Similar Areas & Volumes

Similar Areas & Volumesvideo

Similar areas & volumes

How do I find the length, area or volume scale factors of similar shapes?

• The scale factor (SF) for a given quantity (length, area or volume) between two similar shapes can be found by dividing the quantity on one shape by the quantity on the other shape

  • $\mathrm{scale} \mathrm{factor}=\frac{\mathrm{quantity} \mathrm{on} \mathrm{one} \mathrm{shape}}{\mathrm{corresponding} \mathrm{quantity} \mathrm{on} \mathrm{the} \mathrm{other} \mathrm{shape}}$

• An object could be made either bigger or smaller by a scale factor

  • The object gets bigger if the scale factor is bigger than 1

  • The object gets smaller if the scale factor is bigger than 0 and less than 1

What is the connection between the scale factors for lengths, areas and volumes of similar shapes?

• The length, area and volume scale factors are powers with the same base number

• If the length scale factor is k then

  • The area scale factor is k²

  • The volume scale factor is k³

• If you know one scale factor, you can find the scale factors

  • If you have the length scale factor

    • $\mathrm{Area} \mathrm{scale} \mathrm{factor}={\left(\mathrm{Length} \mathrm{scale} \mathrm{factor}\right)}^2$

    • $\mathrm{Volume} \mathrm{scale} \mathrm{factor}={\left(\mathrm{Length} \mathrm{scale} \mathrm{factor}\right)}^3$

  • If you have the area scale factor

    • $\mathrm{Length} \mathrm{scale} \mathrm{factor}=\sqrt{\mathrm{Area} \mathrm{scale} \mathrm{factor}}$

    • $\mathrm{Volume} \mathrm{scale} \mathrm{factor}={\left(\sqrt{\mathrm{Area} \mathrm{scale} \mathrm{factor}}\right)}^3$

  • If you have the volume scale factor

    • $\mathrm{Length} \mathrm{scale} \mathrm{factor}=\sqrt[3]{\mathrm{Volume} \mathrm{scale} \mathrm{factor}}$

    • $\mathrm{Area} \mathrm{scale} \mathrm{factor}={\left(\sqrt[3]{\mathrm{Volume} \mathrm{scale} \mathrm{factor}}\right)}^2$

How do I find missing lengths, areas and volumes for similar shapes?

• STEP 1
  Identify the equivalent known quantities

  • Recognise if the quantities are lengths, areas or volumes

• STEP 2
  Find the scale factor from two known lengths, areas or volumes

  • $\mathrm{scale} \mathrm{factor}=\frac{\mathrm{second} \mathrm{quantity}}{\mathrm{first} \mathrm{quantity}}$

• STEP 3
  Use the scale factor you have found to find other required scale factor(s)

  • $\mathrm{Length} \mathrm{scale} \mathrm{factor}=k$

  • $\mathrm{Area} \mathrm{scale} \mathrm{factor}=k^2$

  • $\mathrm{Volume} \mathrm{scale} \mathrm{factor}=k^3$

• STEP 4
  Multiply or divide by relevant scale factor to find the missing quantity

  • Think about whether the quantity should be bigger or smaller than the given quantity