--- title: "Similar Shapes: Area and Volume Scale Factors" description: "Use the fact that areas scale by the square of the length scale factor and volumes by its cube, and work the relationship in both directions." canonical: https://lightmysky.com/learn/mathematics/similar-shapes-area-and-volume-scale-factors-mt_v37c_AeLQa source: https://lightmysky.com/learn/mathematics/similar-shapes-area-and-volume-scale-factors-mt_v37c_AeLQa.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Similar Shapes: Area and Volume Scale Factors Use the fact that areas scale by the square of the length scale factor and volumes by its cube, and work the relationship in both directions. Subject: Mathematics · Area: Geometry · Ages 14 to 15 Page: https://lightmysky.com/learn/mathematics/similar-shapes-area-and-volume-scale-factors-mt_v37c_AeLQa ## Ready when they can - Find the area of an enlarged shape from the length scale factor - Work back to the length scale factor from a ratio of volumes - Explain why doubling every length of a solid multiplies its volume by eight ## Lesson: Big copy, bigger area Start with the length scale factor k, new length divided by old length. Areas scale by k squared, since area multiplies two lengths. A photo enlarged with k equal to 3 grows 9 times in area. A 4 cm by 5 cm photo has area 20, and its enlargement has 20 times 9, which is 180. Volumes scale by k cubed, since volume multiplies three lengths. Double every edge and the volume grows 8 times, not twice. Two jugs 10 cm and 15 cm tall give k equal to 1.5, and 1.5 cubed is 3.375, so the taller holds over three times as much. **Example.** Ratios also run backwards. From an area ratio take the square root, and from a volume ratio take the cube root. Volumes in ratio 27 to 1 mean lengths in ratio 3 to 1. Areas of 20 and 180 mean a length factor of 3. **Tip.** The power matches the dimension: 1 for length, 2 for area, 3 for volume. Learn k, k squared and k cubed once, then reuse them for every shape. Scaling treats pyramids and boxes alike, so one rule covers them all. **Recap.** Square k for areas, cube it for volumes, and take roots on the way back. ## Practice 13 questions on this page, each with its working shown. ## Needs first - [Surface Area of Solids](https://lightmysky.com/learn/mathematics/surface-area-of-solids-mt_hdhmurUndc) - [Volume and Surface Area of Prisms and Cylinders](https://lightmysky.com/learn/mathematics/volume-and-surface-area-of-prisms-and-cylinders-mt_JTLJ5LDrYh) - [Similar Triangles and Proving Similarity](https://lightmysky.com/learn/mathematics/similar-triangles-and-proving-similarity-mt_NPRVlr4lrs)