- 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,***k*^{2} - Equivalent
**volumes**are linked by a**volume factor,***k*^{3}

- Equivalent

- 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 k
^{2}= s.f. - If the scale factor, s.f., is from two lengths, write it as k
^{3}= 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)

- Area Scale Factor = (Length Scale Factor)
- Use the scale factor to find a new quantity

- Take extra care not to mix up which shape is which when you have started carrying out the calculations
- It can help to label the shapes and always write an equation
- For example if shape A is similar to shape B:
- length A =
*k*(length B) - area A =
*k*^{2}(area B) - volume A =
*k*^{3}(volume B)

- length A =

- For example if shape A is similar to shape B:

Solid *A *and solid *B *are mathematically similar.

The volume of solid *A *is 32 cm ^{3}.

The volume of solid *B *is 108 cm ^{3}.

The height of solid *A *is 10 cm.

Find the height of solid *B.*

Calculate , the scale factor of enlargement for the volumes, using ,

Or .

For similar shapes, if the volume scale factor is
, then the length scale factor is
.

FInd
.

Substitute into formula for the heights of the similar shapes. ,

**Height of B = 15 **

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