Chapter 8 of 8

All 8 chapters
  1. 1Centimetres and metres, readunderstood
  2. 2Millimetres and kilometres, readunderstood
  3. 3Tenths and the decimal point, readunderstood
  4. 4Grams and kilograms, readunderstood
  5. 5Millilitres and litres, readunderstood
  6. 6Telling the time, readunderstood
  7. 7How long it takes, readunderstood
  8. 8Perimeter and area, readunderstood

M2.6.8 Chapter 8

Perimeter and area

I made two shapes from 12 squares. Is the way round the same?

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

  • Find the perimeter of a shape by adding its sides
  • Find the area of a shape by counting squares
  • Draw two shapes with the same area but different perimeters

A garden needs a fence round it. The garden also covers some ground. These are two different questions.

The distance all the way round the edge is the perimeter. The flat space inside is the area.

Draw it first

Perimeter and area of a rectangle

  1. A rectangle made of 12 squares in 3 rows of 4. Step 2 draws a dashed line round the edge and labels the sides 4 cm, 3 cm, 4 cm and 3 cm. Step 3 adds the words "perimeter" and "4 + 3 + 4 + 3 = 14 cm". Step 4 numbers the squares 1 to 12 and adds the words "area" and "12 squares".

    1 This shape is 4 squares long and 3 squares wide. Each square is 1 cm wide.

  2. A rectangle made of 12 squares in 3 rows of 4. Step 2 draws a dashed line round the edge and labels the sides 4 cm, 3 cm, 4 cm and 3 cm. Step 3 adds the words "perimeter" and "4 + 3 + 4 + 3 = 14 cm". Step 4 numbers the squares 1 to 12 and adds the words "area" and "12 squares".4 cm4 cm3 cm3 cm

    2 Walk round the edge. Write each side: 4, 3, 4 and 3.

  3. A rectangle made of 12 squares in 3 rows of 4. Step 2 draws a dashed line round the edge and labels the sides 4 cm, 3 cm, 4 cm and 3 cm. Step 3 adds the words "perimeter" and "4 + 3 + 4 + 3 = 14 cm". Step 4 numbers the squares 1 to 12 and adds the words "area" and "12 squares".4 cm4 cm3 cm3 cmperimeter4 + 3 + 4 + 3 = 14 cm

    3 Add all the sides. 4 + 3 + 4 + 3 = 14 cm. That is the perimeter.

  4. A rectangle made of 12 squares in 3 rows of 4. Step 2 draws a dashed line round the edge and labels the sides 4 cm, 3 cm, 4 cm and 3 cm. Step 3 adds the words "perimeter" and "4 + 3 + 4 + 3 = 14 cm". Step 4 numbers the squares 1 to 12 and adds the words "area" and "12 squares".4 cm4 cm3 cm3 cmperimeter4 + 3 + 4 + 3 = 14 cmarea12 squares123456789101112

    4 Count the squares inside. There are 12. That is the area.

Perimeter is a length, so it has units like cm. Area is counted in squares.

A rectangle made of squares in 2 rows of 6. The top side is labelled 6 cm. The left side is labelled 2 cm.6 cm2 cm
Each square is 1 cm wide.

Worked example

A rectangle is 6 squares long and 2 squares wide. Each square is 1 cm wide. Find the perimeter.

  1. The sides are 6, 2, 6 and 2.
  2. Add them all. 6 + 2 + 6 + 2 = 16.

Answer: 16 cm

To find an area, count one row. Then add the rows. 2 rows of 6 squares is 6 + 6 = 12 squares.

See it three ways

With things

Lay 12 squares of paper in 2 rows of 6. Count the edges all round. Then count the squares.

As a picture

A rectangle made of 12 squares in 2 rows of 6. Under it are the words "perimeter 16 cm" and "area 12 squares".perimeter 16 cmarea 12 squares

In symbols

6 + 2 + 6 + 2 = 16 cm. 6 + 6 = 12 squares.

Now take the same 12 squares. Make other rectangles. Look at the perimeters.

Three rectangles, each made of 12 squares. The first is 1 by 12 with the words "1 by 12: perimeter 26 cm". The second is 2 by 6 with "2 by 6: perimeter 16 cm". The third is 3 by 4 with "3 by 4: perimeter 14 cm".1 by 12: perimeter 26 cm2 by 6: perimeter 16 cm3 by 4: perimeter 14 cm
All three have an area of 12 squares. The perimeters are not the same.

Try it

Same squares, different edge

You need: 12 small squares of paper, all the same size, a pencil.

Be safe: A grown up cuts the paper if you need scissors. You can also tear it along a fold.

  1. Make a line of 12 squares, touching side to side. Count the edges all round.
  2. Make 2 rows of 6. Count the edges all round.
  3. Make 3 rows of 4. Count the edges all round.
  4. Write the three numbers.

What you will see: The squares stay the same. The way round gets shorter as the shape gets fatter.

If you cannot try it, this is what happens.
A table with three columns: "shape", "squares" and "edges". The rows are: 1 by 12, 12, 26. 2 by 6, 12, 16. 3 by 4, 12, 14.shapesquaresedges1 by 1212262 by 612163 by 41214

A wrong idea many people have

Shapes with the same area have the same perimeter.

Why it feels right: Both shapes use all 12 squares. It seems the edge should match.

The test that breaks it: Make 12 squares into a line. Count the edges all round: 26. Now make a block of 3 by 4. Count again: 14.

What is true: Area counts the squares inside. Perimeter measures the way round. A long thin shape has a longer way round.

Key words

perimeter
the distance all the way round the edge of a shape.
area
the flat space inside a shape. We count it in squares.

Back to your guess

The answer is B: No. The way round can change when the shape changes.

The way round can change. A long thin shape has a longer way round. Two rectangles from 12 squares can have different ways round.

Check yourself

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

  1. Each square is 1 cm wide. What is the perimeter of this rectangle?

    A rectangle made of squares, 5 squares long and 2 squares wide.
    See the answer

    14 cm

    The sides are 5, 2, 5 and 2. 5 + 2 + 5 + 2 = 14.

  2. How many squares cover this shape?

    A shape made of squares. The top row has 3 squares. Two more squares are below the left one.
    See the answer

    5 squares

    Count the top row: 3. Count the two below: 2. 3 + 2 = 5.

  3. Both shapes are made of 8 squares. Each square is 1 cm wide. Which has the longer perimeter?

    Two shapes made of 8 squares. Shape A is 2 rows of 4 squares, labelled "A". Shape B is one row of 8 squares, labelled "B".AB
    Tap your answer.

    1. Not this one. Shape A is 2 by 4. 2 + 4 + 2 + 4 = 12. Shape B is 1 by 8. 1 + 8 + 1 + 8 = 18. B is longer. Try another.

    2. Yes, that is right. Shape A has a perimeter of 12 cm. Shape B has 18 cm. The long thin shape has the longer way round.

    3. Not this one. Both have 8 squares, so the area is the same. The perimeter is not. Count the edges all round. Try another.

    See the answer

    Shape B

    Shape A has a perimeter of 12 cm. Shape B has 18 cm. The long thin shape has the longer way round.

    • If you chose A: Shape A is 2 by 4. 2 + 4 + 2 + 4 = 12. Shape B is 1 by 8. 1 + 8 + 1 + 8 = 18. B is longer.
    • If you chose C: Both have 8 squares, so the area is the same. The perimeter is not. Count the edges all round.

  4. Each square is 1 cm wide. Draw two shapes that cover 6 squares but have different perimeters. Write both perimeters.

    See the answer

    A line of 6 squares has a perimeter of 14 cm. 2 rows of 3 have a perimeter of 10 cm.

    Both cover 6 squares. 1 + 6 + 1 + 6 = 14 and 2 + 3 + 2 + 3 = 10. There are other answers too.

  5. A rectangle has a perimeter of 18 cm. Two opposite sides are each 5 cm. How long is each of the other two sides?

    See the answer

    4 cm

    5 + 5 = 10. 18 − 10 = 8. The other two sides share 8, so each is 4 cm.

  6. From an earlier chapter

    Change 3 m 40 cm into centimetres.

    See the answer

    340 cm

    1 m is 100 cm. 3 m is 300 cm. 300 + 40 = 340.

  7. What is the perimeter of this rectangle?

    A rectangle made of squares in 3 rows of 5. The top side is labelled 5 cm. The left side is labelled 3 cm.5 cm3 cm
    Tap your answer.

    1. Yes, that is right. The sides are 5, 3, 5 and 3. 5 + 3 + 5 + 3 = 16.

    2. Not this one. 15 is the number of squares inside. That is the area. The perimeter is the way round. Try another.

    3. Not this one. You added only two sides, 5 and 3. A rectangle has four sides. Add all of them. Try another.

    See the answer

    16 cm

    The sides are 5, 3, 5 and 3. 5 + 3 + 5 + 3 = 16.

    • If you chose B: 15 is the number of squares inside. That is the area. The perimeter is the way round.
    • If you chose C: You added only two sides, 5 and 3. A rectangle has four sides. Add all of them.

  8. Each square is 1 cm wide. How many squares cover this shape?

    A rectangle made of squares in 4 rows of 6. The label under it says each square is 1 cm wide.each square is 1 cm wide
    Tap your answer.

    1. Not this one. 20 is the way round, 6 + 4 + 6 + 4. The area is all the squares inside. Try another.

    2. Not this one. You counted only the squares along the edge. The squares in the middle count too. Try another.

    3. Yes, that is right. Count one row: 6. There are 4 rows. 6 + 6 + 6 + 6 = 24.

    See the answer

    24 squares

    Count one row: 6. There are 4 rows. 6 + 6 + 6 + 6 = 24.

    • If you chose A: 20 is the way round, 6 + 4 + 6 + 4. The area is all the squares inside.
    • If you chose B: You counted only the squares along the edge. The squares in the middle count too.

  9. Amina puts a fence round her garden. What does she need to measure?

    Tap your answer.

    1. Not this one. The area is the flat space inside. A fence goes round the edge, so she needs the way round. Try another.

    2. Yes, that is right. A fence goes all the way round the edge. The distance round the edge is the perimeter.

    3. Not this one. They are different. The area is the space inside. The perimeter is the way round. Try another.

    See the answer

    the perimeter

    A fence goes all the way round the edge. The distance round the edge is the perimeter.

    • If you chose A: The area is the flat space inside. A fence goes round the edge, so she needs the way round.
    • If you chose C: They are different. The area is the space inside. The perimeter is the way round.

  10. Each shape is made of 12 squares. Which sentence is true?

    Two shapes made of squares. Shape A is 2 rows of 6 squares, labelled "A". Shape B is 3 rows of 4 squares, labelled "B". Under them is the label each square is 1 cm wide.ABeach square is 1 cm wide
    Tap your answer.

    1. Yes, that is right. Both cover 12 squares, so the areas are the same. A has a perimeter of 16 cm and B has 14 cm.

    2. Not this one. The same squares can make different edges. A has a perimeter of 16 cm. B has 14 cm. Try another.

    3. Not this one. A long thin shape has the longer way round. A is long and thin, and it is 16 cm. B is 14 cm. Try another.

    See the answer

    The areas are the same. The perimeters are different.

    Both cover 12 squares, so the areas are the same. A has a perimeter of 16 cm and B has 14 cm.

    • If you chose B: The same squares can make different edges. A has a perimeter of 16 cm. B has 14 cm.
    • If you chose C: A long thin shape has the longer way round. A is long and thin, and it is 16 cm. B is 14 cm.

Keep it

Copy this into your exercise book.

Perimeter is the way round. Add all the sides. Area is how many squares cover a shape. Same area can have different perimeters.

Draw: A rectangle of 3 by 4 squares. Write 3 + 4 + 3 + 4 = 14 and 12 squares.

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 (882 KB). It opens on any phone.