Scaling in Space Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Scaling in Space.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

How length, area, and volume measurements change when a figure is uniformly enlarged or shrunk by a scale factor.

Double the size: length \times 2, area \times 4, volume \times 8.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Length scales linearly; area scales by square; volume scales by cube.

Common stuck point: Area and volume scale differently than length—this catches many students.

Sense of Study hint: Try doubling the side of a square and counting the new unit squares. You will see area grows by 4, not 2.

Worked Examples

Example 1

easy
A square has side length 3 cm. If all lengths are doubled (scale factor k=2), what are the new perimeter and area?

Solution

  1. 1
    Step 1: Original side = 3 cm. New side = 3 \times 2 = 6 cm.
  2. 2
    Step 2: Perimeter scales by k: new perimeter = 4 \times 6 = 24 cm (original was 12 cm, doubled).
  3. 3
    Step 3: Area scales by k^2: original area = 9 cm², new area = 9 \times 4 = 36 cm².
  4. 4
    Step 4: Verify: 6^2 = 36 cm².

Answer

New perimeter = 24 cm; new area = 36 cm².
When a shape is scaled by factor k: all lengths multiply by k, all areas multiply by k^2, and all volumes multiply by k^3. This is because area is two-dimensional (two lengths multiplied) and volume is three-dimensional.

Example 2

medium
A sphere has radius 2 cm and volume V = \frac{4}{3}\pi r^3. If the radius is tripled, how many times larger is the new volume?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
A photo is 4 in × 6 in. It is enlarged with scale factor k = 3. What are the new dimensions and new area?

Example 2

hard
Two similar pyramids have heights 4 m and 10 m. If the smaller pyramid has volume 32 m³, what is the volume of the larger?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

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