiplier that converts one set of corresponding lengths into another.
If a side of length \(a\) becomes a corresponding side of length \(b\), then
\[ k=\frac{b}{a}. \]The new length is
\[ b=ka. \]Enlargements
If
\[ k>1, \]the image is larger than the original.
For example, a scale factor of
\[ 3 \]changes a length of \(5\) units into
\[ 3(5)=15 \]units.
Reductions
If
\[ 0For example, a scale factor of
\[ \frac12 \]changes a length of \(10\) units into
\[ \frac12(10)=5. \]Congruence as a Special Case
If
\[ k=1, \]all corresponding lengths remain the same.
The figures are not merely similar; they are congruent.
Congruence is therefore a special case of similarity with scale factor \(1\).
Corresponding Sides
Correct correspondence is essential.
Suppose triangle \(ABC\) is similar to triangle \(DEF\), with
\[ A\leftrightarrow D, \qquad B\leftrightarrow E, \qquad C\leftrightarrow F. \]Then the corresponding sides are:
\[ AB\leftrightarrow DE, \] \[ BC\leftrightarrow EF, \] \[ AC\leftrightarrow DF. \]Similarity Notation
We may write
\[ \triangle ABC\sim\triangle DEF. \]The order of the letters communicates correspondence:
\[ A\leftrightarrow D, \qquad B\leftrightarrow E, \qquad C\leftrightarrow F. \]Changing the order carelessly can lead to incorrect proportions.
Corresponding-Side Proportions
If
\[ \triangle ABC\sim\triangle DEF, \]then
\[ \frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}. \]Each ratio compares corresponding lengths in the same direction.
Finding a Missing Side
Suppose similar triangles have corresponding sides
\[ 4\leftrightarrow10 \]and
\[ 6\leftrightarrow x. \]Write
\[ \frac{4}{10} = \frac{6}{x}. \]Cross multiply:
\[ 4x=60. \]So
\[ \boxed{x=15}. \]Use the Scale Factor When It Is Obvious
In the previous example,
\[ 4\longrightarrow10 \]has scale factor
\[ \frac{10}{4}=2.5. \]Then
\[ 6(2.5)=15. \]Scaling is often faster than writing a proportion when the multiplier is easy to see.
Find the Scale Factor First
Suppose a side of length \(8\) corresponds to a side of length \(14\).
The scale factor from the smaller figure to the larger is
\[ k=\frac{14}{8} = \frac74. \]Any other corresponding length is multiplied by
\[ \frac74. \]Direction Matters
If the scale factor from Figure A to Figure B is
\[ \frac74, \]then the scale factor from Figure B back to Figure A is
\[ \frac47. \]The reverse scale factor is the reciprocal of the forward scale factor.
Similarity Preserves Angles
Uniform scaling changes lengths but not angle measures.
If two triangles are similar and one has angles
\[ 40^\circ,\quad60^\circ,\quad80^\circ, \]then the corresponding triangle has the same three angle measures.
Similarity Preserves Shape
A square enlarged by any positive scale factor remains a square.
A circle enlarged uniformly remains a circle.
A triangle enlarged uniformly preserves all of its angles.
Similarity is about preserving shape under uniform scaling.
Uniform Scaling
For true geometric similarity, every linear dimension must be multiplied by the same scale factor.
If width is doubled but height is tripled, a rectangle generally changes shape.
That is not a similarity transformation.
Maps and Scale Drawings
A map may use the scale
\[ 1\text{ cm}:5\text{ km}. \]This means each centimeter on the map represents 5 kilometers in reality.
We can write
\[ \frac{5\text{ km}}{1\text{ cm}}. \]This scale is a constant of proportionality.
Map Example
Two locations are 7.2 cm apart on a map with scale
\[ 1\text{ cm}:5\text{ km}. \]The actual distance is
\[ 7.2(5)=36. \]Therefore,
\[ \boxed{36\text{ km}}. \]Architectural and Engineering Drawings
A drawing may use a scale such as
\[ 1:50. \]This means one unit on the drawing corresponds to 50 of the same units in reality.
For example,
\[ 2\text{ cm} \]on the drawing represents
\[ 100\text{ cm} \]in the actual object.
Models
Scale models use the same proportional structure.
If a model car is built at
\[ 1:24 \]scale, then each model length is
\[ \frac1{24} \]of the corresponding real-car length.
Unit Consistency in Scale Problems
Suppose a drawing uses
\[ 1\text{ cm}:2\text{ m}. \]The units differ, so be careful.
We may leave the ratio as a contextual scale, or convert to common units:
\[ 2\text{ m}=200\text{ cm}. \]Then the scale is
\[ 1:200. \]Representative Fraction
A map scale written as
\[ 1:100000 \]means one unit on the map represents 100,000 of the same units in reality.
This type of unitless same-unit scale is sometimes called a representative fraction or ratio scale.
Scale and Perimeter
If all lengths are multiplied by \(k\), then perimeter is also multiplied by \(k\).
For example, if a rectangle's dimensions are doubled, then each side doubles, so the perimeter doubles.
Thus,
\[ P_{\text{new}}=kP_{\text{old}}. \]Why Perimeter Uses the Same Scale Factor
Suppose a polygon has side lengths
\[ a,b,c,\ldots \]and perimeter
\[ P=a+b+c+\cdots. \]After scaling by \(k\):
\[ P'=ka+kb+kc+\cdots. \]Factor out \(k\):
\[ P'=k(a+b+c+\cdots)=kP. \]Scale and Area
Area does not scale by the same factor as length.
If all lengths are multiplied by \(k\), then area is multiplied by
\[ \boxed{k^2}. \]Why Area Scales by \(k^2\)
Consider a rectangle:
\[ A=lw. \]Scale both dimensions by \(k\):
\[ A'=(kl)(kw). \]Then
\[ A'=k^2lw. \]So
\[ \boxed{A'=k^2A}. \]Area-Scale Example
Suppose two similar figures have linear scale factor
\[ 3. \]Their areas differ by the factor
\[ 3^2=9. \]If the smaller area is
\[ 20\text{ cm}^2, \]the larger area is
\[ 20(9)=180\text{ cm}^2. \]Finding Linear Scale from Area Scale
If the area ratio of two similar figures is
\[ 25:49, \]then the linear scale ratio is
\[ \sqrt{\frac{25}{49}} = \frac57. \]Area factors must be square-rooted to recover length factors.
Scale and Volume
If all linear dimensions are multiplied by \(k\), then volume is multiplied by
\[ \boxed{k^3}. \]Why Volume Scales by \(k^3\)
For a rectangular prism,
\[ V=lwh. \]Scale every dimension by \(k\):
\[ V'=(kl)(kw)(kh). \]Thus,
\[ V'=k^3lwh. \]Therefore,
\[ \boxed{V'=k^3V}. \]Volume-Scale Example
If a solid is enlarged by linear scale factor
\[ 2, \]its volume is multiplied by
\[ 2^3=8. \]Doubling every length produces eight times the volume.
Length, Area, and Volume Scale Differently
| Measurement | Scale Multiplier |
|---|---|
| Length | \(k\) |
| Perimeter | \(k\) |
| Area | \(k^2\) |
| Volume | \(k^3\) |
This Is a Dimensional Pattern
Length is one-dimensional, so it scales with
\[ k. \]Area is two-dimensional, so it scales with
\[ k^2. \]Volume is three-dimensional, so it scales with
\[ k^3. \]The exponent reflects the number of independent length dimensions being multiplied.
Similarity and Proportions
If two figures are similar, corresponding sides form proportions.
For example, if
\[ \frac{a}{d} = \frac{b}{e} = \frac{c}{f}, \]then the same scale factor relates each pair of corresponding lengths.
This is one of the most important geometric uses of proportions.
Triangle Similarity
Triangles receive special attention because several conditions can prove similarity.
Common similarity criteria include:
- AA — two corresponding angles are equal;
- SAS similarity — an included angle is equal and surrounding sides are proportional;
- SSS similarity — all corresponding sides are proportional.
AA Similarity
If two angles of one triangle equal two angles of another triangle, then the third angles must also be equal.
The triangles are similar.
This is especially useful because no side lengths are needed to establish similarity.
SSS Similarity
Suppose two triangles have side sets
\[ 3,4,5 \]and
\[ 6,8,10. \]Then
\[ \frac63=\frac84=\frac{10}{5}=2. \]All corresponding sides share the same scale factor, so the triangles are similar.
SAS Similarity
Suppose two triangles have an equal included angle, and the sides surrounding that angle satisfy
\[ \frac{a}{d} = \frac{b}{e}. \]Then the triangles are similar by SAS similarity.
Indirect Measurement
Similarity can be used to measure quantities that are difficult to reach directly.
Examples include:
- building heights;
- tree heights;
- river widths;
- distances across obstacles;
- dimensions in photographs or diagrams.
Shadow Method
Suppose a 2-meter pole casts a 3-meter shadow at the same time a tree casts a 12-meter shadow.
If sunlight creates the same angle, the triangles are similar.
Let \(h\) be the tree height:
\[ \frac{2}{3} = \frac{h}{12}. \]Then
\[ 3h=24, \] \[ \boxed{h=8\text{ m}}. \]Why the Shadow Method Works
The pole and tree are vertical, so each forms a right angle with the ground.
The sun's rays arrive at approximately the same angle for nearby objects.
Thus, the two right triangles share another equal angle and are similar by AA.
Photographs and Images
If a photograph is uniformly resized, its width and height must use the same scale factor to preserve shape.
If a photo changes from
\[ 800\times600 \]pixels to width
\[ 1200, \]the scale factor is
\[ \frac{1200}{800}=1.5. \]The new height should be
\[ 600(1.5)=900. \]So the proportional size is
\[ 1200\times900. \]Aspect Ratio
The aspect ratio of a rectangle compares width and height.
For
\[ 1920\times1080, \]the ratio is
\[ \frac{1920}{1080} = \frac{16}{9}. \]Resizing proportionally preserves this ratio.
Distortion
If width and height are scaled by different factors, the image is distorted.
For example, changing
\[ 800\times600 \]to
\[ 1200\times700 \]does not preserve the same aspect ratio.
The result is not geometrically similar to the original rectangle.
Scale in Coordinate Geometry
A dilation centered at the origin with scale factor \(k\) maps
\[ (x,y) \]to
\[ (kx,ky). \]Every coordinate is multiplied by the same factor.
Coordinate Example
Under a dilation with scale factor
\[ 3, \]the point
\[ (2,-4) \]maps to
\[ (6,-12). \]The direction from the origin is preserved while the distance from the origin is tripled.
Dilations Preserve Angles
A dilation changes lengths but preserves angle measures.
This is one reason dilations generate similar figures.
Dilations Preserve Parallelism
Lines not passing through the center of dilation map to parallel lines.
Lines passing through the center remain on the same line.
This gives dilations a strong geometric structure beyond simple numerical scaling.
Scale Factor and Distance
Under a dilation of scale factor \(k\), any distance between corresponding points is multiplied by
\[ |k|. \]In elementary similarity, scale factors are usually taken as positive.
Similarity Is an Equivalence Relation
Similarity has three useful properties:
- a figure is similar to itself;
- if Figure A is similar to Figure B, then Figure B is similar to Figure A;
- if Figure A is similar to Figure B and Figure B is similar to Figure C, then Figure A is similar to Figure C.
This means similarity organizes figures into families of the same shape.
Composing Scale Factors
Suppose Figure A is scaled by \(2\) to get Figure B, and Figure B is scaled by \(3\) to get Figure C.
The total scale factor from A to C is
\[ 2\cdot3=6. \]Scale factors compose multiplicatively.
Undoing a Scale
If a figure was enlarged by factor
\[ 5, \]the inverse scaling uses
\[ \frac15. \]The two scale factors multiply to
\[ 5\cdot\frac15=1. \]Worked Example: Missing Side
Two similar triangles have corresponding sides
\[ 5\leftrightarrow12 \]and
\[ 8\leftrightarrow x. \]Write
\[ \frac{5}{12} = \frac{8}{x}. \]Cross multiply:
\[ 5x=96. \]Thus,
\[ \boxed{x=19.2}. \]Worked Example: Area Scaling
Two similar figures have linear scale factor
\[ 4. \]If the smaller area is
\[ 7\text{ m}^2, \]the larger area is
\[ 7(4^2) = 7(16) = 112. \]Therefore,
\[ \boxed{112\text{ m}^2}. \]Worked Example: Volume Scaling
Two similar solids have linear scale factor
\[ \frac32. \]If the smaller volume is
\[ 64\text{ cm}^3, \]then the larger volume is
\[ 64\left(\frac32\right)^3. \]Since
\[ \left(\frac32\right)^3=\frac{27}{8}, \]we get
\[ 64\cdot\frac{27}{8}=216. \]Thus,
\[ \boxed{216\text{ cm}^3}. \]Worked Example: Recover Linear Scale from Volume
Suppose two similar solids have volume ratio
\[ 8:125. \]The linear scale ratio is
\[ \sqrt[3]{\frac{8}{125}} = \frac25. \]So the corresponding length ratio is
\[ \boxed{2:5}. \]Worked Example: Map Scale
A map uses the scale
\[ 1\text{ cm}:12\text{ km}. \]A road measures 4.5 cm on the map.
The actual distance is
\[ 4.5(12)=54. \]Therefore,
\[ \boxed{54\text{ km}}. \]Worked Example: Reverse Map Problem
A real distance is 84 km. The map scale is
\[ 1\text{ cm}:7\text{ km}. \]The map distance is
\[ \frac{84}{7}=12. \]Therefore,
\[ \boxed{12\text{ cm}}. \]A Reliable Similarity Workflow
- Identify corresponding points, sides, and angles.
- Confirm that the figures are similar or that similarity is justified.
- Choose a consistent direction for the ratios.
- Find the scale factor when possible.
- Use the scale factor or a proportion to find missing lengths.
- For areas, square the linear scale factor.
- For volumes, cube the linear scale factor.
- Keep units consistent.
- Check whether the answer is reasonable for an enlargement or reduction.
A Reliable Scale-Drawing Workflow
- Write the stated scale clearly.
- Identify the drawing measurement and real measurement.
- Keep corresponding units aligned.
- Convert units if necessary.
- Use multiplication, division, or a proportion to find the missing value.
- Interpret the result in the requested unit.
Common Errors
Matching Noncorresponding Sides
Similarity proportions work only when corresponding measurements are aligned correctly.
Using Different Ratio Directions
If one ratio is small-to-large, all corresponding ratios should use small-to-large.
Using the Linear Scale Factor for Area
Area scales by
\[ k^2, \]not \(k\).
Using the Linear Scale Factor for Volume
Volume scales by
\[ k^3. \]Assuming Equal Angles Alone Prove Similarity for All Polygons
For triangles, AA is sufficient. For general polygons, equal corresponding angles alone do not necessarily guarantee similarity.
Ignoring Units
A map scale involving centimeters and kilometers must be interpreted with those units intact.
Scaling Different Dimensions by Different Factors
Nonuniform scaling distorts shape and destroys similarity.
Practice
1. Find the Scale Factor
A side of length \(6\) corresponds to a side of length \(15\). Find the scale factor from the first figure to the second.
Check Answer
\[ k=\frac{15}{6} = \frac52. \]Answer:
\[ \boxed{\frac52}. \]2. Find a Missing Side
Two similar triangles have corresponding sides
\[ 4\leftrightarrow10 \]and
\[ 7\leftrightarrow x. \]Check Answer
The scale factor is
\[ \frac{10}{4}=2.5. \]So
\[ x=7(2.5)=17.5. \]Answer:
\[ \boxed{17.5}. \]3. Area Scaling
A figure is enlarged by scale factor \(3\). By what factor does its area change?
Check Answer
\[ 3^2=9. \]Answer:
\[ \boxed{9}. \]4. Volume Scaling
A solid is reduced by scale factor
\[ \frac12. \]By what factor does its volume change?
Check Answer
\[ \left(\frac12\right)^3 = \frac18. \]Answer:
\[ \boxed{\frac18}. \]5. Map Scale
A map uses
\[ 1\text{ cm}:8\text{ km}. \]What real distance corresponds to 6.5 cm?
Check Answer
\[ 6.5(8)=52. \]Answer:
\[ \boxed{52\text{ km}}. \]6. Shadow Problem
A 1.5 m object casts a 2 m shadow. At the same time, a tree casts a 10 m shadow. Find the tree height.
Check Answer
\[ \frac{1.5}{2} = \frac{h}{10}. \] \[ 2h=15. \] \[ h=7.5. \]Answer:
\[ \boxed{7.5\text{ m}}. \]7. Similar or Congruent?
Two figures are similar with scale factor \(1\). What stronger relationship do they have?
Check Answer
They are congruent.
Key Ideas
- Similar figures have the same shape but may have different sizes.
- Corresponding angles are equal and corresponding lengths are proportional.
- The scale factor is the common multiplier between corresponding lengths.
- A scale factor greater than 1 enlarges; a factor between 0 and 1 reduces.
- Congruence is similarity with scale factor \(1\).
- Correspondence and ratio direction must remain consistent.
- Maps, drawings, models, photographs, and dilations use the same proportional scaling ideas.
- Perimeter scales by \(k\), area by \(k^2\), and volume by \(k^3\).
- The exponents arise from the dimensional structure of length, area, and volume.
- Similarity allows indirect measurement through proportional triangles.
- Uniform scaling preserves shape; nonuniform scaling causes distortion.
- Scale factors compose multiplicatively and reverse by reciprocals.
Where This Leads
Scale and similarity connect proportional reasoning directly to geometry.
The same ideas will reappear in triangle similarity, trigonometry, coordinate transformations, maps and models, dimensional scaling, area and volume comparisons, perspective, indirect measurement, and many applied problems where shape is preserved while size changes.