Geometry Lab formula guide
Similarity & scale factors
Similar shapes have equal angles and proportional sides. With scale factor k, every length is multiplied by k, every area by k², and every volume by k³.
Length × k · Area × k² · Volume × k³
| Use it to | Formula |
|---|---|
| Lengths | × k |
| Areas | × k² |
| Volumes | × k³ |
| Dilation from the origin | (x, y) → (kx, ky) |
Try it with your own numbers
Similarity & scale factors calculator
Why doubling the size changes more than you might expect. Change a value and the diagram, result, and working update together.
Transformation playgroundLIVE EXPLORATION
Green: original · orange: transformed
Your measurements
The original triangle is A(−2, −1), B(2, −1), C(0, 2). Rotations and dilations are centered at the origin.
Why it works
Rotation: (x, y) → (x cos θ − y sin θ, x sin θ + y cos θ)
The idea
Similar shapes have equal corresponding angles and proportional corresponding lengths. Under a positive uniform scale factor k, lengths multiply by k, areas by k², and volumes by k³. This distinction explains why a bigger package can hold much more while using proportionally less surface material.
Work through an example
1Work through an example
- Scale a 3 cm cube by a factor of 2. Its new edge is 6 cm. Surface area grows from 54 to 216 cm², a factor of 4. Volume grows from 27 to 216 cm³, a factor of 8.
Common mistakes
- The rule assumes uniform scaling in every dimension. Doubling only the height of a cylinder doubles volume; it does not multiply it by eight.
Make the formula move.
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