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Scale factor

A logo has to go on a large poster and on a small badge and still look exactly like itself. What has to happen to its lengths, and what single number describes the change?

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Why this exists

A small company has a logo, drawn 5 cm wide and 3 cm tall. It wants the logo on a poster, much larger, and on a badge, much smaller, and in both places it must look like the same logo: nothing squashed, nothing pulled out of shape.

Adding the same amount to every length does not do it. Adding 35 cm to both sides turns the 5 by 3 logo into one 40 cm wide and 38 cm tall, nearly square. What keeps a shape is multiplying: if the width is multiplied by 8, the height must be multiplied by 8 too, and so must every other length in the drawing. That one multiplier is the subject of this lesson.

The lesson takes two days. Part 1 is what a scale copy is and how to find and use its scale factor. Part 2 is testing whether a figure really is a scale copy, working back from a copy to the original, and writing the scale factor as a ratio.

Definition

A figure B is a scale copy of a figure A when there is one positive number k such that

  • every length in B is k times the matching length in A, and
  • every angle in B is equal to the matching angle in A.

The number k is the scale factor from A to B. Matching lengths are lengths in the same place on the two figures: the bottom edge of one and the bottom edge of the other, the left side and the left side. Where the copy sits, and whether it has been turned or flipped over, does not matter.

Dividing any copy length by its matching original length gives k:

k = \frac{\text{copy length}}{\text{original length}}

Both lengths must be in the same unit, and the scale factor itself has no unit: it is a pure number, “times 8” or “times \tfrac{2}{5}”.

The copy is an enlargement when k > 1, a reduction when k < 1, and when k = 1 it is an exact copy, the same size and shape as the original.

Intuition

4 3 1 8 6 2 original copy
Figure 1Every length of the original is multiplied by 2. The corners and the slant of the sloping edge are unchanged.

Figure 1 shows a shape drawn on a grid and a copy of it. The bottom edge is 4 units in the original and 8 in the copy; the left side is 3 units and becomes 6; the step on the right is 1 unit and becomes 2. Each copy length is twice its original length, so the scale factor is 2.

Put the matching lengths in a table and the pattern is the one from proportional relationships:

Original length4312
Copy length8624

The last column is the top of the step, from the right side across to the slant. The copy lengths are proportional to the original lengths, and the scale factor is the constant of proportionality.

The angles did not change. Every square corner is still square, and the sloping edge, which climbs 2 units while going 1 unit across in the original, climbs 4 while going 2 across in the copy: just as steep. A scale copy is the same shape at a different size, and the scale factor is the size change, written as one multiplier.

Results

Finding the factor. If B is a scale copy of A, then any one pair of matching lengths gives the scale factor, copy length over original length, and every pair gives the same answer.

Using the factor. Any length of the copy is the matching original length times k.

Bigger, smaller or the same. A factor above 1 enlarges, a factor between 0 and 1 reduces, and a factor of exactly 1 keeps the size.

Going back. If the scale factor from A to B is k, then every original length is the copy length divided by k, and the scale factor from B back to A is the reciprocal, \tfrac{1}{k}.

As a ratio. If matching lengths, original to copy, are in the ratio a : b, then the scale factor from the original to the copy is \tfrac{b}{a}, and from the copy back to the original it is \tfrac{a}{b}.

Testing a rectangle. Two rectangles are scale copies of each other exactly when their widths and heights give the same quotient, copy over original. Equivalently, the width divided by the height is the same for both.

Why it is true

Every pair gives the same factor. By definition each copy length is k times its original length. Dividing a copy length by its original length undoes the multiplication and leaves k, whichever pair was chosen. If two pairs give different quotients, there is no single k, and the figure is not a scale copy.

Bigger or smaller. Multiplying a positive length by a number above 1 makes it larger, by a number between 0 and 1 makes it smaller, and by 1 leaves it alone.

Going back. From \text{copy} = k \times \text{original}, divide both sides by k:

\text{original} = \text{copy} \div k = \text{copy} \times \frac{1}{k}

Dividing by a number is multiplying by its reciprocal, so \tfrac{1}{k} is a single multiplier that takes every copy length back to its original length. That makes A a scale copy of B, with scale factor \tfrac{1}{k}. The two factors undo each other: k \times \tfrac{1}{k} = 1.

As a ratio. If original and copy lengths are in the ratio a : b, then \tfrac{\text{copy}}{\text{original}} = \tfrac{b}{a}, which is the scale factor.

Testing a rectangle. Every rectangle has four right angles, so the angle condition holds automatically and only lengths need checking. A rectangle’s lengths are its width and its height, repeated. If the copy’s width is kw and its height is kh, then its width over its height is

\frac{kw}{kh} = \frac{w}{h}

the same as the original’s. In the other direction, if two rectangles have equal width over height, then cross-multiplying shows that the widths and the heights grow by the same factor.

Worked examples

Examples 1 to 3 belong to Part 1, and Examples 4 to 7 to Part 2.

Example 1

A logo for a poster and a badge

Margarethe’s logo is 5 cm wide and 3 cm tall. The poster version is 40 cm wide and the badge version is 2 cm wide, both scale copies. Find each scale factor and each height.

Solution

The widths are matching lengths, so each scale factor is the copy’s width over the original’s width.

For the poster:

k = \frac{40}{5} = 8

Every length is multiplied by 8, so the height is 3 \times 8 = 24 cm. The factor is above 1, and the poster logo is an enlargement.

For the badge:

k = \frac{2}{5} = 0.4

The height is 3 \times 0.4 = 1.2 cm. The factor is below 1, and the badge logo is a reduction.

Check. The original is \tfrac{5}{3} times as wide as it is tall; so is the poster logo, \tfrac{40}{24} = \tfrac{5}{3}, and so is the badge, \tfrac{2}{1.2} = \tfrac{5}{3}.

Example 2

Missing sides of a triangle

In Figure 2 the larger triangle is a scale copy of the smaller one. The original’s sides are 4 cm, 6 cm and 7 cm, and the copy’s side matching the 6 cm side is 9 cm. Find a and b.

7 cm 4 cm 6 cm original a b 9 cm copy
Figure 2The copy’s 9 cm side matches the original’s 6 cm side. Find a and b.
Solution

The only pair of matching sides with both lengths known is the 6 cm side and the 9 cm side, so the scale factor comes from them:

k = \frac{9}{6} = \frac{3}{2}

Side a matches the 7 cm side and side b matches the 4 cm side, so each is its original length times \tfrac{3}{2}:

a = 7 \times \frac{3}{2} = 10.5 \text{ cm} \qquad b = 4 \times \frac{3}{2} = 6 \text{ cm}

The copy’s sides are 6 cm, 9 cm and 10.5 cm, and its angles are the original’s angles, as the matching arcs show.

Example 3

A toy giraffe and a real one

A toy giraffe is a scale copy of a real giraffe. The real giraffe is 5 m tall and the toy is 20 cm tall. Find the scale factor from the real giraffe to the toy, and the length of the toy’s neck if the real neck is 2 m long.

Solution

The two heights are in different units, and a quotient of 20 by 5 would say the toy is 4 times as big as the giraffe. Write both in centimeters first, using 1 \text{ m} = 100 \text{ cm} (metric conversion): the giraffe is 500 cm tall.

k = \frac{20}{500} = \frac{1}{25}

The toy is a reduction: every length of the toy is \tfrac{1}{25} of the real length. The real neck is 2 \text{ m} = 200 \text{ cm}, so the toy’s neck is

200 \times \frac{1}{25} = 8 \text{ cm}
Example 4

A true copy or a stretched one

Figure 3 shows a house shape and two attempts at copying it. Which of P and Q is a scale copy of the original?

2 2 original 4 4 P 4 2 Q
Figure 3Both copies are twice as wide as the original. Only one of them is twice as tall.
Solution

Compare matching lengths, copy over original, and see whether one factor works for all of them.

For P: the width goes from 2 to 4 and the wall from 2 to 4, so both give

\frac{4}{2} = 2

The peak of the roof also goes from 3 units high to 6, again times 2. The roof’s edge climbs 1 unit for each unit across in the original and 2 units for each 2 across in P, so it is equally steep and the angles are unchanged. P is a scale copy with scale factor 2.

For Q: the width gives \tfrac{4}{2} = 2, but the wall gives \tfrac{2}{2} = 1. Two matching pairs give two different quotients, so no single number multiplies every length, and Q is not a scale copy. It has been stretched sideways only, and its roof is visibly flatter: the angles changed too.

Example 5

Going back from the copy to the original

Ottavia had a photo enlarged with a scale factor of 2.5. The enlarged print is 45 cm wide and 30 cm tall. How large was the original photo, and what is the scale factor from the print back to the original?

Solution

Each print length is 2.5 times the original length, so each original length is the print length divided by 2.5:

45 \div 2.5 = 18 \qquad 30 \div 2.5 = 12

The original photo was 18 cm wide and 12 cm tall.

Dividing by 2.5 is multiplying by its reciprocal. Since 2.5 = \tfrac{5}{2}, the reciprocal is \tfrac{2}{5}, and that is the scale factor from the print back to the original:

45 \times \frac{2}{5} = 18 \qquad 30 \times \frac{2}{5} = 12

Check. 18 \times 2.5 = 45 and 12 \times 2.5 = 30.

Example 6

The scale factor as a ratio

Severin copies a shape on a grid so that matching lengths, original to copy, are in the ratio 4 : 7. One side of the original is 12 units long. Another side of the copy is 35 units long. Find the copy’s side matching the first and the original’s side matching the second.

Solution

The ratio 4 : 7 says that for every 4 units of original length there are 7 units of copy length, so the scale factor from original to copy is

k = \frac{7}{4}

The copy’s side matching the 12 unit side is 12 \times \tfrac{7}{4} = 21 units.

Going back, the factor is the reciprocal, \tfrac{4}{7}, so the original’s side matching the 35 unit side is 35 \times \tfrac{4}{7} = 20 units.

The same answer comes from a proportion, since the ratio of original to copy is the same for every pair of matching sides:

\frac{4}{7} = \frac{x}{35} \quad\text{so}\quad x = \frac{4 \times 35}{7} = 20
Example 7

Which print sizes keep the shape

Nils has a photo 15 cm wide and 10 cm tall. The print shop offers 18 cm by 12 cm, 20 cm by 16 cm and 6 cm by 4 cm, each listed width first. Which of these are scale copies of the photo, and with what scale factor?

Solution

All the shapes are rectangles, so only the lengths need testing. A quick test compares each shape’s own width over its height, which a scale copy keeps (Results). The photo has

\frac{15}{10} = \frac{3}{2}

The 18 by 12 print has \tfrac{18}{12} = \tfrac{3}{2}, the same, so it is a scale copy. Its scale factor is \tfrac{18}{15} = \tfrac{6}{5}, an enlargement; the heights agree, \tfrac{12}{10} = \tfrac{6}{5}.

The 20 by 16 print has \tfrac{20}{16} = \tfrac{5}{4}, which is not \tfrac{3}{2}, so it is not a scale copy: the photo would have to be stretched or cut to fill it.

The 6 by 4 print has \tfrac{6}{4} = \tfrac{3}{2}, so it is a scale copy, with scale factor \tfrac{6}{15} = \tfrac{2}{5}, a reduction.

Boundaries

The factor is positive. Lengths are positive, so a scale factor is a positive number. A factor of 0 would shrink every length to nothing and leave no figure at all.

Lengths, not areas. The scale factor multiplies lengths. Areas and volumes change by a different amount: a copy with scale factor 2 has four times the area, not twice. That belongs to scale factor, area and volume.

Same unit on both lengths. The quotient copy over original is the scale factor only when both lengths are measured in the same unit, as the giraffe in Example 3 shows.

Matching lengths only. The quotient is taken between lengths in the same place on both figures. Dividing the copy’s width by the original’s height gives a number, but not the scale factor.

Equal length quotients are not always enough. For a shape with four or more sides, every length can be multiplied by the same number while the angles change. A square with sides 2 cm and a rhombus with sides 3 cm and angles of 60^\circ and 120^\circ have every side in the quotient \tfrac{3}{2}, yet the rhombus is not a scale copy of the square. For a rectangle the angles are always right angles, so the lengths settle it. For triangles, sides in one quotient do force the angles to match; that is the SSS similarity theorem of geometry.

Where this goes next

Once a single factor is known, it applies to every length in the figure, including diagonals and the perimeter: scale factor and length. A map or floor plan is a scale copy of something too large to draw, and its scale is often stated with units, such as one centimeter for every kilometer; using a scale on maps, plans and models turns such a scale into a factor. Areas and volumes follow the square and the cube of the factor, in scale factor and area and later scale factor and perimeter, area and volume.

In geometry the same number is the factor of a dilation, the move that slides every point toward or away from a center to make a scale copy, and scale copies become similar figures. That idea explains why a line has one slope everywhere, in slope triangles.

Exercises

Part 1: finding the scale factor

  1. A side of an original figure is 6 cm, and the matching side of a scale copy is 15 cm. Find the scale factor. Is the copy an enlargement or a reduction?
  2. A side of an original figure is 12 units, and the matching side of a scale copy is 3 units. Find the scale factor.
  3. A side of an original figure is 7.5 cm, and the matching side of a scale copy is 7.5 cm. Find the scale factor and describe the copy.
  4. A side of an original figure is 8 cm, and the matching side of a scale copy is 6 cm. Find the scale factor as a fraction.
  5. A right triangle drawn on a grid has legs of 3 and 4 units and a slanted side of 5 units. It is copied with scale factor 3. Find the copy’s three sides.
  6. Primo’s logo is 12 cm wide and 9 cm tall. He makes a badge from it with scale factor \tfrac{1}{3}. How wide and how tall is the badge logo?

Part 1: using the factor

  1. A real elephant is 3 m tall and a toy elephant, a scale copy of it, is 15 cm tall. Find the scale factor from the real elephant to the toy. The real trunk is 1.8 m long; how long is the toy’s trunk?
  2. A triangle has sides 5 cm, 8 cm and 10 cm. In a scale copy, the side matching the 10 cm side is 25 cm. Find the copy’s other two sides.
  3. Rozalia draws a rectangle 9 units by 6 units on a grid and makes a scale copy whose longer side is 6 units. Find the scale factor and the copy’s shorter side.
  4. A copy is made with scale factor 0.6. Is it larger or smaller than the original? What does a 15 cm side become?

Part 2: scale copy or stretched?

  1. A logo 8 cm wide and 5 cm tall is resized to 12 cm by 7.5 cm and, separately, to 12 cm by 9 cm, width first. Which is a scale copy, and with what factor?
  2. A rectangle is 10 by 4. Is a 25 by 10 rectangle a scale copy of it? Is a 13 by 7 rectangle?
  3. A triangle has sides 4, 5 and 6. Another triangle has sides 8, 10 and 15. Is the second a scale copy of the first?
  4. A photo is 12 cm wide and 9 cm tall. Which of these prints, width first, are scale copies: 16 by 12, 20 by 15, 18 by 12? Give the scale factor of each one that is.

Part 2: going back to the original

  1. A copy made with scale factor 4 has a side of 22 cm. How long is the matching side of the original?
  2. A copy made with scale factor \tfrac{2}{3} has a side of 18 units. How long is the matching side of the original?
  3. A toy climber was made from a real person with scale factor \tfrac{1}{12}. The toy is 14.5 cm tall. How tall is the real person, and what is the scale factor from the toy back to the person?
  4. A poster logo is 54 cm wide and was made from the original with scale factor 4.5. Find the original width and the scale factor from the poster back to the original, as a fraction.

Part 2: the scale factor as a ratio

  1. Matching lengths, original to copy, are in the ratio 3 : 8. Find the scale factor from the original to the copy, and from the copy back to the original.
  2. Matching lengths, original to copy, are in the ratio 5 : 2. A side of the original is 35 cm. How long is the matching side of the copy?
  3. Matching lengths, original to copy, are in the ratio 2 : 5. A side of the copy is 45 units. How long is the matching side of the original?
  4. A copy has scale factor 1.25. Write the ratio of matching lengths, original to copy, using the smallest whole numbers.

Reasoning

  1. A rectangle is 4 cm by 2 cm. A parallelogram has sides of 6 cm and 3 cm and angles of 70^\circ and 110^\circ. Each side of the parallelogram is \tfrac{3}{2} times the matching side of the rectangle. Explain why the parallelogram is still not a scale copy of the rectangle.
  2. Ilse copies a shape with scale factor 4, then copies the copy with scale factor \tfrac{1}{4}. Explain how the final figure compares with the original, and what single scale factor takes the original to it.
  3. A rectangle is 9 cm by 3 cm. One new rectangle is made by adding 3 cm to its width and to its height, another by multiplying its width and its height by 2. Show which of the two is a scale copy of the original, and explain why adding the same length to every side changes the shape while multiplying every side by the same number keeps it.
Answers
  1. k = \tfrac{15}{6} = 2.5, an enlargement, since the factor is above 1.
  2. k = \tfrac{3}{12} = \tfrac{1}{4}, a reduction.
  3. k = \tfrac{7.5}{7.5} = 1. The copy is the same size and shape as the original, an exact copy.
  4. k = \tfrac{6}{8} = \tfrac{3}{4}.
  5. 9, 12 and 15 units: 3 \times 3, 4 \times 3 and 5 \times 3.
  6. 4 cm wide and 3 cm tall: 12 \times \tfrac{1}{3} = 4 and 9 \times \tfrac{1}{3} = 3.
  7. k = \tfrac{1}{20} and the toy’s trunk is 9 cm. In centimeters the elephant is 300 cm tall, so k = \tfrac{15}{300} = \tfrac{1}{20}; the trunk is 180 cm, and 180 \times \tfrac{1}{20} = 9.
  8. 12.5 cm and 20 cm. The factor is \tfrac{25}{10} = 2.5, so 5 \times 2.5 = 12.5 and 8 \times 2.5 = 20.
  9. k = \tfrac{2}{3} and the shorter side is 4 units. The longer sides match, so k = \tfrac{6}{9} = \tfrac{2}{3}, and 6 \times \tfrac{2}{3} = 4.
  10. Smaller, since 0.6 < 1. The side becomes 15 \times 0.6 = 9 cm.
  11. The 12 by 7.5 logo, with factor 1.5: \tfrac{12}{8} = 1.5 and \tfrac{7.5}{5} = 1.5. The 12 by 9 logo is stretched: \tfrac{12}{8} = 1.5 but \tfrac{9}{5} = 1.8.
  12. The 25 by 10 rectangle is, with factor 2.5: \tfrac{25}{10} = 2.5 and \tfrac{10}{4} = 2.5. The 13 by 7 rectangle is not: \tfrac{13}{10} = 1.3 but \tfrac{7}{4} = 1.75.
  13. No. Multiplying by one positive number keeps the sides in order, so the shortest sides match, then the middle, then the longest: \tfrac{8}{4} = 2 and \tfrac{10}{5} = 2 but \tfrac{15}{6} = 2.5, so no single factor works.
  14. 16 by 12, with factor \tfrac{4}{3}, and 20 by 15, with factor \tfrac{5}{3}. The photo has width over height \tfrac{12}{9} = \tfrac{4}{3}; so do \tfrac{16}{12} and \tfrac{20}{15}, and the factors are \tfrac{16}{12} = \tfrac{4}{3} and \tfrac{20}{12} = \tfrac{5}{3}. The 18 by 12 print has \tfrac{18}{12} = \tfrac{3}{2}, so it is not a scale copy.
  15. 5.5 cm, since 22 \div 4 = 5.5.
  16. 27 units, since 18 \div \tfrac{2}{3} = 18 \times \tfrac{3}{2} = 27.
  17. 174 cm, and the factor back is 12. Dividing by \tfrac{1}{12} is multiplying by 12: 14.5 \times 12 = 174.
  18. 12 cm, and the factor back is \tfrac{2}{9}. 54 \div 4.5 = 12, and since 4.5 = \tfrac{9}{2} its reciprocal is \tfrac{2}{9}; check 54 \times \tfrac{2}{9} = 12.
  19. From original to copy \tfrac{8}{3}, and from copy back to original \tfrac{3}{8}.
  20. 14 cm. The factor is \tfrac{2}{5}, and 35 \times \tfrac{2}{5} = 14.
  21. 18 units. The factor back is \tfrac{2}{5}, and 45 \times \tfrac{2}{5} = 18.
  22. 4 : 5. Since 1.25 = \tfrac{5}{4}, every 4 units of original length become 5 units of copy length.
  23. A scale copy must keep every angle as well as multiply every length by one number. The rectangle’s angles are all 90^\circ and the parallelogram’s are 70^\circ and 110^\circ, so the angles have changed: the parallelogram is a different shape, not the rectangle made larger.
  24. It is exactly the same size and shape as the original. Each length is first multiplied by 4 and then by \tfrac{1}{4}, which together multiply it by 4 \times \tfrac{1}{4} = 1, so the single scale factor is 1.
  25. Only the multiplied one, 18 cm by 6 cm. Adding gives 12 cm by 6 cm: the width grows by the factor \tfrac{12}{9} = \tfrac{4}{3} but the height by \tfrac{6}{3} = 2, so no single factor works, and width over height falls from \tfrac{9}{3} = 3 to \tfrac{12}{6} = 2. Multiplying keeps width over height at \tfrac{18}{6} = 3. Adding 3 cm doubles the 3 cm side but adds only a third to the 9 cm side: the same amount is a bigger share of a short side than of a long one. Multiplying grows every side by the same share, so the sides keep their proportions.
kg1128 · updated 2026-10-07 · CC-BY-SA-4.0