Scaling length does not scale everything equally
What happens to area and volume when every length doubles?
Entering midway? Perimeter adds edge lengths. Area counts square units; volume counts cubic units.
Start with a concrete question
Scale a rectangular box uniformly and inspect its front face on equal coordinate axes. Dimensions use metres, square metres and cubic metres.
Try the initial inputs, then write a prediction before changing one setting. All calculations run locally on your CPU.
Go directly to the experiment · Review prerequisite mathematics and return here
What do you expect?
Predict the area and volume multipliers at scale two before changing the control.
Change one thing. Inspect what happens.
- At scale one, check perimeter, face area and volume separately.
- Double the scale and compare ratios with the original.
- Use scale one-half and predict the volume fraction.
- Try zero scale, then explain why it is a degenerate case rather than an ordinary similar shape.
Your settingsMechanism
Calculation belowEvidence
Compare outcomes
Original and scaled rectangular faces — equal coordinate units
Horizontal: Horizontal distance (m). Vertical: Vertical distance (m).
- 1. Original face (solid line)
- 2. Scaled face (dashed line)
Axes use the displayed ranges.
Read plotted values as a table
| Series | Horizontal distance (m) | Vertical distance (m) |
|---|---|---|
| Original face | 0 | 0 |
| Original face | 3 | 0 |
| Original face | 3 | 2 |
| Original face | 0 | 2 |
| Original face | 0 | 0 |
| Scaled face | 0 | 0 |
| Scaled face | 6 | 0 |
| Scaled face | 6 | 4 |
| Scaled face | 0 | 4 |
| Scaled face | 0 | 0 |
Follow the calculation
- Original face: width 3 × height 2 = 6 m²; box depth 2 gives volume 12 m³.
- All lengths multiply by 2; face dimensions become 6 by 4 m.
- Area has two length factors: 6 × 2² = 24 m².
- Volume has three: 12 × 2³ = 96 m³.
- Corresponding length ratios agree, so angles and shape are preserved.
A scale factor multiplies length, area and volume by different powers. The plot shows the rectangular face; depth and volume are calculated separately.
Captures keep your inputs, prediction and observation. Results are recalculated from those inputs. Resetting inputs keeps your captures and challenge history.
Explain the mechanism
Start with the meaning
Perimeter adds edge lengths. Area counts square units; volume counts cubic units.
Scale a rectangular box uniformly and inspect its front face on equal coordinate axes. Dimensions use metres, square metres and cubic metres.
A worked explanation
Similarity preserves angles and ratios of corresponding nonzero lengths. It does not preserve area or volume.
A 3 by 2 face has area 6 m². With depth 2 its box has volume 12 m³. Scaling all lengths by two gives a 6 by 4 face with area 24 and depth 4 with volume 96.
Two length factors produce the square of the scale; three produce its cube. This matters when extrapolating material, capacity or surface coverage from a small model.
Symbols and relationship
Scaled perimeter = sP; area = s²A; volume = s³V. s is a nonnegative scale, P length, A area and V volume.
Use the graph and trace together
The graph and calculations use your current inputs. Read axis labels and units before comparing positions or slopes. Change one input, predict the effect, and use the numerical trace to verify the plotted behavior. A displayed curve is an illustration; the formula and its assumptions determine which conclusions are justified.
Scope and edge cases
- Rectangular dimensions and uniform scaling; nonuniform stretching does not preserve similarity.
- Only the front face is plotted; depth and volume are numerical quantities.
Try a fresh case
A similar solid has original volume 40 cm³. Every length is multiplied by 1.5. What is its new volume?
This is a public practice check. Retries support learning; matching a number does not certify independent proficiency.
Take the investigation further
Choose any extension. These are project prompts for you to carry out; they are not automatically assessed.
School investigation
Design a scaled storage box and distinguish surface coverage from capacity.
College investigation
Derive dimensional scaling and analyze a nonuniform stretch.
Workplace investigation
Audit a scale-up estimate that incorrectly applies a length factor to material area or capacity.
Keep the evidence, choose your next step
Completion records your own decision that you have worked through this investigation. Practice checks and independent project evidence are separate.