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Perimeter

Measurement · level 2, lesson 6 of 6

A question to start

A farmer has 12 pieces of fence, each piece 1 metre long. She wants to make a pen for her goats in the shape of a rectangle, using all 12 pieces, with no pieces left over and no gaps.

How many different rectangle pens can she make? And does it matter which one she picks?

Try it with 12 toothpicks or 12 matchsticks on the table before you read on. You will find more than one answer, and here is the surprise: all the pens use exactly the same fence, but some give the goats much more room than others. By the end of this lesson you will know every answer, and why.

The idea

Walking around the edge

Imagine an ant walking once all the way around the edge of a shape, starting at a corner and stopping when it gets back to the same corner. The distance the ant walks is called the perimeter of the shape.

The word comes from two old Greek words: "peri" means around, and "metre" means measure. So perimeter means "the measure around".

5 cm 5 cm 3 cm 3 cm start and finish at the green dot

To find the perimeter, add up the lengths of all the sides. Go round the shape in order so you do not miss one.

5 cm and 3 cm and 5 cm and 3 cm: 5 and 3 is 8, 8 and 5 is 13, 13 and 3 is 16 cm.

Perimeter is a length, the length of the ant's walk. So it is measured in length units: centimetres or metres.

Where do we need perimeters in real life? A fence around a garden, a ribbon around the edge of a card, the trim around a picture frame, the white line around a soccer field, a running track. Each time, the question is "how long is the way around?"

Rectangles and squares: a shortcut

In a rectangle, the opposite sides are the same length: the top is as long as the bottom, and the left side is as long as the right side. So if you know one long side and one short side, you know all four.

For the 5 cm by 3 cm rectangle: one long side and one short side make 5 and 3, which is 8. The other two sides make another 8. So the perimeter is 8 and 8, which is 16 cm. That is the same answer, with less adding.

A square has 4 sides all the same length. A square with sides of 6 cm has perimeter 6, 12, 18, 24: 24 cm. Counting by the side length 4 times is the same as adding it 4 times.

Perimeter on a grid

On squared paper, each little square has sides of 1 unit. To find the perimeter of a shape on a grid, count the little edges around the outside.

1 2 3 4 5 6 7 8 9 10 11 12 8 squares inside 12 edges around

Walk around and count the edges: 4 along the top, 2 down the right, 4 along the bottom, 2 up the left. That is 12 units.

But careful! There are only 8 squares inside. The perimeter is not the number of squares. Squares are the inside of the shape; edges are the way around. The corner squares, for example, each have 2 edges on the outside, so if you counted squares instead of edges, you would count the corners only once and get the wrong answer.

Finding a missing side

Sometimes you know the perimeter but not all the sides. Then you work backwards.

A rectangle has a perimeter of 20 cm, and its long side is 6 cm. How long is the short side?

  1. One long side and one short side make half of the way around. Half of 20 is 10.
  2. So the long side and the short side together make 10 cm.
  3. The long side is 6 cm. 6 and what make 10? 4. The short side is 4 cm.
  4. Check: 6, 4, 6, 4. 6 and 4 is 10, and 10 and 10 is 20. ✓

The same perimeter, different shapes

Now back to the farmer with 12 metres of fence. Here are all the rectangles with a perimeter of 12 on a grid.

5 by 1 4 by 2 3 by 3
  • 5 by 1: 5 and 1 is 6, twice is 12.
  • 4 by 2: 4 and 2 is 6, twice is 12.
  • 3 by 3 (a square, which is a special rectangle): 3 and 3 is 6, twice is 12.

Why are there only three? For any rectangle, one long side and one short side make half the perimeter, which is 6. The pairs of whole numbers that make 6 are 5 and 1, 4 and 2, 3 and 3. (6 and 0 would be no pen at all: just a flat fence with no room inside!) So there are exactly three pens.

But do they give the goats the same room? Count the squares inside. The 5 by 1 pen has 5 squares. The 4 by 2 pen has 8. The 3 by 3 pen has 9. Same fence, very different room! The farmer should choose the square pen, 3 m by 3 m. The amount of room inside is called area, and you will learn all about it in level 3.

Cutting a corner does not always change the perimeter

Here is a puzzle that surprises many grown-ups. Take a 4 by 3 rectangle. Its perimeter is 4 and 3 and 4 and 3, which is 14. Now cut one square out of the corner.

perimeter 14 corner cut out: still 14

The new shape has fewer squares (11 instead of 12). But count the edges: still 14! Why? When you cut out the corner square, you lose 2 edges of the outside (the 2 edges that were on the outside of that corner). But you also make 2 new edges (the 2 sides of the notch). You lose 2 and gain 2, so the walk around is just as long. The ant just takes a little detour step inward instead of going round the corner.

Worked examples

Example 1. A triangle. Find the perimeter of this triangle.

4 cm 3 cm 5 cm
  1. List the sides: 3 cm, 4 cm, 5 cm.
  2. Add: 3 and 4 is 7, and 7 and 5 is 12.
  3. The perimeter is 12 cm.

Example 2. An L-shaped garden. Find the perimeter of this garden.

10 m 5 m 4 m 3 m 6 m 8 m
  1. Go round and list every side: 10 m, 5 m, 4 m, 3 m, 6 m, 8 m. That is 6 sides; an L shape always has 6.
  2. Check the lengths fit. The left side (8 m) should be the right side and the step up together: 5 and 3 is 8. ✓ The bottom (10 m) should be the top and the step across together: 6 and 4 is 10. ✓
  3. Add, making easy tens where you can: 10, then 5 and 4 and 3 is 12, so 22; then 6 and 8 is 14, so 22 and 14 is 36 m.
  4. Fun check: a plain 10 m by 8 m rectangle also has perimeter 10 and 8 and 10 and 8, which is 36 m. Cutting the corner inward did not change the perimeter, just like the grid puzzle above.

Example 3. A missing side. A rectangle has a perimeter of 18 m and a short side of 2 m. How long is the long side?

  1. Half of 18 is 9. So one short side and one long side make 9 m.
  2. 2 and what make 9? 7. The long side is 7 m.
  3. Check: 7, 2, 7, 2 makes 18. ✓

Example 4. A fence and units. A square sandbox has sides of 2 metres. Fence comes in 50 cm pieces. How many pieces?

  1. The perimeter is 2, 4, 6, 8: 8 metres.
  2. Each metre is 100 cm, which is 2 pieces of 50 cm.
  3. So 8 metres needs 8 lots of 2, which is 16 pieces.

Try it

Questions 5 and 6 use these two shapes.

shape A: staircase shape B: plus
  1. What is the perimeter of a square with sides of 5 cm?
  2. A rectangle is 7 m long and 2 m wide. What is its perimeter?
  3. A triangle has sides of 6 cm, 6 cm and 6 cm. What is its perimeter?
  4. On a grid, a rectangle is 3 squares long and 2 squares tall. How many squares are inside? What is its perimeter?
  5. Find the perimeter of shape A, the staircase, in the picture above.

  6. Find the perimeter of shape B, the plus. Is it more than, less than, or the same as the staircase?

  7. A rectangle has a perimeter of 14 cm and a long side of 5 cm. How long is its short side?

  8. Draw (or build with toothpicks) every rectangle with a perimeter of 16, using whole numbers. Which one has the most squares inside?
Answers
  1. 5, 10, 15, 20: 20 cm.
  2. 7 and 2 is 9; twice is 18 m.
  3. 6 and 6 and 6 is 18 cm.
  4. 6 squares inside (2 rows of 3). Perimeter: 3 and 2 is 5, twice is 10 units. (Not 6: that is the number of squares.)
  5. Walk around and count edges: up the left side 3, along the top 1, down 1, across 1, down 1, across 1, down the right side 1, along the bottom 3. That is 3, 1, 1, 1, 1, 1, 1, 3: 12 units. Notice that a 3 by 3 square also has perimeter 12. Every step-in on the staircase is matched by a step-out, so it walks as far as the square around it.
  6. The plus shape has 12 edges around it: each of the 4 arms has 3 outside edges, and 4 lots of 3 is 12 units. It is the same as the staircase, even though the staircase has 6 squares and the plus only 5.
  7. Half of 14 is 7. The long side and the short side make 7. 5 and what make 7? 2 cm. Check: 5, 2, 5, 2 is 14.
  8. Half of 16 is 8. Pairs of whole numbers that make 8: 7 and 1, 6 and 2, 5 and 3, 4 and 4. So the rectangles are 7 by 1, 6 by 2, 5 by 3 and 4 by 4. Squares inside: 7, 12, 15 and 16. The 4 by 4 square has the most room, just like the goat pen.

Common mistake

Counting squares instead of edges. On a grid, the perimeter is the number of little edges around the outside, not the number of squares. A corner square has 2 outside edges, and both count.

Missing a side. In a rectangle, it is easy to add only two sides (5 and 3 is 8) and forget the other two. Go all the way round, or remember that a rectangle's perimeter is twice "one long side and one short side".

Where it leads