MTH1W How Changing Dimensions Changes Area and Volume: Notes and Practice

A pizza place sells a medium pizza 20 cm across and a large one 40 cm across. The large is twice as wide, so is it twice as much pizza? No: it is four times as much. Picture four medium pizzas' worth of area spread over the large one. A model car that is $\frac{1}{24}$ the length of the real car needs far less than $\frac{1}{24}$ of the paint, and holds far less than $\frac{1}{24}$ of the air. Lengths, areas and volumes respond to scaling in different ways. Knowing exactly how lets you compare prices, estimate materials and predict how much a bigger container holds, without measuring everything again.

Scale factor

When a shape is enlarged or reduced so that every length is multiplied by the same number $k$ , the new shape is similar to the original (same shape, different size), and $k$ is the scale factor. If $k > 1$ the shape gets bigger (an enlargement); if $0 < k < 1$ it gets smaller (a reduction); $k = 1$ leaves the size unchanged. Every length is multiplied: the sides, the height, the diagonal, the radius, the diameter.

Fig 1A 4 cm by 3 cm rectangle with a 5 cm diagonal, reduced with k = 1/2 to 2 cm by 1.5 cm and enlarged with k = 2 to 8 cm by 6 cm, drawn to scale; the diagonal and the perimeter are multiplied by k too

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A photo 10 cm wide is reduced so that it is 4 cm wide, keeping its shape. What is the scale factor?

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A triangle with sides 3 cm, 5 cm and 7 cm is enlarged with a scale factor of 3. How long is the longest side of the new triangle?

Perimeter, area and volume

Perimeter (and circumference) is a sum of lengths, so it is multiplied by $k$ . Area is found by multiplying two lengths, such as length $\times$ width; each one is multiplied by $k$ , so the area is multiplied by $k \times k = k^{2}$ . Surface area is a total of areas, so it is also multiplied by $k^{2}$ . Volume multiplies three lengths, length $\times$ width $\times$ height, so it is multiplied by $k \times k \times k = k^{3}$ . The figure shows this with cubes: doubling every edge of a 1 cm cube makes each face hold 4 unit squares and the whole cube hold 8 unit cubes.

Fig 2Cubes with edges of 1, 2 and 3 centimetres built from unit cubes, showing face areas of 1, 4 and 9 square centimetres and volumes of 1, 8 and 27 cubic centimetres

new perimeter = k \times P new area = k^{2} \times A new volume = k^{3} \times V

Scale factor k	Perimeter multiplied by	Area multiplied by	Volume multiplied by
2	2	4	8
3	3	9	27
10	10	100	1000
$\frac{1}{2}$	$\frac{1}{2}$	$\frac{1}{4}$	$\frac{1}{8}$

Effect of a scale factor on similar shapes

Circles follow the same rule

Doubling the radius of a circle doubles its circumference (

2 π r

becomes

2 π (2 r) = 2 \times 2 π r

) but multiplies its area by 4 (

π r^{2}

becomes

π (2 r)^{2} = 4 π r^{2}

). That is the pizza: diameter

\times 2

, pizza

\times 4

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Every length of a shape is multiplied by 5. Its area is multiplied by what number?

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Every edge of a cube is halved. What happens to its volume?

VideoMario shows that when lengths are multiplied by a scale factor, areas are multiplied by its square and volumes by its cube.Watch on YouTube

Changing only some dimensions

The $k^{2}$ and $k^{3}$ rules need every length to be scaled. If only some dimensions change, multiply by the factor for each dimension that changed and by 1 for each that did not. For a box, $V = lw h$ : doubling only the length doubles the volume; doubling the length and the width multiplies it by $2 \times 2 = 4$ ; doubling all three gives $2 \times 2 \times 2 = 8$ . For a cylinder, $V = π r^{2} h$ : the radius appears twice (it is squared), so doubling only the radius multiplies the volume by 4, while doubling only the height multiplies it by 2. Perimeter is trickier because it adds lengths: a 5 cm by 3 cm rectangle has perimeter 16 cm, but doubling only its length gives a 10 cm by 3 cm rectangle with perimeter 26 cm, which is not double.

Fig 3Boxes of unit cubes: a 2 by 1 by 1 box (2 cubes), then the length doubled (4 cubes), the length and width doubled (8 cubes) and all three doubled (16 cubes), multiplying the volume by 2, 4 and 8

Change to a box (V = lwh)	Volume multiplied by
Length × 2 only	2
Length × 2 and width × 2	4
Length, width and height all × 2	8
Length $\times 3$ , height $\times \frac{1}{2}$	$3 \times \frac{1}{2} = 1.5$

Multiply the factors of the dimensions that change

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A cylinder's radius is doubled and its height is tripled. Its volume is multiplied by what number?

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A box's length is doubled, its width is doubled and its height is halved. Its volume is multiplied by:

Working backwards to the scale factor

If you know how much an area or volume changed for similar shapes, undo the power. Area multiplied by 9 means $k^{2} = 9$ , so $k = 9 = 3$ : every length tripled. Volume multiplied by 64 means $k^{3} = 64$ , so $k = 4$ , because $4 \times 4 \times 4 = 64$ . The scale factor does not have to be a whole number: doubling the area of a square means $k^{2} = 2$ , so each side is multiplied by $2 \approx 1.414$ , not by 2.

Scale models

A 1 to 24 scale model car has

k = \frac{1}{24}

. Its length is

\frac{1}{24}

of the real car's, its painted surface is

(\frac{1}{24})^{2} = \frac{1}{576}

of the real car's, and its volume is

(\frac{1}{24})^{3} = \frac{1}{13 824}

of the real car's.

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Two cans are similar. The bigger one holds 27 times as much as the smaller one. Every length of the bigger can is how many times the matching length of the smaller can?

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The area of a square is multiplied by 4. Each side is multiplied by:

How Changing Dimensions Changes Area and Volume

Key things to remember

Scale factor

Perimeter, area and volume

Changing only some dimensions

Working backwards to the scale factor

Worked examples

Where marks get lost

Flashcards 8

Practice