Grade by Grade
E. Spatial Sense · 3D Objects, Directions and Measurement

Perimeter: The Distance Around

E2.1 A1About 35 minutes
Learning goals
  • I can explain what perimeter means.
  • I can estimate and measure the perimeter of polygons and curved shapes.
  • I can find perimeter by adding side lengths or counting units on a grid.
  • I can make different shapes that have the same perimeter.

Words to know

perimeter
The total distance around the outside of a shape.
side length
How long one side of a polygon is.
estimate
A thoughtful guess about a measurement, made before measuring.
unit
The amount we measure with, like 1 centimetre (cm) or 1 metre (m).
polygon
A closed flat shape with straight sides, like a triangle, rectangle or hexagon.

Let's start with a fence

The Patels want to put a little fence around their vegetable garden so the rabbits stay out. How much fence do they need?

They need to know the distance all the way around the garden. That distance is called the perimeter.

A rectangular vegetable garden with a low wooden picket fence around it and a rabbit sitting on the grass outside.
The fence goes all the way around the garden to keep the rabbit out. Its length is the garden's perimeter. AI-generated image

What is perimeter?

The perimeter is the total length around the outside of a shape. Imagine an ant walking along the edge of a shape, starting at one corner and walking all the way around until it is back where it started. The distance the ant walks is the perimeter.

Perimeter is a length, so we measure it in units of length like centimetres (cm) or metres (m).

Adding the sides

For a polygon, measure each side, then add all the side lengths.

5 cm5 cm3 cm3 cm5 + 3 + 5 + 3= 16 cm
A rectangle 5 cm long and 3 cm wide (drawing not actual size). Start at the dot and go all the way around: 5 + 3 + 5 + 3 = 16 cm.
Worked example

Example 1: A rectangle

Find the perimeter of a rectangle that is 5 cm long and 3 cm wide.

  1. A rectangle has 4 sides. The opposite sides are the same length: two sides are 5 cm and two sides are 3 cm.
  2. Add all four sides: 5 + 3 + 5 + 3.
  3. 5 + 5 = 10 and 3 + 3 = 6. Then 10 + 6 = 16.

The perimeter is 16 cm.

4 cm6 cm7 cm4 + 6 + 7= 17 cm
A triangle with sides 4 cm, 6 cm and 7 cm (drawing not actual size). Its perimeter is 4 + 6 + 7 = 17 cm.
Worked example

Example 2: A triangle

A triangle has sides 4 cm, 6 cm and 7 cm. What is its perimeter?

  1. Add the three sides: 4 + 6 + 7.
  2. 4 + 6 = 10. Then 10 + 7 = 17.

The perimeter is 17 cm.

Don't miss a side

Put your finger on a starting corner. Go around the shape and touch each side as you add it. Stop when you are back at the start. This way you don't skip a side or count one twice.

Perimeter on a grid

On grid paper, each little line segment along the outside is 1 unit long. Count the units all the way around the shape. If the grid squares are 1 cm across, each unit is 1 cm.

perimeter:14 units
An L-shape on a grid. Counting every unit along the outside gives a perimeter of 14 units.
Worked example

Example 3: Count around an L-shape

Find the perimeter of the L-shape on the grid.

  1. Start at the top-left corner and go clockwise.
  2. Top: 2. Right side going down: 2. Step to the right along the top of the bottom row: 2. Down: 1.
  3. Bottom: 4. Left side going up: 3.
  4. Add: 2 + 2 + 2 + 1 + 4 + 3 = 14.

The perimeter is 14 units.

Perimeter of curved shapes

A ruler is straight, but the edge of a plate or a leaf is curved. Here's a trick:

  1. Wrap a string carefully around the edge.
  2. Pinch the string where it meets the start.
  3. Straighten the string and measure it with a ruler or measuring tape.
jar lidwrap05101520string: 22 cm
Wrap a string around a jar lid, straighten it, and measure it on a ruler. This lid's perimeter is about 22 cm.
Worked example

Example 4: Estimate, then measure

Ravi wants the perimeter of a round dinner plate. He estimates it is about 80 cm. With a string, he measures 77 cm. Was his estimate reasonable?

  1. An estimate doesn't have to be exact. It should be close.
  2. 80 cm and 77 cm are only 3 cm apart.

Yes. 80 cm is close to 77 cm, so it was a good estimate.

Making shapes with a given perimeter

Different shapes can have the same perimeter. Here are three rectangles drawn on 1-unit grid squares. Each one has a perimeter of 12 units.

3 by 34 by 25 by 1Each shape has a perimeter of 12 units.
A 3 by 3 square, a 4 by 2 rectangle and a 5 by 1 rectangle. Each has a perimeter of 12 units.
Worked example

Example 5: Build a shape

Make a triangle with a perimeter of 12 cm.

  1. We need three side lengths that add to 12.
  2. Try 4 + 4 + 4 = 12. All sides 4 cm.
  3. Another try: 3 + 4 + 5 = 12.

A triangle with sides 4 cm, 4 cm and 4 cm (or 3 cm, 4 cm and 5 cm) has a perimeter of 12 cm.

Thinking like a mathematician

When you find one answer, ask: "Is there another way?" Many perimeter problems have more than one correct answer.

Just for fun

Why was the fence so good at math?

It always went all the way around the problem!

Hands-on activity

Perimeter hunt

You need: a ruler or measuring tape, a piece of string about 1 metre long, and paper.

  1. Estimate the perimeter of a book. Write your estimate.
  2. Measure each side with a ruler and add. How close was your estimate?
  3. Find something round (a plate, a mug, a lid). Estimate, then measure its perimeter with the string.
  4. Use 12 toothpicks or 12 pieces of uncooked spaghetti, all the same length. Make as many different shapes as you can with a perimeter of 12 pieces. Draw each one.

Grown-up note: toothpicks are sharp, and young children should be supervised.

Math talk: "Which shape used the most space inside, even though all of them had the same perimeter?"

Your turn

1

Find the perimeter of a rectangle 6 cm long and 2 cm wide.

2

A square has sides of 4 cm. What is its perimeter?

3

A triangle has sides 3 cm, 4 cm and 5 cm. What is its perimeter?

4

A garden is a rectangle 5 m long and 3 m wide. How much fence goes around it?

5

What tools could you use to measure the perimeter of a round plate?

6

On grid paper, draw a rectangle with a perimeter of 10 units. What size is it?

7

Which has the bigger perimeter: a square with 5 cm sides, or a rectangle 6 cm by 3 cm?

More practice: warm-up

1

What does perimeter mean?

2

A square has sides of 2 cm. Find its perimeter.

3

Find the perimeter of a rectangle 4 cm by 1 cm.

4

A triangle has three sides of 5 cm. Find its perimeter.

5

Would you measure the perimeter of a soccer field in cm or m?

More practice: on your own

1

A pentagon has 5 sides, each 3 cm long. What is its perimeter?

2

A picture is 30 cm wide and 20 cm tall. How much ribbon goes all the way around it?

3

On grid paper, a rectangle is 3 squares long and 2 squares wide. What is its perimeter?

4

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

5

Lee estimates that the perimeter of his reading book is 3 cm. Is that reasonable? Explain.

Challenge

1

Draw or describe all the rectangles with whole-number sides that have a perimeter of 16 cm.

2

Mr. Leblanc has 20 m of fence for a rectangular dog pen. One side will be 6 m. How long is the other side?

3

A square and a triangle with three equal sides each have a perimeter of 12 cm. How long is one side of each?

Check your understanding

Quiz

1.

What is perimeter?

2.

What is the perimeter of a rectangle 7 cm by 2 cm?

3.

A square has sides of 6 m. What is its perimeter?

4.

How can you measure the perimeter of a round lid?

5.

Which rectangle has a perimeter of 12 units?

6.

A triangle has sides 5 cm, 5 cm and 4 cm. What is its perimeter?

For parents and teachers

Big idea: Perimeter is the distance around a shape. Children estimate, measure and compare perimeters of polygons and curved shapes using appropriate units, and build polygons with a given perimeter (Ontario E2.1). Looking for more than one answer builds reasoning (strand A1).

Common difficulties:

  • Adding only two sides of a rectangle (5 + 3 = 8 instead of 16). Have your child trace around the shape with a finger as they add.
  • Counting squares instead of edges on a grid. Perimeter counts the unit lengths along the outside, not the squares inside (that's area, coming later in this unit).
  • Curved shapes. String works well, but it must follow the edge snugly without stretching.

Safety: supervise toothpicks with young children.

How to help: Measure real perimeters together: a placemat, a picture frame, a rug. Ask your child to estimate first.