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Cube proportions: length, area and volume

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      Solids · Proportion

      Question

      The edge lengths of a large cube and a small cube are in the ratio 3:2. Find the ratios of their total edge lengths, base areas and volumes.

      Cube length, area and volume ratiosThe left cube has edge 3 and the right cube edge 2. Edge lengths are in ratio 3:2, areas 9:4 and volumes 27:8.Large cube (edge 3)Small cube (edge 2)① Edge lengths: 3 : 2② Base areas: 9 : 4③ Volumes: 27 : 8
      Compare edge lengths 3:2, base areas 9:4 and volumes 27:8.

      Step-by-step solution

      1. Total edge lengths

      A cube has twelve equal edges. The ratio is (12 × 3):(12 × 2) = 3:2. Lengths scale by the original scale factor.

      2. Base areas

      A square base has area edge × edge. The ratio is 3²:2² = 9:4. Areas scale by the square of the scale factor.

      3. Volumes

      A cube has volume edge × edge × edge. The ratio is 3³:2³ = 27:8. Volumes scale by the cube of the scale factor.

      4. State the ratios in order

      Total edge lengths: 3:2. Base areas: 9:4. Volumes: 27:8.

      The three ratios are 3:2; 9:4; 27:8.

      Which method does this problem use?

      For similar shapes with length scale factor k, lengths scale by k, areas by k² and volumes by k³. Identify the dimension of the quantity before forming its ratio.