Year 10 Standard · Unit 5 Solids, similarity and transformations · Lesson 4 of 6

If every length doubles, what happens to the area and the volume?

Two solids are similar when one is an enlargement of the other. Lengths, areas and volumes each change by a different amount, so a small change in size can make a large difference to paint and to material.

Geometry and TrigonometryGDC optional

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Year 9: a shape is enlarged by scale factor 3. A side was 4 cm. How long is it now?
    Answer12 cm, because 4 × 3 = 12.
  2. From Year 9: a rectangle is 5 cm by 8 cm. Find its area.
    Answer40 cm², because 5 × 8 = 40.
  3. From the last lesson: find the surface area of a sphere of radius 2 cm in terms of π.
    Answer16π cm², because 4π × 2² = 16π.
Words used in this lesson
similar
the same shape, with all lengths in the same ratio. For example: a photo and its enlargement.
scale factor
the number every length is multiplied by. For example: k.
corresponding
in the same position in each solid. For example: the two heights.
enlargement
a transformation that changes size and keeps shape. For example: a copy on a photocopier at 150%.
ratio
a comparison of two amounts by division. For example: 1 : 2.
scale model
a small copy of a real object with every length reduced by the same factor. For example: a toy train.

Two clay figures

A potter in Oaxaca makes a clay figure 20 cm tall, then a second copy exactly the same shape but twice as tall. She will paint both and wants to know how much more paint, and how much more clay, the large one needs.

Predict first: is the large copy 2 times, 4 times, 6 times or 8 times as much for the paint, and the same for the clay? Write down a reason, then use the first figure with a brick.

1
1area scale factor, k²
1volume scale factor, k³

Scale factor 1: the brick 1 × 2 × 3 becomes 1 × 2 × 3. Surface area 22 becomes 22, which is 1 time as much. Volume 6 becomes 6, which is 1 time as much.

Show me the steps
  1. Scale factor 1: nothing changes, and all three factors are 1.
  2. Scale factor 3: the brick is 3 × 6 × 9, the surface area is 198 (9 times) and the volume is 162 (27 times).
  3. Scale factor 5: lengths are 5 times, areas 25 times and volumes 125 times. The grid of squares shows why: 5 rows of 5 squares.
  4. Scale factor 2, the clay figure's: the brick becomes 2 × 4 × 6. The surface area 22 becomes 88 (4 times) and the volume 6 becomes 48 (8 times).

Similar solids have the same shape and one is an enlargement of the other. If the linear scale factor is k, lengths are multiplied by k, areas by k² and volumes by k³.

So the ratio of lengths a : b gives the ratio of areas a² : b² and the ratio of volumes a³ : b³. To go the other way, take a square root of the area ratio or a cube root of the volume ratio to get k.

2
3
4 : 9ratio of areas
8 : 27ratio of volumes

Lengths 2 : 3 in simplest form. Areas 4 : 9. Volumes 8 : 27. Scale factor from small to large: 1.50.

Show me the steps
  1. Lengths 2 and 3: the ratio is 2 : 3, the areas are 4 : 9 and the volumes are 8 : 27. The scale factor is 1.5.
  2. Lengths 2 and 4: simplify first to 1 : 2. Then areas 1 : 4 and volumes 1 : 8.
  3. Lengths 4 and 4: the ratio is 1 : 1, so the solids are the same size.
  4. Lengths 6 and 9: simplify to 2 : 3 before squaring or cubing: areas 4 : 9, volumes 8 : 27 again.

Match the factor to the quantity. If you are finding a volume, multiply by k³, not k or k². And if you are given an area ratio, take its square root to get k before you use it on a length.

Worked examples

1. Two similar tins have a linear scale factor of 2.5. The small tin has surface area 48 cm². Find the large tin's.

  1. The area scale factor is 2.5² = 6.25.
  2. Large area = 48 × 6.25 = 300 cm².

2. A model boat at a scale of 1 : 20 holds 0.3 litres. How much does the real boat hold?

  1. Every length is 20 times as large, so the volume factor is 20³ = 8000.
  2. Real volume = 0.3 × 8000 = 2400 litres.

3. Two similar solids have volumes 16 cm³ and 54 cm³. The small one has surface area 36 cm². Find the large one's.

  1. k³ = 54 ÷ 16 = 27/8, so k = 3/2.
  2. Area factor = (3/2)² = 9/4, so the large area is 36 × 9/4 = 81 cm².

Your go. The first step is given. Two similar bottles are 12 cm and 18 cm tall. The small one holds 240 ml. How much does the large one hold?

  1. The scale factor is 18 ÷ 12 = 1.5.
  2. Now cube it and multiply the volume.
Check the answer

1.5³ = 3.375, and 240 × 3.375 = 810 ml. Multiplying by 1.5 only would give 360 ml, far too small.

With a GDC (optional)

A GDC is handy for a decimal scale factor. To cube 1.5, enter 1.5 × 1.5 × 1.5, or use the cube key if there is one. Then use that factor on the volume. Work out the exact fraction first when you can: 3/2 cubed is 27/8.

Check the direction: if you are going from the smaller solid to the larger, the factor is more than 1. Everything here can be done by hand with fractions.

Check before you go on

1. Two similar solids have surface areas in the ratio 1 : 9. What is the ratio of their volumes?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. A large cone is an enlargement of a small cone with scale factor 3. The small cone has surface area 20 cm². What is the large cone's surface area, in cm²?

3. A small solid has volume 12 cm³. A similar solid has every length twice as big. What is its volume, in cm³?

Level 2 · Building

4. Two similar vases are 10 cm and 25 cm tall. The small vase holds 0.4 litres. How much does the large vase hold, in litres?

5. Two similar statues have surface areas of 25 cm² and 100 cm². The larger is 18 cm tall. How tall is the smaller, in cm?

6. A model of a longship is made at a scale of 1 : 20. The model needs 0.6 m² of paint. How much paint does the real ship's surface need, in m²?

Level 3 · Stretch

7. Two similar jars hold 54 cm³ and 128 cm³. The small jar has surface area 108 cm². What is the large jar's surface area, in cm²?

8. A tall cone has volume 270 cm³. A small cone, similar to it, is cut off the top. Its height is one third of the tall cone's. What is the volume that is left, in cm³?

Something to think about

Elephants and mice are roughly similar in shape. An elephant is about 50 times as tall as a mouse. By what factor is its weight greater, if weight is proportional to volume? And by what factor is the cross-section of its leg bones greater? What problem does that suggest?

One way to think about it

Weight goes with volume, so it is about 50³ = 125 000 times the mouse's. The bone's cross-section is an area, so only 50² = 2500 times. Each square unit of bone must carry 50 times the load, so a scaled-up mouse would break. That is why large animals have much thicker legs for their size.

Exit ticket

9. A statue is 10 times as tall as its exact scale model. Which statement is correct?