Perimeter and Area

Perimeter and Area: What's the Difference? 📐

Imagine you have a rectangular garden. Two very different questions could come up:

• How much fence do I need to go all the way around it? → that's perimeter.
• How much grass do I need to cover the ground inside? → that's area.

These are the two big ideas in this lesson:

The key thing to remember: perimeter is the outline, area is the inside. Mixing them up is the most common mistake, so keep that picture in your head!

Perimeter: The Distance Around 🚶

Perimeter is the total distance around the outside of a shape. To find it, you simply add up all the side lengths.

Rectangle — a rectangle has two long sides (the length, $l$) and two short sides (the width, $w$). So:

$P = 2l + 2w$

which is the same as

$P = 2(l + w)$

Example: A rectangle is $l = 8$ cm long and $w = 3$ cm wide.

• $P = 2(8 + 3) = 2(11) = 22$ cm ✓

Square — a square has $4$ equal sides, each of length $s$:

$P = 4s$

Example: A square has sides of $5$ in.

• $P = 4 \times 5 = 20$ in ✓

Because perimeter is a length, we measure it in linear units like cm, m, in, or ft.

Area: The Space Inside 🟦

Area is the amount of space inside a shape. We measure it in square units (like $cm^{2}$ or $ft^{2}$) because we are counting how many little squares fit inside.

Rectangle — multiply length by width:

$A = l \times w$

Example: A rectangle that is $8$ cm long and $3$ cm wide has

• $A = 8 \times 3 = 24$ $cm^{2}$ ✓ (that's $24$ little $1$-cm squares inside!)

Square — since all sides are equal, multiply a side by itself:

$A = s \times s = s^2$

Example: A square with side $5$ in has

• $A = 5^2 = 25$ $in^{2}$ ✓

Triangle — a triangle takes up exactly half of a rectangle with the same base and height:

$A = \frac{1}{2}bh$

where $b$ is the base and $h$ is the height.

Watch the units! Perimeter uses plain units (cm). Area uses square units $(cm^{2})$.

Concept Check ✅

Let's make sure the difference between perimeter and area is clear before we go further.

Worked Examples: Step by Step ✏️

Let's work through perimeter and area one step at a time. Each example follows the same plan:

1. Write the correct formula.
2. Plug in the numbers from the shape.
3. Compute and label with the right unit (linear for perimeter, square for area).

Example 1 — Rectangle perimeter

A rectangle is $l = 9$ m long and $w = 4$ m wide. Find the perimeter.

• Formula: $P = 2(l + w)$
• Plug in: $P = 2(9 + 4)$
• Compute: $P = 2(13) = 26$ m ✓

Example 2 — Rectangle area

Same rectangle ($l = 9$ m, $w = 4$ m). Find the area.

• Formula: $A = l \times w$
• Plug in: $A = 9 \times 4$
• Compute: $A = 36$ $m^{2}$ ✓

Notice how the same shape gives a perimeter of $26$ m but an area of $36$ $m^{2}$ — different ideas, different units!

Worked Examples: Squares and Triangles 🔺

Example 3 — Square

A square has a side length of $7$ ft. Find both the perimeter and the area.

• Perimeter: $P = 4s = 4 \times 7 = 28$ ft ✓
• Area: $A = s^2 = 7 \times 7 = 49$ $ft^{2}$ ✓

Example 4 — Triangle area

A triangle has a base of $b = 10$ cm and a height of $h = 6$ cm. Find the area.

• Formula: $A = \frac{1}{2}bh$
• Plug in: $A = \frac{1}{2} \times 10 \times 6$
• Compute the easy way: first $10 \times 6 = 60$, then half of that is $\frac{1}{2} \times 60 = 30$
• So $A = 30$ $cm^{2}$ ✓

Tip for triangles: multiply the base and height first, then take half. It keeps the numbers simple!

Now you try a few in the boxes below.

Your Turn — Fill in the Boxes ✍️

A rectangle is $\textbf{12}$ cm long and $\textbf{5}$ cm wide. Use $P = 2(l + w)$ and $A = l \times w$.

Box 1: The perimeter in cm Box 2: The area in $cm^{2}$ Box 3: A triangle has base $8$ cm and height $5$ cm. Its area in $cm^{2}$ (use $A = \frac{1}{2}bh$)

Guided Practice: Choose the Answer 🎯

Remember: perimeter adds the sides (linear units), area multiplies (square units).

Match the Formula 🔽

For each measurement, choose the correct formula and the correct unit type.

Perimeter and Area in Real Life 🏡

Perimeter and area show up constantly in the real world. The trick is reading the problem and asking: am I going AROUND the shape (perimeter) or COVERING the inside (area)?

Story example: Jada is tiling a rectangular kitchen floor that is $7$ ft long and $5$ ft wide. Tiles cover the inside, so she needs the area:

• $A = l \times w = 7 \times 5 = 35$ $ft^{2}$ of tile. 🧱

Now use this idea on the problems below!

Garden Project 🌻

The Patel family is building a rectangular vegetable garden that is $\textbf{10}$ ft long and $\textbf{6}$ ft wide.

Box 1: They want a fence around the garden. How many feet of fencing? (perimeter) Box 2: They want soil to cover the ground inside. How many square feet? (area) Box 3: A triangular flower bed has base $4$ ft and height $3$ ft. Its area in $ft^{2}$?

Word Problem Challenge 🖼️

Review: Everything in One Place 🌟

You can now find the perimeter and area of rectangles, squares, and triangles! Here is your formula summary:

The two big ideas to never forget:

• Perimeter is the distance around → add the sides → linear units (cm, ft).
• Area is the space inside → multiply → square units ($cm^{2}$, $ft^{2}$).

Quick gut-check: Going around the outside? Perimeter. Covering the inside? Area. Get that right and the formula does the rest! 💪

Final Challenge 🏆

These mix together perimeter and area for different shapes. Decide around or inside first!