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§ Geometry·Grades 7–8

Similarity & Scale Factors Worksheets

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Easy

10 problems

Medium

20 problems

Hard

20 problems

Mixed

30 problems

Free printable similarity & scale factors worksheets with step-by-step answer keys. Every worksheet is uniquely generated so students never see the same problems twice. Topics covered range from find scale factor from two corresponding sides at the easy level through to linear scale factor from area ratio (sqrt) at the advanced level.

CCSS.7.GCCSS.8.G

What is similarity & scale factors?

Similar shapes have identical angles but proportional sides, with the scale factor representing the constant ratio between corresponding measurements. A scale factor of 3 means each side of the larger shape is exactly 3 times longer than the corresponding side of the smaller shape. This relationship extends to areas, which scale by the square of the linear scale factor.

Why it matters

Scale factors appear throughout architecture, engineering, and design when creating blueprints, maps, and models. A 1:50 architectural drawing means 1 inch on paper represents 50 inches in reality. Video game designers use scale factors to resize characters and objects while maintaining proportions. In manufacturing, scale factors help resize patterns for different product sizes — a medium shirt scaled by factor 1.2 becomes a large shirt. The concept extends to advanced mathematics in coordinate geometry transformations and similarity proofs. Medical imaging relies on scale factors when magnifying microscopic structures by factors of 100 or 1000. Even photography uses scale factors in crop ratios and print sizing, where an 8×10 photo scaled by factor 0.6 becomes a 4.8×6 print.

Common mistakes to watch for

  • ✗Confusing scale factor direction: writing scale factor as 4÷8 = 0.5 when comparing an 8-unit side to a 4-unit side, instead of the correct 8÷4 = 2
  • ✗Adding scale factors to sides instead of multiplying: if original side is 6 and scale factor is 3, writing 6+3 = 9 instead of 6×3 = 18
  • ✗Applying linear scale factor to area: if scale factor is 2, writing new area as 12×2 = 24 instead of 12×4 = 48 (area scales by 2²)

Questions teachers ask

What is the difference between scale factor and ratio?+
Scale factor is a single number representing how many times larger one measurement is than another, while ratio compares two quantities using a colon (like 3:1) or fraction form. A scale factor of 4 means the new length is 4 times the original, equivalent to a ratio of 4:1.
How do you find scale factor from area?+
Take the square root of the area ratio. If one triangle has area 36 and a similar triangle has area 144, the area ratio is 144÷36 = 4. The linear scale factor is √4 = 2, meaning each corresponding side is 2 times longer.
Can scale factors be fractions?+
Yes, scale factors can be any positive number, including fractions. A scale factor of 1/2 means the new shape is half the size of the original. For example, if an original side length is 8 units, scaling by 1/2 gives a new side length of 4 units.
What happens when the scale factor is 1?+
When the scale factor equals 1, the shapes are congruent — they have the same size and shape. All corresponding sides are equal, and the shapes are identical except possibly for position or orientation. This means no scaling has occurred.
How do you check if two shapes are similar?+
Verify that all corresponding angles are equal and all corresponding side ratios are the same. Calculate the ratio between several pairs of corresponding sides — if all ratios equal the same value, that's your scale factor and the shapes are similar.
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Beginner

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Concepts
Find scale factor from two corresponding sides
Range
sides 2–30, integer scale factors 2–5
Steps
1 step
Example
Similar squares with sides 4 and 12. Scale factor?

Easy

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Concepts
Scale all sides of a triangle by a given factor
Range
sides 3–13, scale 2–4
Steps
1 step
Example
Triangle 3,4,5 scaled by 3. Find sides

Medium

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Concepts
Find missing side of similar rectangles
Range
sides 3–8, scale 2–4
Steps
2 steps
Example
Rectangles 4×7 and 12×?. Find missing side

Hard

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Concepts
Linear scale factor from area ratio (sqrt)
Range
areas 4–900, linear scale 2–5
Steps
2 steps
Example
Areas 9 and 81. Linear scale factor?

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