Area and Perimeter

📏 Area and Perimeter

Part 1 of 5 — What Do They Even Mean?

Topics in This Part

🔑 Key Idea: Perimeter is the distance around the edge of a shape. Area is the amount of flat space inside it. They answer two different questions, so we measure them in two different ways.

Perimeter — The Distance Around

Imagine walking all the way around the edge of a garden and counting your steps. That total is the perimeter.

To find a perimeter, you add up the lengths of all the sides.

Example: a rectangle that is 5 cm long and 3 cm wide

A rectangle has 4 sides — two long sides and two short sides:

$5 + 3 + 5 + 3 = 16 \text{ cm}$

So the perimeter is 16 cm.

💡 Perimeter is a length, so it uses normal length units: cm, m, inches, feet — never "square" units.

Area — The Space Inside

Area measures how much flat space is covered by a shape — like how much carpet you'd need to cover a floor.

We measure area by counting unit squares: little squares that are 1 unit on each side.

Example: a rectangle on a grid

If a rectangle is 5 squares across and 3 squares down, we can count the squares row by row:

$3 \text{ rows} \times 5 \text{ squares each} = 15 \text{ squares}$

So the area is 15 square centimeters, written $15 \text{ cm}^2$.

🔑 Big difference: Perimeter counts the edge (a line). Area counts the inside (the squares). The same rectangle above has a perimeter of $16 \text{ cm}$ but an area of $15 \text{ cm}^2$.

Concept Check 🎯

Pick the Right Unit 🔽

Perimeter is a length, so it uses units like $\text{cm}$. Area is a count of squares, so it uses square units like $\text{cm}^2$.

One Shape, Two Different Numbers

Here's the idea that trips students up the most: the same rectangle has both a perimeter and an area, and they are almost always different numbers.

Think about the $5$-by-$3$ rectangle from before:

• Walk around the edge → perimeter $= 16$ (a length)
• Count the squares inside → area $= 15$ (a count of squares)

So whenever you read a problem, the very first question to ask yourself is: "Do I want the edge, or the inside?" Let's practice spotting both on the same shape.

Count on the Grid 🧮

A rectangle is drawn on grid paper. It is 6 squares across and 2 squares down.

1) How many unit squares fit inside? (Add the rows: $6 + 6 = ?$) 2) How far is it all the way around the edge? ($6 + 2 + 6 + 2 = ?$)

Wrapping Up Part 1

You now know the two big ideas:

Counting squares one by one works, but it's slow. In Part 2 we'll learn a fast formula for perimeter, and in Part 3 a fast formula for area.

📏 Area and Perimeter

Part 2 of 5 — The Perimeter of a Rectangle

🔑 The Idea: Every rectangle has two pairs of equal sides. Instead of adding four numbers, we can use a shortcut formula to find the perimeter quickly.

Two Ways to Find a Rectangle's Perimeter

A rectangle has a length ($l$) and a width ($w$). The two long sides are both $l$, and the two short sides are both $w$.

Way 1 — Add all four sides:

$P = l + w + l + w$

Way 2 — The shortcut formula:

$P = 2 \times l + 2 \times w$

Both give the same answer! The shortcut just notices there are two lengths and two widths.

Worked Example: $l = 8$ m, $w = 5$ m

$P = 2 \times 8 + 2 \times 5 = 16 + 10 = 26 \text{ m}$

✅ Check by adding all sides: $8 + 5 + 8 + 5 = 26$ m ✓ Same answer.

Worked Example: a square with side $7$ cm

A square is a special rectangle where all 4 sides are equal. So we can just add the side four times, or multiply by 4:

$P = 7 + 7 + 7 + 7 = 4 \times 7 = 28 \text{ cm}$

Worked Example: $l = 10$ ft, $w = 3$ ft

$P = 2 \times 10 + 2 \times 3 = 20 + 6 = 26 \text{ ft}$

💡 Square shortcut: For a square, $P = 4 \times \text{side}$. For any rectangle, $P = 2 \times l + 2 \times w$.

Concept Check 🎯

Build the Perimeter 🔽

A rectangle is $12$ m long and $5$ m wide. Fill in each step.

Find the Perimeter 🧮

Enter just the number (the unit is shown for you).

1) Rectangle: $l = 7$ cm, $w = 6$ cm. $\;P = ?$ cm 2) Rectangle: $l = 15$ m, $w = 10$ m. $\;P = ?$ m 3) Square: side $= 9$ ft. $\;P = ?$ ft

📏 Area and Perimeter

Part 3 of 5 — The Area of a Rectangle

🔑 The Idea: Counting squares one by one is slow. But the squares are arranged in neat rows and columns — and that's exactly what multiplication is for!

The Area Formula

Look at a rectangle that is $5$ squares across and $3$ squares down. There are $3$ rows, and each row has $5$ squares. Instead of counting all $15$, we multiply:

$A = l \times w$

That is, area = length × width.

Worked Example: $l = 5$ cm, $w = 3$ cm

$A = 5 \times 3 = 15 \text{ cm}^2$

So the rectangle covers 15 square centimeters.

🔑 Why "square" units? Each little square is $1 \text{ cm}$ on each side, so it is $1$ square centimeter ($1 \text{ cm}^2$). The area tells you how many of those squares fit inside.

Worked Example: $l = 8$ m, $w = 4$ m

$A = 8 \times 4 = 32 \text{ m}^2$

Worked Example: a square with side $6$ in

For a square, length and width are equal, so:

$A = 6 \times 6 = 36 \text{ in}^2$

⚠️ Watch out! Don't mix up the formulas:

• Perimeter uses adding: $P = 2\times l + 2\times w$
• Area uses multiplying: $A = l \times w$

For an $8 \times 4$ rectangle, $P = 24$ but $A = 32$ — very different numbers!

Concept Check 🎯

Area or Perimeter? 🔽

For each rectangle, pick the correct value. Remember: area multiplies, perimeter adds.

Find the Area 🧮

Enter just the number.

1) Rectangle: $l = 9$ cm, $w = 6$ cm. $\;A = ?$ 2) Rectangle: $l = 12$ m, $w = 5$ m. $\;A = ?$ 3) Square: side $= 8$ in. $\;A = ?$

📏 Area and Perimeter

Part 4 of 5 — Missing Sides & Real-World Problems

🔑 Big Skill: Sometimes you know the area or perimeter but are missing a side. You can work backwards using division. And often the math is hidden inside a story — a garden, a room, a poster.

Working Backwards to Find a Missing Side

Since area = length × width, you can find a missing side by dividing.

Example: A rectangle has area $24 \text{ cm}^2$ and length $6$ cm. What is the width?

$\text{width} = \text{area} \div \text{length} = 24 \div 6 = 4 \text{ cm}$

✅ Check: $6 \times 4 = 24 \text{ cm}^2$ ✓

Example: A rectangle has perimeter $20$ m and length $7$ m. What is the width?

First, two lengths use up $2 \times 7 = 14$ m. That leaves $20 - 14 = 6$ m for the two widths, so each width is $6 \div 2 = 3$ m.

$\text{width} = (20 - 2\times 7) \div 2 = (20 - 14) \div 2 = 3 \text{ m}$

✅ Check: $7 + 3 + 7 + 3 = 20$ m ✓

Find the Missing Side 🧮

Enter just the number.

1) Area $= 30 \text{ cm}^2$, length $= 5$ cm. Width $= ?$ cm 2) Area $= 56 \text{ m}^2$, width $= 7$ m. Length $= ?$ m

Reading a Word Problem

Real problems hide the numbers in a story. The trick is to ask: am I covering the inside (area) or going around the edge (perimeter)?

Example: Maria's Garden

Maria is planting a rectangular garden that is $10$ ft long and $4$ ft wide. She wants to put a small fence around it AND cover the ground with mulch. How much fence and how much mulch does she need?

• Fence goes around the edge → perimeter: $P = 2\times 10 + 2\times 4 = 20 + 8 = 28 \text{ ft of fence}$
• Mulch covers the inside → area: $A = 10 \times 4 = 40 \text{ ft}^2 \text{ of mulch}$

💡 Strategy: Underline the action. "Around / border / fence / frame" → perimeter. "Cover / fill / paint / tile / carpet" → area.

Word-Problem Check 🎯

Spot the Clue Word 🔽

For each job, decide whether you need area or perimeter.

📏 Area and Perimeter

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) tell perimeter from area, (2) find a rectangle's perimeter, (3) find its area, and (4) solve word problems and missing-side puzzles. Let's put it all together.

Quick Reference

⚠️ The #1 mistake is swapping the two: perimeter ADDS (and uses plain units), area MULTIPLIES (and uses square units). When in doubt, ask: around the edge, or inside?

Mixed Practice 🧮

A rectangle is $11$ cm long and $4$ cm wide.

1) Find its perimeter (in cm). 2) Find its area (the number of $\text{cm}^2$). 3) A square has a perimeter of $20$ cm. What is the length of one side (in cm)?

Mixed Practice 🎯

Exit Quiz ✅

Answer all three to finish the lesson.