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Grade 7 · lesson 2 of 4

Scale Both Dimensions before Finding Area

Area is built from two lengths, so scaling both by the same factor multiplies area by the factor squared.

About 20 minutes, plus any extended practice

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Supplied materials

  • Paper or a browser scratch area; a calculator is optional unless rounding is requested.
  • Supplied teaching case: A scale drawing uses 1 cm for 3 m. A rectangular region is 2 cm by 3 cm on the drawing. What actual area is represented? Candidate answers: (A) 15 m² (B) 18 m² (C) 54 m² (D) 6 m².
  • Independent check inputs: A scale drawing uses 1 cm for 3 m. A rectangular region is 3 cm by 4 cm on the drawing. What actual area is represented? Candidate answers: (A) 108 m² (B) 12 m² (C) 21 m² (D) 36 m².

Learn the idea

A scale drawing links represented and actual lengths through a fixed factor. Geometric relationships connect angles through straight lines and triangles, so a justified equation can replace guessing from a sketch. Area is built from two lengths, so scaling both by the same factor multiplies area by the factor squared. Converting the side lengths first makes this relationship visible.

Here is the method on the supplied teaching case. Actual sides are 6 m and 9 m. Area is their product, 54 m². A length scale of 3 makes an area scale of 9.

A length scale factor is not automatically the area factor. Label the converted sides before multiplying them; square units describe the resulting two-dimensional region, not the perimeter around it.

  1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
  2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
  3. Solve the independent check with its changed inputs before revealing feedback.
  4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
Worked example

See the method in action

Teaching case: A scale drawing uses 1 cm for 3 m. A rectangular region is 2 cm by 3 cm on the drawing. What actual area is represented? Candidate answers: (A) 15 m² (B) 18 m² (C) 54 m² (D) 6 m². Solution: Actual sides are 6 m and 9 m. Area is their product, 54 m². A length scale of 3 makes an area scale of 9. The final answer is 54 m². A length scale factor is not automatically the area factor. Label the converted sides before multiplying them; square units describe the resulting two-dimensional region, not the perimeter around it.

Your turn

Check your reasoning

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Helpful tips
  • Area is built from two lengths, so scaling both by the same factor multiplies area by the factor squared.
  • A length scale factor is not automatically the area factor.
A scale drawing uses 1 cm for 3 m. A rectangular region is 3 cm by 4 cm on the drawing. What actual area is represented?

Feedback appears here after you check. This is practice, not a graded school assessment.

See the answer and explanation

108 m²

Actual sides are 9 m and 12 m. Area is their product, 108 m². A length scale of 3 makes an area scale of 9.

Take it further

Changed-input task: A scale drawing uses 1 cm for 3 m. A rectangular region is 9 cm by 10 cm on the drawing. What actual area is represented? Candidate answers: (A) 270 m² (B) 57 m² (C) 810 m² (D) 90 m². Show the mathematical reason, not only the selected answer. Expected result: 810 m². Actual sides are 27 m and 30 m. Area is their product, 810 m². A length scale of 3 makes an area scale of 9.

Nothing here is saved to an account or sent to a teacher. The answer download includes your current choice and any checked feedback. It does not include choices from the separate More practice section. Download that section separately.

More practice: try new cases

Choose an answer for each problem. Explain your method on paper, then use the feedback to decide what to revisit.

More practice: try new cases
1. A scale drawing uses 1 cm for 3 m. A rectangular region is 4 cm by 5 cm on the drawing. What actual area is represented?
Need a hint?
  • Area is built from two lengths, so scaling both by the same factor multiplies area by the factor squared.
  • A length scale factor is not automatically the area factor.
2. A scale drawing uses 1 cm for 3 m. A rectangular region is 5 cm by 6 cm on the drawing. What actual area is represented?
Need a hint?
  • Area is built from two lengths, so scaling both by the same factor multiplies area by the factor squared.
  • A length scale factor is not automatically the area factor.
3. A scale drawing uses 1 cm for 3 m. A rectangular region is 6 cm by 7 cm on the drawing. What actual area is represented?
Need a hint?
  • Area is built from two lengths, so scaling both by the same factor multiplies area by the factor squared.
  • A length scale factor is not automatically the area factor.

Your choices are temporary. Feedback appears after you check all the answers.

Nothing here is saved to an account or sent to a teacher. Keep your own notes or download them before leaving. The score describes these questions; it does not certify subject mastery.

Check the subject sources

These LifeStarter passages, scenarios and exercises are authored for this Academy with AI assistance and review. External resources keep their publishers’ terms. A grade label suggests a starting level; it does not establish alignment with your school’s complete requirements.