Geometry

Similarity

Scale lengths, areas and volumes.

Predict, test and explain

Step 1 of 3

Predict

Similar shapes have length scale factor 2. What is the area scale factor?

length multiplier=2\text{length multiplier}=2
3 cm3\,\text{cm}
4 cm4\,\text{cm}
6 cm6\,\text{cm}
8 cm8\,\text{cm}

Scroll the diagram sideways to see every label.

3↦63\mapsto 64↦84\mapsto 85↦105\mapsto 10

Both drawings use the same length scale. The second shape is similar to the first.

InvestigateDouble a length. What happens to area and volume?

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Try it in the simulation. Change one thing at a time.

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Hints, self-check and connections
Try a self-check

Fixed values, separate from the diagram controls.

A shape with area 9 square centimetres is enlarged by a length scale factor of 3. Find the enlarged area.

Hint

Area scales by the square of the length scale factor.

Reveal answer

The area multiplier is 9, giving 81 square centimetres.

A′=9×32=81 cm2A^{\prime}=9\times3^2=81\,\text{cm}^2

Build an explanation

Hint 1

Compare corresponding lengths in the two similar shapes.

Hint 2

Every length uses the same scale factor k.

Hint 3

Areas use k² because two dimensions scale; volumes use k³ because three dimensions scale.

Worked example

This example uses fixed values, separate from the diagram controls.

A rectangle has area 6 cm²; a cuboid has volume 6 cm³. Both are enlarged by length scale factor 2. Find their new area and volume respectively.

  1. The area multiplier is 2² = 4; multiply 6 by 4.
  2. The volume multiplier is 2³ = 8; multiply 6 by 8.

Answer: Rectangle area 24 cm²; cuboid volume 48 cm³.

Connect this idea

Watch out: Doubling every length quadruples area and multiplies volume by eight.