Geometry and measurement
Similar Shapes and Scale Factors
Find corresponding lengths, enlarge and reduce shapes, and distinguish length scale factors from area scale factors.
Grades 6–8 practice · 6 questions · Free to print · Answers included
Make sense of the method
Similar shapes have matching angles and one consistent scale factor for corresponding lengths. A larger-looking shape is not enough: every corresponding length must fit the same multiplier.
The sketches here illustrate labelled dimensions. Use the numbers and correspondence labels rather than measuring the screen.
- Match corresponding sides or vertices.
- Divide image length by original length to find the scale factor.
- Multiply every original length by that same factor.
- For area, use the square of the length factor; do not apply a length factor only once.
A worked example
A rectangle is 3 cm by 2 cm. A similar rectangle has corresponding length 9 cm. Find its width.
- The length factor is 9 ÷ 3 = 3.
- Multiply the original width by 3: 2 × 3 = 6 cm.
- Both length ratios are 3, so the rectangles are similar.
The width is 6 cm.
Try the worksheet
Start with the first two questions, then build toward the final challenge. Show your method with calculations, drawings or a short explanation. These are practice questions with published answers, not a full-topic readiness check.
Build the foundation
1.A rectangle is 4 cm by 3 cm. Enlarge it by scale factor 2. Find the new dimensions and explain why both must change.
Original rectangle Show your work on paper.Build the foundation
2.A 15 cm by 10 cm photograph is reduced by scale factor 0.4. Find its new dimensions.
Photograph dimensions Show your work on paper.Strengthen your method
3.Triangle A has sides 5 cm, 7 cm and 8 cm. Triangle B has corresponding sides 15 cm, 21 cm and 24 cm. Are they similar? Show all three ratios.
Show your work on paper.Strengthen your method
4.A rectangle is 6 cm by 4 cm. Another is 9 cm by 7 cm, with long sides corresponding. Are they similar? Explain.
Compare corresponding sides Show your work on paper.Apply and explain
5.Dilate triangle A(1, 1), B(3, 1), C(1, 2) from the origin by scale factor 3. Plot the image and compare the horizontal and vertical leg lengths.
Dilation from the origin: accessible grid. Columns give horizontal x positions; rows give vertical y positions. Vertical y / horizontal x x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 y 10 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 9 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 8 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 7 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 6 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 5 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 4 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 3 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 2 Empty C Empty Empty Empty Empty Empty Empty Empty Empty Empty y 1 Empty A Empty B Empty Empty Empty Empty Empty Empty Empty y 0 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Show your work on paper.Apply and explain
6.A 5 cm by 4 cm rectangle is enlarged by length scale factor 1.5. Find the new dimensions, both areas and the area scale factor.
Original rectangle for area comparison Show your work on paper.
Worked answers
Try the questions first. Then compare the reasoning, not just the final result.
Show all six worked solutions
1. The enlarged rectangle is 8 cm by 6 cm.
- Length: 4 × 2 = 8 cm.
- Width: 3 × 2 = 6 cm.
- Changing only one dimension would change the side ratio and usually the shape.
Answer: factor two 2. The reduced photograph is 6 cm by 4 cm.
- Multiply both dimensions by 0.4.
- 15 × 0.4 = 6 cm and 10 × 0.4 = 4 cm.
- A factor between 0 and 1 makes every length smaller.
Answer: reduced photograph 3. Yes. Each corresponding side ratio is 3.
- 15 ÷ 5 = 3.
- 21 ÷ 7 = 3 and 24 ÷ 8 = 3.
- All three side lengths have the same multiplier, so the triangles are similar.
4. No: the length factors are 1.5 and 1.75.
- Long-side factor: 9 ÷ 6 = 1.5.
- Short-side factor: 7 ÷ 4 = 1.75.
- There is no single scale factor for both dimensions.
5. A′(3, 3), B′(9, 3), C′(3, 6). The legs change from 2 and 1 units to 6 and 3 units.
- Multiply both coordinates of each vertex by 3.
- A′ = (3, 3), B′ = (9, 3), C′ = (3, 6).
- Horizontal leg: 2 × 3 = 6. Vertical leg: 1 × 3 = 3. Matching length ratios preserve the triangle’s shape.
Answer: dilated vertices: accessible grid. Columns give horizontal x positions; rows give vertical y positions. Vertical y / horizontal x x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 y 10 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 9 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 8 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 7 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 6 Empty Empty Empty C′ Empty Empty Empty Empty Empty Empty Empty y 5 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 4 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 3 Empty Empty Empty A′ Empty Empty Empty Empty Empty B′ Empty y 2 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 1 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty y 0 Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty Empty 6. New dimensions: 7.5 cm by 6 cm. Areas: 20 cm² and 45 cm². Area factor: 2.25.
- Multiply lengths: 5 × 1.5 = 7.5 and 4 × 1.5 = 6.
- Original area: 5 × 4 = 20 cm². New area: 7.5 × 6 = 45 cm².
- 45 ÷ 20 = 2.25 = 1.5² because two dimensions each scale by 1.5.
Answer: enlarged rectangle
Mistakes worth catching
Using different multipliers for corresponding sides.
Compute every ratio and check for one shared factor.
Using the length factor for area.
Area contains two length factors, so square the scale factor.
Comparing sides that do not correspond.
Match long sides, short sides or labelled vertices before dividing.
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Skills covered
Geometry and measurement
- Use corresponding lengths to enlarge or reduce similar shapes.
- Distinguish a length scale factor from an area scale factor.
Choose this practice by the skills you need. Grade placement and strand names vary between school systems; one worksheet covers selected skills rather than every expectation in a topic.
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