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 a factor between zero and one. 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
Worked example
Similar solids have volumes 24 cm³ and 81 cm³. The first has a 4 cm edge. Find the corresponding edge of the second
- \(\displaystyle \frac{V_B}{V_A}=\frac{81}{24}=\frac{27}{8}\)Find the volume factor in the requested direction
- \(\displaystyle k=\sqrt[3]{\frac{27}{8}}=\frac32\)Recover the positive linear factor
- \(\displaystyle L_B=4\times\frac32=6\,\mathrm{cm}\)Scale the corresponding edge
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 a factor between zero and one, square or cube the same factor. At a linear factor of one half, the area factor is one quarter and the volume factor is one eighth
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
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 11, VCE Units 1 & 2, General Mathematics, chapter 10: Measurement and similarity