Library / Geometry, coordinates, surveying and trigonometry / GEO-21
Geometry, coordinates, surveying and trigonometry
Area after a similarity transformation
Area of an image that is k times as long as the original in every direction.
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- GEO-21
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- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- A2 [cm²]
When a shape is enlarged by a length scale factor k, every area is multiplied by k². The reason is that area is a product of two lengths. A scale factor above 1 enlarges, below 1 reduces, and a reflection afterwards does not change the area. The factor is always positive here.
Area of an image that is k times as long as the original in every direction. It also holds for an enlargement followed by a reflection.
Calculator
- Area scale factor, k²2.25
- Change in area30 cm²
- Length scale factor needed to double the area, √21.414
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.
- Enlarging 24 cm² by a factor of 1.5 gives 54 cm², a multiplier of 2.25 on area.
- To double the area you need a length factor of only √2 ≈ 1.414, not 2.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (cm²) | Local step | Change in result |
|---|---|---|---|---|---|---|
| k Length scale factor | 0.2 to 5 | 0.96 | 600 | 599 | +10% | +21 % |
| A1 Original area | 1 to 200 cm² | 2.25 | 450 | 447.8 | +10% | +10 % |
Sweep of Length scale factor
| Length scale factor | Image area (cm²) |
|---|---|
| 0.2 | 0.96 |
| 0.8 | 15.36 |
| 1.4 | 47.04 |
| 2 | 96 |
| 2.6 | 162.2 |
| 3.2 | 245.8 |
| 3.8 | 346.6 |
| 4.4 | 464.6 |
| 5 | 600 |
Sweep one variable
Variables and how each one contributes
Image area
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. The area scales with the square of k, so a 10 % larger k gives a 21 % larger area. Raising it by 0.1 (to 1.6) moves the result by +13.8 %, from 54 to 61.44 cm².
Original area
The area of the original shape, in cm². A larger original gives a larger image in direct proportion. Raising it by 10 % (to 26.4) moves the result by +10 %, from 54 to 59.4 cm².
Worked example with the values above
- Square the length factor: k² = 1.5² = 2.25.
- Multiply by the original area 24 cm².
- A₂ = 54 cm².
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