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Lesson · Grade 11 · Foundations of Mathematics

Scale Factor: Lengths by k, Areas by k², Volumes by k³

  • Grade 11
  • Foundations of Mathematics
  • 2 min 10 s
  • British Columbia
  • BC11-MA-35

A 1 cm cube is enlarged by scale factor k = 3: each edge becomes 3 cm, a face holds 9 squares (3²) and 27 small cubes fill it layer by layer (3³), which gives the rule lengths × k, areas × k², volumes × k³. A 1 : 20 model car then uses the rule (a 2.5 cm wheel becomes 50 cm, 10 cm² of paint becomes 4000 cm², 1 cm³ of plastic becomes 8000 cm³), shows a reduction with k = 1/20, corrects the common mistake of multiplying an area by k alone, and ends on a one-line recap.

Concept: Scale factor: lengths by k, areas by k squared, volumes by k cubed

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In one sentence

The question this lesson answers

When we enlarge a 1 centimetre cube, what happens to its area and volume?

Enlarging multiplies every length by one number, the scale factor k.

Transcript

What the lesson says

When we enlarge a 1 centimetre cube, what happens to its area and volume?

Enlarging multiplies every length by one number, the scale factor k.

Let k be 3. Each edge becomes 1 times 3, which is 3 centimetres.

One face is 3 by 3, so it holds 9 squares of 1 square centimetre.

So the area is 9 times bigger, and 9 is 3 squared, 3 times 3.

Now fill it with 1 centimetre cubes, layer by layer: 9 cubes, then 18, then 27.

So the volume is 27 times bigger, and 27 is 3 cubed, 3 times 3 times 3.

So lengths are multiplied by k, areas by k squared, and volumes by k cubed.

Now a 1 to 20 model car. Every real length is 20 times the model's, so k is 20.

The model wheel is 2.5 centimetres across. 2.5 times 20 is 50 centimetres.

Paint covers an area: 10 square centimetres times 20 squared, 400, is 4000 square centimetres.

Plastic fills a volume: 1 cubic centimetre times 20 cubed, 8000, is 8000 cubic centimetres.

Real car to model is a reduction, so k is less than 1: one twentieth. 50 centimetres times one twentieth is 2.5 centimetres.

A common mistake is to multiply an area or a volume by just k: 10 times 20 is only 200.

Areas need k squared and volumes need k cubed, so the paint covers 4000 square centimetres.

Lengths scale by k, areas by k squared, and volumes by k cubed.

Curriculum

Where this fits

British Columbia, Grade 11

Foundations of Mathematics 11, Geometry: scale modelsBC11-MA-35

Previous: Proving the angle sums of triangles and polygonsNext: The Square-Cube Law: Why Giants Can't Just Be Scaled Up

The outcomes and strands follow the British Columbia curriculum.

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