Similar Shapes | Mathematics for GCSE/IGCSE - Class 10 PDF Download

Similar Lengths

• Equivalent lengths in two similar shapes are connected by a scale factor, indicating that they are in the same ratio. 
  • Typically, the initial step involves determining the scale factor.
  • Step 1: Recognize equivalent known lengths.
  • Step 2: Establish the direction of change:
    • If the scale factor is greater than 1, the shape is enlarging.
    • If the scale factor is less than 1, the shape is shrinking.
  • Step 3: Calculate the scale factor by dividing the second length by the first length.
  • Step 4: Utilize the scale factor to ascertain the required length.

Similar Areas & Volumes

What are similar shapes?

• Two shapes are mathematically similar if one is an enlargement of the other.
• If two similar shapes are linked by the scale factor, k
  • Equivalent areas are linked by an area factor, k²
  • Equivalent volumes are linked by a volume factor, k³

How do I work with similar shapes involving area or volume?

• Step 1: Identify the equivalent known quantities
  • These could be for lengths, areas or volumes
• Step 2: Establish direction
  • Are they getting bigger or smaller?
• Step 3: Find the Scale Factor from two known lengths, areas or volumes
  • Second Quantity ÷ First Quantity
  • Check the scale factor is > 1 if getting bigger and < 1 if getting smaller
  • If the scale factor, s.f., is from two lengths, write it as k = s.f.
  • If the scale factor, s.f., is from two areas, write it as k2 = s.f.
  • If the scale factor, s.f., is from two volumes, write it as k3 = s.f.
• Step 4: Use the value of the scale factor you have found to convert other corresponding lengths, areas or volumes using
  • Area Scale Factor = (Length Scale Factor)2 
    • Or Length Scale Factor = √(Area Scale Factor)
  • Volume Scale Factor = (Length Scale Factor)3 
    • Or Length Scale Factor = ∛(Volume Length Factor) 
• Use the scale factor to find a new quantity