When two shapes are similar, one shape is an enlargement or reduction of the other. This applies to both 2-D and 3-D shapes and is based on a single consistent scale factor.

 

 

What It Means for Shapes to Be Similar

Two shapes are similar if:
• all corresponding angles are equal
• corresponding lengths are in the same ratio

 

This means the shapes have the same shape but not necessarily the same size.

 

Because the shapes are similar, one can be obtained from the other by an enlargement.

 

 

Enlargement in 2-D Shapes

For 2-D shapes, enlargement means:
• all lengths are multiplied by the same scale factor
• angles stay the same
• the shape keeps the same proportions

 

If the scale factor is greater than 1, the image is larger.
If the scale factor is between 0 and 1, the image is smaller.

 

For example, if every side of one shape is twice the length of the corresponding side of another, the scale factor is 2.

 

The same scale factor must apply to every side

 

 

Enlargement in 3-D Shapes

For 3-D shapes, the same idea applies.

 

When one 3-D shape is an enlargement of another:
• all corresponding lengths are multiplied by the same scale factor
• all corresponding angles remain equal
• the shapes remain similar

 

This applies to solids such as cubes, cuboids, prisms, pyramids and spheres.

 

 

Identifying the Scale Factor

To identify the scale factor between two similar shapes:
• choose a pair of corresponding lengths
• divide the larger length by the smaller length

 

If the same ratio works for all corresponding lengths, the shapes are similar and one is an enlargement of the other.

 

In 3-D shapes, edge lengths are usually compared.

 

 

Using Enlargement to Solve Problems

Knowing that one shape is an enlargement of another allows you to:
• find missing lengths
• compare sizes
• work out relationships between shapes

 

For example, if a small shape is enlarged by a scale factor, all corresponding lengths change by that factor.

 

Always check that angles match before assuming enlargement

 

 

Common Errors to Avoid

Common mistakes include:
• assuming shapes are similar just because they look alike
• using different scale factors for different sides
• forgetting that angles must be equal
• confusing congruent shapes with similar shapes

 

Congruent shapes have a scale factor of 1.

 

 

Key Points to Remember

Similar shapes are related by enlargement or reduction.
All corresponding angles are equal.
All corresponding lengths change by the same scale factor.
This applies to both 2-D and 3-D shapes.
Finding the scale factor helps solve missing length problems.

 

Understanding that similar shapes are enlargements of one another allows you to compare shapes accurately and solve a wide range of geometric problems.