When shapes are similar, their dimensions change in a predictable way. The ratios of lengths, areas, volumes and capacities are all linked. Understanding these relationships allows you to scale measurements correctly.

 

 

Length Ratio

The length ratio compares corresponding lengths in two similar shapes.

 

If one shape is an enlargement of another, every corresponding length changes by the same scale factor.

 

For example, if a shape is enlarged so that each length is multiplied by the same number, that number is the linear scale factor.

 

All other ratios are based on this length ratio.

 

 

Area Ratio

The area ratio compares the areas of two similar shapes.

 

When lengths change by a given scale factor, areas change by the square of that scale factor.

 

This means:
• if lengths double, areas become four times as large
• if lengths triple, areas become nine times as large

 

Area grows faster than length because it depends on two dimensions.

 

Do not use the length ratio directly for area

 

 

Volume and Capacity Ratio

The volume ratio compares the volumes of two similar 3-D shapes.

 

Capacity follows the same rule as volume.

 

When lengths change by a given scale factor, volumes and capacities change by the cube of that scale factor.

 

This means:
• if lengths double, volume becomes eight times as large
• if lengths triple, volume becomes twenty seven times as large

 

Volume increases fastest because it depends on three dimensions.

 

 

Relationship Between All Ratios

The relationships can be summarised as:

• lengths change in the scale factor
• areas change in the square of the scale factor
• volumes and capacities change in the cube of the scale factor

 

Each step adds another dimension.

 

 

Using These Relationships in Problems

To solve problems involving similar shapes:

• identify the length scale factor first
• square it to find the area ratio
• cube it to find the volume or capacity ratio

 

If you are given an area or volume ratio, you may need to work backwards to find the length scale factor.

 

 

Common Errors to Avoid

Common mistakes include:
• using the length ratio for area or volume
• forgetting to square or cube the scale factor
• mixing up area and volume ratios
• assuming capacity scales like area

 

Always think about how many dimensions are involved.

 

 

Key Points to Remember

Similar shapes have consistent scaling rules.
Lengths scale by the linear scale factor.
Areas scale by the square of the scale factor.
Volumes and capacities scale by the cube of the scale factor.
Finding the length ratio first makes problems easier.

 

Understanding how length, area, volume and capacity ratios are linked is essential for solving problems involving similar shapes accurately and confidently.