Chapter 6 of 8

All 8 chapters
  1. 1Reflections and rotations, readunderstood
  2. 2Translations and combined moves, readunderstood
  3. 3Enlargements, readunderstood
  4. 4Congruent triangles, readunderstood
  5. 5Similar shapes, readunderstood
  6. 6Similar areas and volumes, readunderstood
  7. 7Arcs and sectors, readunderstood
  8. 8Angle rules in circles, readunderstood

M4.4.6 Chapter 6

Similar areas and volumes

A jar is made twice as high and twice as wide. Does it hold twice as much?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Use k, k² and k³ for length, area and volume scale factors
  • Find the volume of a larger jar of the same shape

Two solids of the same shape are similar, like two jars made from the same mould. Every length of the large jar is the same number of times the matching length of the small jar. That number is the scale factor, k. Here k = 2.

Two jars of the same shape. The small jar is 8 cm high and 5 cm wide. The large jar is 16 cm high and 10 cm wide. The labels are small jar, large jar, 8 cm, 5 cm, 16 cm and 10 cm.8 cm5 cmsmall jar16 cm10 cmlarge jar
Every length of the large jar is 2 times the matching length of the small jar.

Lengths go up by k. Areas and volumes do not go up by k. Start with areas.

A square 1 cm by 1 cm with an area of 1 cm², beside a square 2 cm by 2 cm made of 4 small squares with an area of 4 cm². The labels are 1 cm, 2 cm, 1 cm² and 4 cm².1 cm1 cm²2 cm4 cm²
The sides are 2 times as long. The area is 4 times as big.

The big square has 2 rows of 2 small squares, so 2 × 2 = 4 of them. For scale factor k, the big square has k rows of k small squares, so k × k = k² of them. Any flat shape can be cut into small squares, so every area is multiplied by k². This is the area scale factor.

A cube 1 cm on each edge with a volume of 1 cm³, beside a block 2 cm on each edge made of 8 small cubes with a volume of 8 cm³. The labels are 1 cm, 2 cm, 1 cm³ and 8 cm³.1 cm1 cm³2 cm8 cm³
The edges are 2 times as long. The volume is 8 times as big.

The big block has 2 layers. Each layer has 2 rows of 2 small cubes. That is 2 × 2 × 2 = 8 cubes. For scale factor k there are k × k × k = k³ small cubes. Every volume is multiplied by k³. This is the volume scale factor.

A table with the head k, Len., Area and Vol., short for length, area and volume. Row 1: 2, × 2, × 4, × 8. Row 2: 3, × 3, × 9, × 27. Row 3: 10, × 10, × 100, × 1 000.kLen.AreaVol.2× 2× 4× 83× 3× 9× 2710× 10× 100× 1 000
Lengths multiply by k, areas by k × k, volumes by k × k × k.
A square 1 cm by 1 cm with an area of 1 cm², beside a square 3 cm by 3 cm made of 9 small squares with an area of 9 cm². The labels are 1 cm, 3 cm, 1 cm² and 9 cm².1 cm1 cm²3 cm9 cm²
With k = 3, there are 9 small squares in the big square.

Worked example

A photo frame has an area of 20 cm². A similar frame has all its lengths 3 times as long. What is the area of the larger frame?

  1. The length scale factor is k = 3.
  2. The area scale factor is k² = 3 × 3 = 9.
  3. The new area is 20 × 9 = 180 cm².

Answer: 180 cm²

Two jars of the same shape. The small jar is 12 cm high and 4 cm wide. The large jar is 24 cm high and 8 cm wide. The labels are small jar, large jar, 12 cm, 4 cm, 24 cm and 8 cm.12 cm4 cmsmall jar24 cm8 cmlarge jar
The large jar is twice as high and twice as wide.

Worked example

A small jar holds about 150 ml. A larger jar of the same shape is twice as high. About how much does the larger jar hold?

  1. Twice as high means the length scale factor is k = 2.
  2. The volume scale factor is k³ = 2 × 2 × 2 = 8.
  3. The larger jar holds about 150 × 8 = 1 200 ml.

Answer: about 1 200 ml

You can also go backwards. If you know how areas compare, take a square root to find k. If you know how volumes compare, take a cube root.

Two jars of the same shape. The small jar is 10 cm high and 6 cm wide. The large jar is 20 cm high and 12 cm wide. The labels are small jar, large jar, 10 cm, 6 cm, 20 cm and 12 cm.10 cm6 cmsmall jar20 cm12 cmlarge jar
The whole outsides of the jars have areas of about 245 cm² and 980 cm².

Worked example

Two jars have the same shape. Their whole outsides, with lid and base, have areas of about 245 cm² and 980 cm². The small jar holds about 283 ml. Find k and the volume of the large jar.

  1. The area scale factor is 980 ÷ 245 = 4.
  2. The area scale factor is k², so k is the square root of 4, which is 2.
  3. The volume scale factor is k³ = 2 × 2 × 2 = 8.
  4. The large jar holds about 283 × 8 = 2 264 ml.

Answer: k is 2 and the large jar holds about 2 264 ml

See it three ways

With things

Build a cube from 27 small cubes, 3 along each edge. Its edge is 3 times the edge of one small cube. Count the small cubes: 3 × 3 × 3.

As a picture

A cube 1 cm on each edge with a volume of 1 cm³, beside a block 3 cm on each edge made of 27 small cubes with a volume of 27 cm³. The labels are 1 cm, 3 cm, 1 cm³ and 27 cm³.1 cm1 cm³3 cm27 cm³

In symbols

Scale factor k for lengths. Scale factor k × k for areas. Scale factor k × k × k for volumes. With k = 3 these are 3, 9 and 27.

Try it

Count the squares

You need: squared paper, a pencil.

  1. Draw a square 1 cm by 1 cm. Draw a square 4 cm by 4 cm beside it.
  2. Count the 1 cm squares inside the big square.
  3. Divide the number of squares in the big square by the number in the small square.

What you will see: The big square has 16 small squares. The sides are 4 times as long, and 4 × 4 = 16.

If you cannot try it, this is what happens.
A square 1 cm by 1 cm with an area of 1 cm², beside a square 4 cm by 4 cm made of 16 small squares with an area of 16 cm². The labels are 1 cm, 4 cm, 1 cm² and 16 cm².1 cm1 cm²4 cm16 cm²

A wrong idea many people have

If a shape is enlarged by scale factor 3, its area is 3 times as big.

Why it feels right: The lengths are 3 times as long, so it seems the area should be 3 times as big too.

The test that breaks it: Take a square 1 cm by 1 cm with an area of 1 cm². Enlarge it by scale factor 3. It becomes 3 cm by 3 cm, with an area of 9 cm². That is 9 times as big, not 3 times.

What is true: Area is length times width, and both lengths are multiplied by k. So the area is multiplied by k × k = k². Volume has three lengths, so it is multiplied by k³.

Key words

area scale factor
the number every area is multiplied by. It is k², where k is the length scale factor.
volume scale factor
the number every volume is multiplied by. It is k³, where k is the length scale factor.

Back to your guess

The answer is C: It holds 8 times as much.

Doubling the height, the width and the depth makes 2 × 2 × 2 = 8 times the space inside. A jar that is twice as big in every direction holds 8 times as much. This chapter shows why.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. A small box has a volume of 40 cm³. A similar large box has every length 3 times as long. What is the volume of the large box?

    Tap your answer.

    1. Not this one. You multiplied the volume by 3. A volume has three lengths, so it is multiplied by 3 × 3 × 3. Go to M4.4.6. Try another.

    2. Not this one. You multiplied by 3 × 3 = 9, which is for an area. A volume needs three lengths, so use 3 × 3 × 3. Go to M4.4.6. Try another.

    3. Yes, that is right. The volume scale factor is k³ = 3 × 3 × 3 = 27. The large box has a volume of 40 × 27 = 1 080 cm³.

    See the answer

    1 080 cm³

    The volume scale factor is k³ = 3 × 3 × 3 = 27. The large box has a volume of 40 × 27 = 1 080 cm³.

    • If you chose A: You multiplied the volume by 3. A volume has three lengths, so it is multiplied by 3 × 3 × 3. Go to M4.4.6.
    • If you chose B: You multiplied by 3 × 3 = 9, which is for an area. A volume needs three lengths, so use 3 × 3 × 3. Go to M4.4.6.

  2. A poster has an area of 50 cm². It is enlarged by scale factor 4. What is the area of the enlarged poster?

    Tap your answer.

    1. Not this one. You multiplied the area by 4, the length scale factor. An area has two lengths, so multiply by 4 × 4. Go to M4.4.6. Try another.

    2. Yes, that is right. The area scale factor is k² = 4 × 4 = 16. The new area is 50 × 16 = 800 cm².

    3. Not this one. You multiplied by 4 × 4 × 4, which is for a volume. An area has only two lengths, so use 4 × 4. Go to M4.4.6. Try another.

    See the answer

    800 cm²

    The area scale factor is k² = 4 × 4 = 16. The new area is 50 × 16 = 800 cm².

    • If you chose A: You multiplied the area by 4, the length scale factor. An area has two lengths, so multiply by 4 × 4. Go to M4.4.6.
    • If you chose C: You multiplied by 4 × 4 × 4, which is for a volume. An area has only two lengths, so use 4 × 4. Go to M4.4.6.

  3. Two similar shapes have areas of 12 cm² and 108 cm². What is the length scale factor from the small shape to the large one?

    Tap your answer.

    1. Yes, that is right. The area scale factor is 108 ÷ 12 = 9. The length scale factor is the square root of 9, which is 3.

    2. Not this one. You divided 108 by 12. That is the area scale factor. The length scale factor is its square root. Go to M4.4.6. Try another.

    3. Not this one. You took 12 away from 108. Scale factors are found by dividing, and then a square root is needed for lengths. Go to M4.4.6. Try another.

    See the answer

    3

    The area scale factor is 108 ÷ 12 = 9. The length scale factor is the square root of 9, which is 3.

    • If you chose B: You divided 108 by 12. That is the area scale factor. The length scale factor is its square root. Go to M4.4.6.
    • If you chose C: You took 12 away from 108. Scale factors are found by dividing, and then a square root is needed for lengths. Go to M4.4.6.

  4. A small jar holds 100 ml. A similar large jar holds 800 ml. How many times as high as the small jar is the large jar?

    Tap your answer.

    1. Not this one. You divided 800 by 100. That is the volume scale factor. The length scale factor is its cube root. Go to M4.4.6. Try another.

    2. Not this one. You halved 8. Volume scales by k × k × k, so you need the number k with k × k × k = 8. Go to M4.4.6. Try another.

    3. Yes, that is right. The volume scale factor is 800 ÷ 100 = 8. The length scale factor k has k³ = 8, so k = 2, because 2 × 2 × 2 = 8.

    See the answer

    2

    The volume scale factor is 800 ÷ 100 = 8. The length scale factor k has k³ = 8, so k = 2, because 2 × 2 × 2 = 8.

    • If you chose A: You divided 800 by 100. That is the volume scale factor. The length scale factor is its cube root. Go to M4.4.6.
    • If you chose B: You halved 8. Volume scales by k × k × k, so you need the number k with k × k × k = 8. Go to M4.4.6.

  5. A shape is enlarged by scale factor 5. Which statement about its area is true?

    Tap your answer.

    1. Not this one. Five times is the scale factor for lengths. Area has two lengths, so it grows by more than that. Go to M4.4.6. Try another.

    2. Yes, that is right. The area scale factor is k² = 5 × 5 = 25.

    3. Not this one. You added 5 and 5. Areas are found by multiplying, so the factor is 5 × 5. Go to M4.4.6. Try another.

    See the answer

    The area is 25 times as big.

    The area scale factor is k² = 5 × 5 = 25.

    • If you chose A: Five times is the scale factor for lengths. Area has two lengths, so it grows by more than that. Go to M4.4.6.
    • If you chose C: You added 5 and 5. Areas are found by multiplying, so the factor is 5 × 5. Go to M4.4.6.

  6. From an earlier chapter

    Triangle ABC has sides of 4 cm and 5 cm. A similar triangle has a side of 12 cm that matches the 4 cm side. How long is the side that matches 5 cm, in centimetres?

    See the answer

    15

    The scale factor is 12 ÷ 4 = 3. Multiply 5 by it: 5 × 3 = 15 cm.

  7. A shape has an area of 6 cm². It is enlarged by scale factor 3. What is the new area, in cm²?

    See the answer

    54

    The area scale factor is 3 × 3 = 9. The new area is 6 × 9 = 54 cm².

  8. A jar holds 15 ml. A jar of the same shape has every length 2 times as long. How many ml does it hold?

    See the answer

    120

    The volume scale factor is 2 × 2 × 2 = 8. The larger jar holds 15 × 8 = 120 ml.

  9. A small jar holds 10 ml. A similar large jar holds 270 ml. How many times as high as the small jar is the large jar?

    See the answer

    3

    The volume scale factor is 270 ÷ 10 = 27. The length scale factor k has k × k × k = 27, so k = 3.

Keep it

Copy this into your exercise book.

Scale factor k: lengths × k, areas × k², volumes × k³. Doubling every length gives 4 times the area and 8 times the volume.

Draw: A 1 cm cube beside a 2 cm cube made of 8 small cubes, with the three scale factors written under them.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (1.0 MB). It opens on any phone.