Chapter 2 of 8

All 8 chapters
  1. 1Changing metric units, readunderstood
  2. 2Perimeter, readunderstood
  3. 3Area of rectangles, readunderstood
  4. 4Area of triangles and parallelograms, readunderstood
  5. 5Trapeziums and joined shapes, readunderstood
  6. 6Circles, readunderstood
  7. 7Volume of cuboids, readunderstood
  8. 8Surface area, readunderstood

M3.6.2 Chapter 2

Perimeter

How much fence do I need to go all the way round my garden?

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.

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

By the end you can

  • Find the perimeter of a shape with straight sides
  • Find a missing side when you know the perimeter

A farmer wants a fence round a field. She needs to know how long the fence must be. The distance all the way round a shape is its perimeter.

To find a perimeter, add the lengths of all the sides. Every length needs its unit, such as cm or m.

Draw it first

Add every side

  1. A rectangle on a grid, 6 cm long and 4 cm wide. The sides are labelled 6 cm, 4 cm, 6 cm and 4 cm. Below it says 6 + 4 + 6 + 4 = 20 cm.

    1 Draw the shape. This rectangle is 6 cm long and 4 cm wide.

  2. A rectangle on a grid, 6 cm long and 4 cm wide. The sides are labelled 6 cm, 4 cm, 6 cm and 4 cm. Below it says 6 + 4 + 6 + 4 = 20 cm.6 cm4 cm6 cm4 cm

    2 Write the length of every side beside it, with its unit.

  3. A rectangle on a grid, 6 cm long and 4 cm wide. The sides are labelled 6 cm, 4 cm, 6 cm and 4 cm. Below it says 6 + 4 + 6 + 4 = 20 cm.6 cm4 cm6 cm4 cm6 + 4 + 6 + 4 = 20 cm

    3 Add all four sides: 6 + 4 + 6 + 4 = 20 cm.

A rectangle has two long sides and two short sides. The opposite sides are equal. So the perimeter is 2 × length + 2 × width. You can also add the length and width, then double it: 2 × (length + width).

Worked example

A garden is a rectangle 8 m long and 5 m wide. How much fence goes round it?

  1. The fence goes round all four sides. We need the perimeter.
  2. The garden has two sides of 8 m and two sides of 5 m.
  3. 8 + 5 + 8 + 5 = 26 m.

Answer: 26 m

A rectangle that is the garden. Its sides are labelled 8 m, 5 m, 8 m and 5 m.8 m5 m8 m5 m
The garden. The fence is 8 + 5 + 8 + 5 = 26 m long.

A square has four sides that are all equal. So its perimeter is 4 × the side. A square with sides of 7 m has a perimeter of 4 × 7 = 28 m.

Sometimes a side is missing, and you know the perimeter instead. Add the sides you know. Then take that total away from the perimeter.

Worked example

A triangle has sides of 5 cm and 6 cm, and a third side. Its perimeter is 18 cm. How long is the third side?

  1. The three sides add to the perimeter: 5 + 6 + the third side = 18.
  2. The two known sides add to 5 + 6 = 11 cm.
  3. The third side is 18 − 11 = 7 cm.
  4. Check: 5 + 6 + 7 = 18 cm.

Answer: 7 cm

A triangle with sides labelled 6 cm, 5 cm and ? cm. Below it says perimeter 18 cm.6 cm5 cm? cmperimeter 18 cm
The side marked ? cm is 18 − 5 − 6 = 7 cm.

Some shapes hide a side. Look at an L shape. Two of its sides have no label. You can find each one from the sides on the opposite edge.

Worked example

Find the perimeter of this L shape. Two sides are marked ? m.

  1. Find the two missing sides first.
  2. The flat missing side is 11 − 4 = 7 m.
  3. The upright missing side is 8 − 3 = 5 m.
  4. Add all six sides: 11 + 3 + 7 + 5 + 4 + 8 = 38 m.
  5. A rectangle 11 m by 8 m has 2 × (11 + 8) = 38 m. The same.

Answer: 38 m

An L shape. The bottom is 11 m, the right side is 3 m, the top is 4 m and the left side is 8 m. The two other sides are marked ? m.11 m3 m? m? m4 m8 m
The flat ? side is 11 − 4 = 7 m. The upright ? side is 8 − 3 = 5 m.

Look again at the L shape. Its perimeter is the same as the rectangle that would hold it. That works for any L shape with one corner cut out.

See it three ways

With things

Wrap a string round a book cover. Pull the string straight and measure it with a ruler.

As a picture

A rectangle on a grid, 6 cm long and 4 cm wide. The sides are labelled 6 cm, 4 cm, 6 cm and 4 cm.6 cm4 cm6 cm4 cm

In symbols

Perimeter = 6 + 4 + 6 + 4 = 20 cm. Or 2 × (6 + 4) = 20 cm.

Try it

The string round a book

You need: a book or an exercise book, a piece of string, a ruler, a pencil.

  1. Measure the long side and the short side of the cover in cm.
  2. Add the four sides to find the perimeter.
  3. Lay the string round the edge of the cover. Cut it where it meets.
  4. Straighten the string and measure it with the ruler.

What you will see: The string and the sum give the same length. A cover is 21 cm by 15 cm. Its perimeter is 21 + 15 + 21 + 15 = 72 cm.

If you cannot try it, this is what happens.
A rectangle for a book cover. Its sides are labelled 21 cm, 15 cm, 21 cm and 15 cm. Each square on the grid is 3 cm by 3 cm.21 cm15 cm21 cm15 cm

A wrong idea many people have

The perimeter of a rectangle is its length plus its width.

Why it feels right: Those are the two numbers you see on the picture.

The test that breaks it: Walk round a rectangle that is 5 m by 3 m. You walk 5 m, then 3 m, then 5 m, then 3 m. That is 16 m, not 8 m.

What is true: A rectangle has four sides. Add all four, or add the length and width and then double the total.

Key words

perimeter
the distance all the way round a shape.
rectangle
a four sided shape with four right angles. Its opposite sides are equal.
missing side
a side with no length written. You work it out from the sides you know.

Back to your guess

The answer is A: Add the lengths of all the sides.

A fence goes round every side. So you add the lengths of all the sides. This total is the perimeter. Multiplying or counting squares gives the space inside, which is a different job.

Check yourself

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

  1. What is the perimeter of this rectangle?

    A rectangle with sides labelled 9 cm, 4 cm, 9 cm and 4 cm.9 cm4 cm9 cm4 cm
    Tap your answer.

    1. Not this one. You added only one length and one width. A rectangle has four sides, so add all four. Go to M3.6.2. Try another.

    2. Yes, that is right. Add the four sides: 9 + 4 + 9 + 4 = 26 cm. Or 2 × (9 + 4) = 26 cm.

    3. Not this one. You multiplied 9 × 4. That finds the space inside, not the distance round. For the perimeter, add the sides. Go to M3.6.2. Try another.

    See the answer

    26 cm

    Add the four sides: 9 + 4 + 9 + 4 = 26 cm. Or 2 × (9 + 4) = 26 cm.

    • If you chose A: You added only one length and one width. A rectangle has four sides, so add all four. Go to M3.6.2.
    • If you chose C: You multiplied 9 × 4. That finds the space inside, not the distance round. For the perimeter, add the sides. Go to M3.6.2.

  2. A square has sides of 7 m. What is its perimeter?

    A square with all four sides labelled 7 m.7 m7 m7 m7 m
    Tap your answer.

    1. Not this one. You added only two sides. A square has four equal sides. Go to M3.6.2. Try another.

    2. Not this one. You multiplied 7 × 7. That is a different measure, the space inside. The perimeter adds the four sides. Go to M3.6.2. Try another.

    3. Yes, that is right. A square has four equal sides. 4 × 7 = 28 m.

    See the answer

    28 m

    A square has four equal sides. 4 × 7 = 28 m.

    • If you chose A: You added only two sides. A square has four equal sides. Go to M3.6.2.
    • If you chose B: You multiplied 7 × 7. That is a different measure, the space inside. The perimeter adds the four sides. Go to M3.6.2.

  3. A triangle has sides of 9 cm and 7 cm and a third side. Its perimeter is 24 cm. How long is the third side?

    A triangle with sides labelled 9 cm, 7 cm and ? cm. Below it says perimeter 24 cm.9 cm7 cm? cmperimeter 24 cm
    Tap your answer.

    1. Yes, that is right. The known sides add to 9 + 7 = 16 cm. The third side is 24 − 16 = 8 cm.

    2. Not this one. You added 9 + 7, the two sides you know. The third side is what is left of the perimeter after that. Go to M3.6.2. Try another.

    3. Not this one. You added the perimeter to the two sides. The sides are parts of the perimeter, so take them away. Go to M3.6.2. Try another.

    See the answer

    8 cm

    The known sides add to 9 + 7 = 16 cm. The third side is 24 − 16 = 8 cm.

    • If you chose B: You added 9 + 7, the two sides you know. The third side is what is left of the perimeter after that. Go to M3.6.2.
    • If you chose C: You added the perimeter to the two sides. The sides are parts of the perimeter, so take them away. Go to M3.6.2.

  4. A rectangle has a perimeter of 30 cm. Its length is 9 cm. How wide is it?

    A rectangle with its bottom labelled 9 cm and its right side labelled ? cm. Its perimeter is 30 cm.9 cm? cmperimeter 30 cm
    Tap your answer.

    1. Not this one. You took away only one length. The perimeter holds two lengths and two widths. Go to M3.6.2. Try another.

    2. Yes, that is right. The two lengths take 9 + 9 = 18 cm. That leaves 30 − 18 = 12 cm for the two widths. Each width is 12 ÷ 2 = 6 cm.

    3. Not this one. You took 18 from 30 and stopped. The 12 cm is for both widths together, so each width is half of it. Go to M3.6.2. Try another.

    See the answer

    6 cm

    The two lengths take 9 + 9 = 18 cm. That leaves 30 − 18 = 12 cm for the two widths. Each width is 12 ÷ 2 = 6 cm.

    • If you chose A: You took away only one length. The perimeter holds two lengths and two widths. Go to M3.6.2.
    • If you chose C: You took 18 from 30 and stopped. The 12 cm is for both widths together, so each width is half of it. Go to M3.6.2.

  5. What is the perimeter of this L shape? Two sides are marked ? m.

    An L shape. The bottom is 12 m, the right side is 2 m, the top is 3 m and the left side is 8 m. The two other sides are marked ? m.12 m2 m? m? m3 m8 m
    Tap your answer.

    1. Not this one. You added only the four sides that have a label. The two sides marked ? are part of the perimeter too. Go to M3.6.2. Try another.

    2. Not this one. You multiplied 12 × 8. The perimeter is the distance round, so add the sides. The missing sides are 12 − 3 = 9 m and 8 − 2 = 6 m. Go to M3.6.2. Try another.

    3. Yes, that is right. The flat missing side is 12 − 3 = 9 m. The upright missing side is 8 − 2 = 6 m. The perimeter is 12 + 2 + 9 + 6 + 3 + 8 = 40 m.

    See the answer

    40 m

    The flat missing side is 12 − 3 = 9 m. The upright missing side is 8 − 2 = 6 m. The perimeter is 12 + 2 + 9 + 6 + 3 + 8 = 40 m.

    • If you chose A: You added only the four sides that have a label. The two sides marked ? are part of the perimeter too. Go to M3.6.2.
    • If you chose B: You multiplied 12 × 8. The perimeter is the distance round, so add the sides. The missing sides are 12 − 3 = 9 m and 8 − 2 = 6 m. Go to M3.6.2.

  6. Which job needs the perimeter?

    Tap your answer.

    1. Yes, that is right. A fence goes all the way round the edge of the field. Its length is the perimeter.

    2. Not this one. Tiles cover the space inside a floor. That is area, not the distance round the edge. Go to M2.6.8. Try another.

    3. Not this one. Sand fills the space inside a box. That is about how much a box holds, not the distance round an edge. Go to M2.6.5. Try another.

    See the answer

    Putting a fence round a field

    A fence goes all the way round the edge of the field. Its length is the perimeter.

    • If you chose B: Tiles cover the space inside a floor. That is area, not the distance round the edge. Go to M2.6.8.
    • If you chose C: Sand fills the space inside a box. That is about how much a box holds, not the distance round an edge. Go to M2.6.5.

  7. A rectangle is 12 m long and 7 m wide. What is its perimeter?

    See the answer

    38 m

    Add the four sides: 12 + 7 + 12 + 7 = 38 m.

  8. From an earlier chapter

    A rectangle is 1.5 m long and 80 cm wide. What is its perimeter in centimetres?

    See the answer

    460 cm

    1.5 m is 1.5 × 100 = 150 cm. The perimeter is 150 + 80 + 150 + 80 = 460 cm.

Keep it

Copy this into your exercise book.

Perimeter is the distance all the way round. Add every side and write the unit. Rectangle: 2 × (length + width). Square: 4 × side. Missing side: perimeter minus the sides you know.

Draw: A rectangle 6 cm by 4 cm with each side labelled. Write 6 + 4 + 6 + 4 = 20 cm under it.

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 (825 KB). It opens on any phone.