One length factor, three different effects
Similar solids have the same shape, with every corresponding length multiplied by one positive factor k. A reduction has 0<k<1. The image need not be larger
- Lengths Multiply corresponding lengths by k. \(\displaystyle L_B=kL_A\)
- Surface areas Each face scales in two directions. Multiply its area, and the total surface area, by k squared \(\displaystyle A_B=k^2A_A\)
- Volumes Three length directions contribute to volume. Multiply volume by k cubed \(\displaystyle V_B=k^3V_A\)
- Check the premise Do not apply these powers to dissimilar solids. For cuboids compare all three corresponding edges; for right circular cylinders or cones compare radius and perpendicular height
- Capacity needs internal similarity Use internal dimensions and consistent capacity units. Similar outside shapes do not guarantee similar interiors when wall thickness differs or fails to scale
- Evaluate uncertainty For a cube, cube both plausible edge bounds. The resulting volume range is not obtained by simply adding the length uncertainty to the volume
Worked example
A solid cube has measured edge 2.0 cm, with possible error up to 0.1 cm. Use a cube model to estimate its volume and an approximate plausible range
- \(\displaystyle V=2^3=8\text{ cm}^3\)Use the central measurement for the estimate
- \(\displaystyle a_{\min}=1.9\text{ cm}\)Subtract the possible edge error
- \(\displaystyle V_{\min}=1.9^3\)Volume increases with positive edge length
- \(\displaystyle V_{\min}=6.859\text{ cm}^3\)Calculate the lower volume endpoint
- \(\displaystyle a_{\max}=2.1\text{ cm}\)Add the possible edge error
- \(\displaystyle V_{\max}=2.1^3\)Cube the upper edge bound
- \(\displaystyle V_{\max}=9.261\text{ cm}^3\)Calculate the upper volume endpoint
- \(\displaystyle V\approx8\text{ cm}^3\)Report approximately 6.86–9.26 cm³ under the cube-shape and edge-error assumptions. More digits do not remove measurement uncertainty
Common mistakes
Difficulty levels
Questions about this topic
Are all solids of the same kind similar?+
No. Cuboids need matching ratios for all three corresponding edges. Right circular cylinders and cones need matching radius and perpendicular-height ratios. All spheres are similar
Why is the volume factor cubed?+
Every one of the three length dimensions contributes a factor k, giving k × k × k = k³
How do reductions work?+
For 0<k<1, square or cube the same factor. At k = 1/2, the area factor is 1/4 and the volume factor is 1/8.
How do I recover a length from a volume ratio?+
First establish similarity, then take the positive cube root of the volume ratio. Use that linear factor on the corresponding length
Does doubled capacity always mean a doubled length?+
No. Even for similar containers, doubled volume means a length factor equal to the cube root of 2. Real containers also need matching internal shapes and dimensions for a capacity comparison
What if the dimensions use different factors?+
The solids are not necessarily similar. For a cuboid, multiply the three directional factors to obtain its volume factor; one cube factor cannot replace unequal directional factors
Does a small edge error give the same percentage volume error?+
No. Cubing changes the relative uncertainty. Calculate the volumes at the plausible edge endpoints and report an appropriately rounded range
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 7: Geometry and similarity