Library / Geometry, coordinates, surveying and trigonometry / GEO-22
Geometry, coordinates, surveying and trigonometry
Volume after a similarity transformation
Volume of a solid that is k times as long in all three directions.
- Ref
- GEO-22
- Rev
- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- V2 [cm³]
A solid enlarged by the same length scale factor k in all three directions has k³ times the volume. If the material is the same, mass scales by k³ too, while strength depends on cross-section and scales by k². That is why large copies of a design are relatively weaker. A stretch that differs between directions does not follow this rule.
Volume of a solid that is k times as long in all three directions. A stretch that is not the same in each direction does not follow this rule.
Calculator
- Volume scale factor, k³3.375
- Area scale factor, k²2.25
- Length scale factor needed to double the volume, ∛21.26
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- Scaling a 24 cm³ solid by 1.5 gives 81 cm³, a volume multiplier of 3.375 but an area multiplier of only 2.25.
- A scale factor of 2 gives eight times the volume, so a model one half size holds just one eighth of the contents.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (cm³) | Local step | Change in result |
|---|---|---|---|---|---|---|
| k Length scale factor | 0.2 to 5 | 0.192 | 3,000 | 3,000 | +10% | +33.1 % |
| V1 Original volume | 1 to 200 cm³ | 3.375 | 675 | 671.6 | +10% | +10 % |
Sweep of Length scale factor
| Length scale factor | Image volume (cm³) |
|---|---|
| 0.2 | 0.192 |
| 0.8 | 12.29 |
| 1.4 | 65.86 |
| 2 | 192 |
| 2.6 | 421.8 |
| 3.2 | 786.4 |
| 3.8 | 1,317 |
| 4.4 | 2,044 |
| 5 | 3,000 |
Sweep one variable
Variables and how each one contributes
Image volume
What the equation gives you. Everything else feeds it.
Length scale factor
The ratio of an image length to the matching original length, a plain number above 0. Volume scales with the cube of k, so a 10 % larger k gives a 33 % larger volume. Raising it by 0.1 (to 1.6) moves the result by +21.4 %, from 81 to 98.3 cm³.
Original volume
The volume of the original solid, in cm³. A larger original gives a larger image in direct proportion. Raising it by 10 % (to 26.4) moves the result by +10 %, from 81 to 89.1 cm³.
Worked example with the values above
- Cube the length factor: k³ = 1.5³ = 3.375.
- Multiply by the original volume 24 cm³.
- V₂ = 81 cm³.
Assumptions and limits
- Similar shapes: every length changes by the same factor k.
- Coordinates in a right-handed system with y upwards, and angles measured anticlockwise.
Common mistakes
- Applying the length scale factor to an area or volume without squaring or cubing it.
- Forgetting that a negative scale factor puts the image on the opposite side of the centre.
- Using degrees for a rotation where radians are needed, or the wrong turn direction.
Statistics and uncertainty
Scale factors act on all lengths, so area errors double and volume errors triple the error in k. For coordinate transformations, the result is as accurate as the measured positions and angles.
Where it is used
- Scaling models, maps and drawings, and finding how material, mass and cost change with size.
- Moving, turning and mirroring points in drawings and CAD.
Maths you need
- Scale factor: ratio of corresponding lengths
- Coordinates: a point’s position as (x, y)
- Sine and cosine: the horizontal and vertical parts of a rotation