What Is Perimeter?

The perimeter of a two-dimensional shape is the total distance around the outside of the shape.

Imagine walking all the way around the edge of a playground. The total distance you walk is the perimeter of the playground.

To find perimeter:

Add the lengths of all the outside sides.

For any polygon:

Perimeter = sum of all side lengths

For example, if a triangle has side lengths:

4 cm, 6 cm, and 7 cm

then:

P = 4 + 6 + 7

P = 17 cm

Perimeter is a length, so it is measured in units such as:

• mm
• cm
• m
• km

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Perimeter of a Rectangle

A rectangle has:

• Two equal lengths.
• Two equal widths.

Suppose a rectangle has:

length = 8 cm

width = 5 cm

We could add all four sides:

P = 8 + 5 + 8 + 5

P = 26 cm

Because opposite sides are equal, we can use:

P = 2l + 2w

or:

P = 2(l + w)

where:

P = perimeter

l = length

w = width

Worked Example

A rectangular garden is:

12 m long

and:

7 m wide

Find its perimeter.

P = 2(l + w)

P = 2(12 + 7)

P = 2(19)

P = 38 m

The garden has a perimeter of 38 m.

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Perimeter of a Square

A square has four equal sides.

If each side has length s:

P = 4s

Worked Example

A square has a side length of:

6 cm

Therefore:

P = 4 × 6

P = 24 cm

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Perimeter of a Triangle

A triangle has three sides.

To find its perimeter:

P = a + b + c

Worked Example

A triangle has side lengths:

5 cm, 8 cm, and 10 cm

Therefore:

P = 5 + 8 + 10

P = 23 cm

Notice that the sides do not need to be equal.

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Perimeter of Other Polygons

The same basic rule works for any polygon:

Add every outside side.

Suppose a pentagon has side lengths:

3 cm, 5 cm, 4 cm, 6 cm, and 7 cm

Then:

P = 3 + 5 + 4 + 6 + 7

P = 25 cm

This method works for:

• Triangles.
• Quadrilaterals.
• Pentagons.
• Hexagons.
• Regular polygons.
• Irregular polygons.

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Regular Polygons

A regular polygon has sides of equal length.

If a regular polygon has:

n sides

and each side has length:

s

then:

P = ns

Example: Regular Pentagon

A regular pentagon has five sides.

Each side is:

7 cm

Therefore:

P = 5 × 7

P = 35 cm

Example: Regular Octagon

A regular octagon has eight equal sides.

If each side is:

4.5 m

then:

P = 8 × 4.5

P = 36 m

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Irregular Polygons

An irregular polygon does not have all sides equal.

There may not be a shortcut formula.

Instead:

Add all the outside side lengths.

Example

An irregular shape has side lengths:

4 m, 7 m, 3 m, 5 m, 6 m, and 8 m

Therefore:

P = 4 + 7 + 3 + 5 + 6 + 8

P = 33 m

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Finding Missing Side Lengths

Sometimes a diagram does not give every side length directly.

You may need to determine a missing side before calculating the perimeter.

Example

A rectangle has:

length = 11 cm

width = 4 cm

Only three sides are labelled:

11 cm, 4 cm, 11 cm

What is the missing side?

Opposite sides of a rectangle are equal.

Therefore:

missing side = 4 cm

Now:

P = 11 + 4 + 11 + 4

P = 30 cm

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Finding a Missing Side from the Perimeter

Sometimes the perimeter is known, but one side is missing.

Worked Example

A triangle has a perimeter of:

30 cm

Two sides measure:

8 cm and 12 cm

Find the third side.

Let the missing side be x.

8 + 12 + x = 30

20 + x = 30

x = 10 cm

The missing side is 10 cm.

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Another Missing-Side Example

A rectangle has a perimeter of:

42 m

Its length is:

13 m

Find its width.

Use:

P = 2l + 2w

Substitute the known values:

42 = 2(13) + 2w

42 = 26 + 2w

Subtract 26:

16 = 2w

Divide by 2:

w = 8 m

The rectangle is 8 m wide.

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Compound Shapes

A compound shape is made by combining simpler shapes.

When calculating its perimeter, include only the sides around the outside boundary.

Do not include internal lines.

A useful strategy is:

1. Start at one corner.
2. Move around the outside of the shape.
3. Write down each side length.
4. Determine any missing lengths.
5. Add the outside lengths.
6. Check that you have returned to your starting point.

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Finding Missing Lengths in Compound Shapes

Consider an L-shaped figure.

Suppose its total height is:

10 cm

One vertical section is:

6 cm

The missing vertical section must be:

10 − 6 = 4 cm

This missing length can then be included when calculating the perimeter.

A similar method works horizontally.

If the total width is:

15 cm

and one horizontal section is:

9 cm

then the missing section is:

15 − 9 = 6 cm

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Perimeter in Real Life

Perimeter is useful whenever we need to know the distance around something.

Examples include:

• Fencing a garden.
• Building a fence around a property.
• Installing a border around a garden bed.
• Putting trim around a window.
• Framing a picture.
• Marking the boundary of a sports field.
• Installing baseboards around a room.
• Decorating the edge of a poster.

The important question is:

Do I need to know the distance around the object?

If yes, perimeter is probably involved.

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Worked Example: Fencing a Garden

A rectangular garden is:

15 m long

and:

9 m wide

How much fencing is needed to completely surround it?

P = 2(l + w)

P = 2(15 + 9)

P = 48 m

Therefore, 48 m of fencing is required.

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Worked Example: Cost of Fencing

Suppose fencing costs:

$12 per metre

The garden has a perimeter of:

48 m

Cost:

48 × $12 = $576

Therefore, the fencing costs:

$576

This is a common real-world use of perimeter.

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Worked Example: Picture Frame

A photograph is:

30 cm by 20 cm

How much framing material is needed?

P = 2(30 + 20)

P = 2(50)

P = 100 cm

Therefore, 100 cm of framing material is required.

This could also be written as:

1 m

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Perimeter and Units

Because perimeter measures distance, we use linear units.

Correct examples:

P = 25 cm

P = 18 m

P = 3.4 km

We do not use square units for perimeter.

Incorrect:

P = 25 cm²

Correct:

P = 25 cm

Square units such as cm² and m² are used for area.

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Converting Units

Sometimes side lengths are given in different units.

Convert them to the same unit before adding.

Example

A rectangle has:

length = 2 m

width = 50 cm

We should not calculate:

2 + 50

because the units are different.

Convert:

2 m = 200 cm

Then:

P = 2(200 + 50)

P = 500 cm

or:

P = 5 m

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Perimeter vs Area

Perimeter and area describe different properties of a shape.

Perimeter

Measures the distance around a shape.

Area

Measures the amount of surface inside a shape.

Think of a garden.

Fence around the garden → perimeter

Grass covering the garden → area

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Comparing Perimeter and Area

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Worked Example: Perimeter and Area

A rectangle measures:

8 m × 5 m

Perimeter

P = 2(8 + 5)

P = 26 m

Area

A = 8 × 5

A = 40 m²

Notice the different units:

Perimeter = 26 m

Area = 40 m²

They represent completely different measurements.

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Shapes Can Have the Same Perimeter but Different Areas

Consider two rectangles.

Rectangle A

8 cm × 2 cm

Perimeter:

P = 2(8 + 2) = 20 cm

Area:

A = 8 × 2 = 16 cm²

Rectangle B

5 cm × 5 cm

Perimeter:

P = 4 × 5 = 20 cm

Area:

A = 5 × 5 = 25 cm²

Both shapes have the same perimeter:

20 cm

but their areas are different.

This demonstrates that knowing the perimeter does not necessarily tell you the area.

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Shapes Can Have the Same Area but Different Perimeters

Consider:

Rectangle A

12 cm × 2 cm

Area:

24 cm²

Perimeter:

28 cm

Rectangle B

6 cm × 4 cm

Area:

24 cm²

Perimeter:

20 cm

Both have the same area but different perimeters.

Therefore:

same area ≠ same perimeter

and:

same perimeter ≠ same area

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Finding a Missing Dimension from Perimeter

Perimeter can be used to work backward.

Example

A rectangular field has a perimeter of:

100 m

Its length is:

30 m

Find its width.

P = 2l + 2w

100 = 2(30) + 2w

100 = 60 + 2w

40 = 2w

w = 20 m

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Real-World Example: Running Track

A student runs around the outside of a rectangular field measuring:

80 m × 50 m

One lap is:

P = 2(80 + 50)

P = 260 m

If the student completes four laps:

distance = 4 × 260

distance = 1040 m

Therefore, the student runs:

1.04 km

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Real-World Example: Garden Border

A square garden has sides of:

6.5 m

The owner wants to install a decorative border around the entire garden.

P = 4 × 6.5

P = 26 m

If the border material comes in 2 m sections:

26 ÷ 2 = 13

Therefore, 13 sections are required.

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Real-World Example: A Gate in a Fence

Sometimes part of the perimeter does not require the same material.

A rectangular yard measures:

20 m × 12 m

Total perimeter:

P = 2(20 + 12)

P = 64 m

A gate takes up:

3 m

Therefore, fencing required:

64 − 3 = 61 m

This shows why understanding the situation is important instead of automatically using a formula and stopping.

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A Reliable Problem-Solving Strategy

When solving a perimeter problem:

Step 1: Identify the shape.

Step 2: Identify all known side lengths.

Step 3: Determine any missing lengths.

Step 4: Convert units if necessary.

Step 5: Add all outside sides or use an appropriate formula.

Step 6: Include the correct linear unit.

Step 7: Ask whether the answer is reasonable.

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Estimating Perimeter

Estimation can help check your calculations.

Suppose a rectangle measures:

19.8 m × 10.2 m

Estimate:

20 m × 10 m

Estimated perimeter:

2(20 + 10) = 60 m

Exact calculation:

P = 2(19.8 + 10.2)

P = 2(30)

P = 60 m

The estimate confirms that the answer is reasonable.

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Common Mistakes

Confusing Perimeter and Area

Remember:

Perimeter = around

Area = inside

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Using Square Units

Incorrect:

P = 32 cm²

Correct:

P = 32 cm

Perimeter is a length.

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Forgetting a Side

For an irregular or compound shape, trace around the entire outside boundary.

Make sure every outside side is included.

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Counting Internal Lines

Only the outside boundary contributes to perimeter.

Lines inside a compound shape should not be included.

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Assuming All Sides Are Equal

A square has four equal sides.

A rectangle has equal opposite sides.

An irregular polygon may have no equal sides.

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Adding Different Units

Do not calculate:

2 m + 40 cm = 42

Convert to the same unit first.

For example:

2 m = 200 cm

so:

200 cm + 40 cm = 240 cm

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Multiplying Instead of Adding

For a rectangle:

8 × 5 = 40

calculates the area, not the perimeter.

Perimeter:

2(8 + 5) = 26

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Forgetting to Determine Missing Sides

Some compound-shape problems intentionally leave side lengths unlabelled.

Use the geometry of the shape to determine them before calculating the perimeter.

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Check Your Understanding

1. Define perimeter.

2. What units are used for perimeter?

3. Find the perimeter of a square with side length 9 cm.

4. Find the perimeter of a rectangle measuring 12 cm by 7 cm.

5. Find the perimeter of a triangle with sides 6 cm, 8 cm, and 11 cm.

6. A regular pentagon has sides of 5 cm. Find its perimeter.

7. A regular hexagon has sides of 7.5 cm. Find its perimeter.

8. An irregular polygon has side lengths 3 m, 7 m, 4 m, 8 m, and 6 m. Find its perimeter.

9. A triangle has a perimeter of 40 cm. Two sides measure 12 cm and 15 cm. Find the third side.

10. A rectangle has a perimeter of 50 cm and a length of 15 cm. Find its width.

11. A square has a perimeter of 36 m. Find its side length.

12. A rectangular garden measures 18 m by 11 m. How much fencing is required?

13. The fencing in Question 12 costs $8 per metre. Find the total cost.

14. A picture measures 40 cm by 25 cm. How much framing material is required?

15. Explain the difference between perimeter and area.

16. What units would you use for the area of a classroom? What units would you use for its perimeter?

17. A rectangle measures 10 cm × 4 cm. Calculate both its perimeter and area.

18. A rectangular field measures 100 m × 60 m. A runner completes three laps around its edge. How far does the runner travel?

19. A rectangular yard has a perimeter of 70 m and a width of 12 m. Find its length.

20. A rectangle measures 2 m by 75 cm. Find its perimeter in metres.

21. Explain why 35 cm² cannot be a correct unit for perimeter.

22. Two rectangles have the same perimeter. Must they have the same area? Explain using an example.

23. Two rectangles have the same area. Must they have the same perimeter? Explain.

24. A square garden has a perimeter of 48 m. Find its area.

25. A rectangular garden measures 14 m × 8 m. A 2 m-wide gate is installed. How many metres of fencing are required for the rest of the boundary?

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Key Terms

• Perimeter – total distance around a two-dimensional shape.
• Polygon – closed two-dimensional shape made from straight sides.
• Regular polygon – polygon with equal side lengths and equal angles.
• Irregular polygon – polygon whose sides and/or angles are not all equal.
• Rectangle – quadrilateral with four right angles and equal opposite sides.
• Square – quadrilateral with four equal sides and four right angles.
• Compound shape – shape made from two or more simpler shapes.
• Side length – distance from one vertex of a polygon to the next.
• Area – amount of two-dimensional space inside a shape.
• Linear unit – unit used to measure length, such as cm or m.
• Square unit – unit used to measure area, such as cm² or m².

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Key Takeaways

• Perimeter is the total distance around a two-dimensional shape.
• To find the perimeter of any polygon, add all the outside side lengths.
• Rectangle: P = 2(l + w)
• Square: P = 4s
• Regular polygon: P = number of sides × side length
• Missing side lengths may need to be determined before calculating perimeter.
• Opposite sides of a rectangle are equal.
• Compound shapes require careful tracing of the entire outside boundary.
• Internal lines are not included in perimeter.
• All measurements must use compatible units before they are added.
• Perimeter is measured in linear units such as cm, m, and km.
• Area is measured in square units such as cm² and m².
• Perimeter measures the distance around a shape.
• Area measures the space inside a shape.
• Shapes can have the same perimeter but different areas.
• Shapes can have the same area but different perimeters.
• Perimeter has many practical applications involving fencing, borders, frames, trim, and travel around boundaries.

A simple way to remember the difference is:

Perimeter = around.

Area = inside.