Lesson 4 ~25 min Unit 3 · Measurement & Geometry +85 XP

Perimeter of Polygons

Add all outer sides — and find missing lengths in composite shapes first.

Today's hook: A council builds a walking track around an L-shaped park. How do they calculate the total fencing when some side lengths aren't given directly?

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Think First

warm-up

A square has a perimeter of 36 cm. What is its side length? Now imagine the same perimeter but shaped as a rectangle that is twice as long as it is wide. What are its dimensions?

Record your answer in your workbook.

The Big Idea

+5 XP

Perimeter is the total distance around the outside of a shape — add every outer side length. For composite shapes, use opposite-side relationships to find any missing lengths before adding.

Trace the boundary of the shape with your finger. Every side you cross gets added. For L-shapes and step shapes, some sides are not given — find them first using opposite sides equal the total rule before adding all sides together.

$P = \text{sum of ALL outer sides}$

Trace the boundary

Physically trace the outline — count every side, even if it's not labelled.

Opposite sides rule

In composite shapes: missing horizontal = total width − partial width.

Count the sides

Write out every side length before adding — avoids missed sides.

What You'll Master

objectives

Know

Perimeter = sum of all outer side lengths
Regular polygon formula: $P = n \times s$
Rectangle formula: $P = 2l + 2w$

Understand

How opposite-side relationships reveal missing lengths in L-shapes
How to work backwards from perimeter to find an unknown side

Can Do

Find the perimeter of regular polygons, rectangles and composites
Calculate missing side lengths in L-shaped figures
Find an unknown side given the perimeter and other sides

Words You Need

vocabulary

PerimeterThe total length of the boundary (outer edge) of a 2D shape.

PolygonA closed 2D shape made entirely of straight sides (triangle, square, hexagon, etc.).

Regular polygonA polygon where all sides are equal length and all angles are equal.

Composite figureA shape made by combining two or more simpler shapes (e.g. L-shape = two rectangles).

Missing sideA side whose length is not given; found using opposite-side relationships.

Opposite-sides ruleIn a composite shape with right angles: opposite parallel sides sum to the same total length.

Spot the Trap

heads-up

Wrong: For an L-shape, adding only the 4 labelled sides and ignoring the 2 missing sides.

Right: Find missing sides first using the opposite-sides rule, then add all 6 sides.

Wrong: Rectangle perimeter $= l + w = 15 + 8 = 23$ m.

Right: $P = 2l + 2w = 2(15) + 2(8) = 46$ m — there are two of each side!

Regular Polygons

+5 XP

In a regular polygon, all sides are equal. Perimeter $= n \times s$, where $n$ is the number of sides and $s$ is the side length.

Examples:

Equilateral triangle ($n=3$, $s=7$): $P = 3 \times 7 = 21$ cm
Square ($n=4$, $s=9$): $P = 4 \times 9 = 36$ cm
Regular hexagon ($n=6$, $s=8$): $P = 6 \times 8 = 48$ cm
Regular octagon ($n=8$, $s=5$): $P = 8 \times 5 = 40$ cm

Reverse: if $P$ and $n$ are known, $s = P \div n$.

$$P = n \times s$$

L-Shaped Composite Figures

+5 XP

Trace the boundary of an L-shape — you will find 6 sides. Two are usually unlabelled; use the opposite-sides rule to find them before adding.

Opposite-sides rule:

Missing horizontal $=$ total width $-$ partial width
Missing vertical $=$ total height $-$ partial height

Then: $P = $ sum of all 6 sides.

Find missing sides first, then add all 6.

Finding Missing Sides in Composite Shapes

+5 XP

Apply the opposite-sides rule: sides facing each other (and parallel) must account for the full length of the shape in that direction.

For an L-shape with total width 15 cm and a notch 9 cm wide, and total height 12 cm with a notch 7 cm tall:

Missing horizontal $= 15 - 9 = 6$ cm
Missing vertical $= 12 - 7 = 5$ cm

$P = 12 + 15 + 5 + 9 + 7 + 6 = 54$ cm

Missing $h = $ total $-$ partial; Missing $v = $ total $-$ partial

Working Backwards from Perimeter

+5 XP

If you know the perimeter and some sides, find an unknown side by subtracting the known sides from the perimeter.

Rectangle example: $P = 46$ m, $l = 15$ m. Find $w$.

$P = 2l + 2w$
$46 = 2(15) + 2w = 30 + 2w$
$2w = 46 - 30 = 16$
$w = 8$ m

Or: half-perimeter method: $\frac{P}{2} = l + w$, so $w = \frac{P}{2} - l = 23 - 15 = 8$ m.

Unknown side $= \frac{P}{2} - $ known side (rectangles)

Common Pitfalls

heads-up

Missing sides in composite shapes

Adding only the 4 labelled sides of an L-shape and omitting the 2 unlabelled sides.

Fix: Always count sides by tracing the boundary. An L-shape has 6 sides — find the missing 2 before adding.

Forgetting to double in rectangle formula

Writing $P = l + w$ instead of $P = 2l + 2w$ for a rectangle.

Fix: A rectangle has 4 sides — two lengths and two widths. Always multiply by 2.

Adding missing sides instead of subtracting

Finding missing horizontal as $15 + 9 = 24$ instead of $15 - 9 = 6$.

Fix: The missing side plus the given partial side must equal the total. Subtract, not add.

Copy Into Your Books

Perimeter Formulas

Any polygon: $P = $ sum of sides
Regular: $P = n \times s$
Rectangle: $P = 2l + 2w$
Square: $P = 4s$

Composite Shapes

Trace the boundary
Count all sides (L-shape = 6)
Missing $h =$ total $-$ partial
Missing $v =$ total $-$ partial

Working Backwards

$P = 2l + 2w$
$l + w = P \div 2$
Unknown $= (P \div 2) - $ known

Check

Did you count every side?
Did you double for rectangles?
Did you find missing sides first?

How are you completing this lesson?

Watch Me Solve It · 3 examples

Watch Me Solve It · Regular Hexagon

+15 XP per step

PROBLEM

A regular hexagon has a side length of 8 cm. Find its perimeter.

1
Identify $n$ and $s$
$n = 6$ (hexagon), $s = 8$ cm
A regular hexagon has 6 equal sides.
2
Apply the formula
$P = n \times s = 6 \times 8$
3
Calculate
$P = 48$ cm
Six equal sides of 8 cm each — simple multiplication.

Answer$P = 48$ cm

Watch Me Solve It · L-Shaped Figure

+15 XP per step

PROBLEM

An L-shape has total width 15 cm, total height 12 cm. A rectangular notch 9 cm wide and 7 cm tall is cut from the bottom-right. Find the perimeter.

1
Find missing sides
Missing horizontal $= 15 - 9 = 6$ cm; Missing vertical $= 12 - 7 = 5$ cm
The two missing sides complete the L-shape boundary.
2
List all 6 sides
12, 15, 5, 9, 7, 6 (all in cm)
3
Add all sides
$P = 12 + 15 + 5 + 9 + 7 + 6 = 54$ cm
All 6 outer sides traced and summed.

Answer$P = 54$ cm

Watch Me Solve It · Working Backwards

+15 XP per step

PROBLEM

A rectangle has perimeter 46 m and length 15 m. Find the width.

1
Write the formula
$P = 2l + 2w$
A rectangle has two lengths and two widths.
2
Substitute and simplify
$46 = 2(15) + 2w = 30 + 2w$, so $2w = 16$
3
Solve for $w$
$w = 8$ m
Check: $2(15) + 2(8) = 30 + 16 = 46$ m ✓

AnswerWidth $= 8$ m

Brain Trainer · 4 problems

Brain Trainer · Perimeter Drills

4 problems

Find the perimeter or missing side for each problem. Work it, then reveal the answer.

1 Equilateral triangle, $s = 7$ cm. Perimeter?
$P = 3 \times 7 =$ 21 cm
2 Rectangle $P = 56$ cm, $l = 18$ cm. Find $w$.
$w = (56 \div 2) - 18 = 28 - 18 =$ 10 cm
3 Regular hexagon $P = 60$ cm. Find the side length.
$s = 60 \div 6 =$ 10 cm
4 $P = 44$ m, $l = 14$ m. Find $w$ for a rectangle.
$w = (44 \div 2) - 14 = 22 - 14 =$ 8 m

Complete in your workbook.

Quick Check · 5 questions

Equilateral triangle with side 7 cm. Perimeter?

+10 XP

Rectangle $P = 56$ cm, $l = 18$ cm. Find $w$.

+10 XP

Regular hexagon $P = 60$ cm. Find the side length.

+10 XP

L-shape: total 10 cm × 8 cm with a 4 cm × 3 cm notch cut from one corner. Perimeter?

+10 XP

$P = 44$ m, $l = 14$ m. Find $w$ for a rectangle.

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

ApplyMedium3 MARKS

Q6. A pentagon has sides 8 cm, 7 cm, 6 cm, 9 cm and 5 cm. Find the perimeter and state whether this is a regular pentagon.

Answer in your workbook.

UnderstandEasy2 MARKS

Q7. A square has a perimeter of 28 m. Find its side length and then calculate its area.

Answer in your workbook.

ReasonHard4 MARKS

Q8. A square and a rectangle have the same perimeter. The square has side 20 m. The rectangle has length 25 m. (a) Find the perimeter. (b) Find the rectangle's width. (c) Compare the two areas — which is larger?

Answer in your workbook.

Comprehensive Answers

Quick Check

1. C — $P = 3 \times 7 = 21$ cm.

2. B — $w = 28 - 18 = 10$ cm.

3. A — $s = 60 \div 6 = 10$ cm.

4. C — Missing sides: $10-4=6$, $8-3=5$. $P = 8+10+5+4+3+6 = 36$ cm.

5. B — $w = 22 - 14 = 8$ m.

Model Answers

Q6 (3 marks): $P = 8+7+6+9+5 = 35$ cm. Not regular — the sides are not all equal.

Q7 (2 marks): Side $= 28 \div 4 = 7$ m. Area $= 7^2 = 49$ m².

Q8 (4 marks): (a) $P = 4 \times 20 = 80$ m. (b) $w = 80/2 - 25 = 40 - 25 = 15$ m. (c) Square area $= 20^2 = 400$ m². Rectangle area $= 25 \times 15 = 375$ m². The square has the larger area.

Stretch Challenge · +25 XP, +10 coins

Regular Octagon Deep Dive

A regular octagon has a perimeter of 64 cm. (a) Find the side length. (b) The apothem (distance from centre to the middle of a side) of a regular octagon is approximately $1.207 \times s$. Find the apothem to 2 decimal places. (c) If you were to fence this octagonal garden, how many full metres of fencing do you need to buy?

Reveal solution

(a) $s = 64 \div 8 = 8$ cm. (b) Apothem $= 1.207 \times 8 = 9.656 \approx 9.66$ cm. (c) Perimeter $= 64$ cm $= 0.64$ m, so 1 full metre of fencing is needed.

Quick Review

$P = $ sum of sides

Add every outer side — don't skip any.

Regular: $P = n \times s$

All sides equal — multiply the side length by the number of sides.

Rectangle: $P = 2l + 2w$

Two lengths plus two widths — always double both dimensions.

Composite shapes

Trace the boundary, count 6 sides for L-shapes, find missing sides before adding.

Missing sides

Missing $=$ total $-$ partial, using the opposite-sides rule.

Working backwards

$w = \frac{P}{2} - l$ for rectangles. Subtract known sides from half-perimeter.

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