Perimeter and Area

📐 Perimeter and Area

Part 1 of 5 — Perimeter: The Distance Around

Topics in This Part

🔑 Key Concept: Perimeter is the total distance around a flat shape — imagine walking along every edge once. You find it by adding up the lengths of all the sides.

What Perimeter Means

If you put a fence around a yard or trim around a picture, you care about the perimeter — the length of the boundary.

To find it, add every side:

$P = s_1 + s_2 + s_3 + \cdots$

Example: A Triangle

A triangle has sides $5\text{ cm}$, $7\text{ cm}$, and $9\text{ cm}$:

$P = 5 + 7 + 9 = 21\text{ cm}$

💡 Units matter. Perimeter is a length, so the answer keeps the same unit as the sides — $\text{cm}$, $\text{m}$, $\text{in}$, $\text{ft}$ — never squared.

Rectangles and Squares Have Shortcuts

A rectangle has two equal lengths $\ell$ and two equal widths $w$, so instead of adding four numbers you can use:

$P = 2\ell + 2w \quad\text{or}\quad P = 2(\ell + w)$

A square is a rectangle with all four sides equal to $s$:

$P = 4s$

🔑 The shortcuts and "add every side" give the same answer — the formulas just save time.

Concept Check 🎯

Find the Perimeter 🧮

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

1) A square with side $7\text{ cm}$: $P = \,?\ \text{cm}$ 2) A rectangle, $\ell = 12\text{ in}$, $w = 5\text{ in}$: $P = \,?\ \text{in}$ 3) A triangle with sides $6$, $8$, $10\text{ ft}$: $P = \,?\ \text{ft}$

Working Backward: A Missing Side

If you know the perimeter, you can find a missing side by subtracting the sides you know.

Example

A rectangle has perimeter $30\text{ m}$ and length $9\text{ m}$. Find the width.

$2(9) + 2w = 30 \;\Rightarrow\; 18 + 2w = 30 \;\Rightarrow\; 2w = 12 \;\Rightarrow\; w = 6\text{ m}$

💡 Perimeter problems are really just addition (going forward) and subtraction/division (working backward). In Part 2 we tackle area — a different kind of measurement.

Work Backward 🧮

1) A square has perimeter $36\text{ cm}$. How long is one side? $\,?\ \text{cm}$ 2) A rectangle has perimeter $26\text{ in}$ and width $4\text{ in}$. Find the length. $\,?\ \text{in}$

📐 Perimeter and Area

Part 2 of 5 — Area of Rectangles, Squares & Parallelograms

🔑 The Idea: Area measures the space inside a flat shape — how many unit squares fit. Because it covers a 2-D region, area is always in square units ($\text{cm}^2$, $\text{ft}^2$, …).

Area of a Rectangle

Cover a rectangle with a grid of $1\times 1$ squares. The number that fit is the area:

$A = \ell \times w \quad(\text{length} \times \text{width})$

Example: $\ell = 8\text{ cm}$, $w = 3\text{ cm}$

$A = 8 \times 3 = 24\text{ cm}^2$

A grid $8$ across and $3$ down holds exactly $24$ unit squares. ✓

A square is the special case $\ell = w = s$:

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

⚠️ Perimeter vs. Area — don't mix them up! For that $8\times 3$ rectangle, the perimeter is $22\text{ cm}$ but the area is $24\text{ cm}^2$. Perimeter is the border; area is the inside.

Perimeter or Area? 🎯

Area of a Parallelogram

Slide a triangle from one end of a parallelogram to the other and it becomes a rectangle. So the area uses the base and the perpendicular height (the straight-up distance between the two parallel sides — not the slanted side):

$A = b \times h$

Example: $b = 10\text{ m}$, $h = 4\text{ m}$

$A = 10 \times 4 = 40\text{ m}^2$

⚠️ The height is always perpendicular to the base. If a problem gives you the slanted side length, that is not the height — use the vertical distance.

Pick the Right Formula 🔽

Match each shape to its area formula.

Find the Area 🧮

Enter each area (the unit is shown).

1) Rectangle, $\ell = 9\text{ ft}$, $w = 5\text{ ft}$: $A = \,?\ \text{ft}^2$ 2) Square, side $7\text{ cm}$: $A = \,?\ \text{cm}^2$ 3) Parallelogram, base $12\text{ m}$, height $6\text{ m}$: $A = \,?\ \text{m}^2$

📐 Perimeter and Area

Part 3 of 5 — Triangles & Trapezoids

🔑 Why the halves? A triangle is half of a rectangle (or parallelogram). A trapezoid averages its two parallel sides. Both formulas have a $\dfrac{1}{2}$ for exactly this reason.

Area of a Triangle

Put two copies of a triangle together and you get a parallelogram with the same base and height. So a triangle is half:

$A = \frac{1}{2}\, b \, h$

The height is the perpendicular distance from the base to the opposite corner.

Example: $b = 10\text{ cm}$, $h = 6\text{ cm}$

$A = \frac{1}{2}(10)(6) = \frac{1}{2}(60) = 30\text{ cm}^2$

💡 Tip: Multiply $b \times h$ first, then take half. $10 \times 6 = 60$, half of $60$ is $30$.

Concept Check 🎯

Area of a Trapezoid

A trapezoid has two parallel sides — call them $b_1$ and $b_2$. You average them and multiply by the height:

$A = \frac{1}{2}\,(b_1 + b_2)\,h$

Example: $b_1 = 6\text{ in}$, $b_2 = 10\text{ in}$, $h = 4\text{ in}$

$A = \frac{1}{2}(6 + 10)(4) = \frac{1}{2}(16)(4) = \frac{1}{2}(64) = 32\text{ in}^2$

🔑 If the two parallel sides were equal, a trapezoid would just be a parallelogram — and the formula would collapse back to $b \times h$.

Trapezoid, Step by Step 🔽

Find the area of a trapezoid with $b_1 = 5\text{ cm}$, $b_2 = 9\text{ cm}$, height $6\text{ cm}$.

Triangles & Trapezoids 🧮

Enter each area.

1) Triangle, base $14\text{ ft}$, height $9\text{ ft}$: $A = \,?\ \text{ft}^2$ 2) Triangle, base $7\text{ cm}$, height $4\text{ cm}$: $A = \,?\ \text{cm}^2$ 3) Trapezoid, $b_1 = 8\text{ m}$, $b_2 = 12\text{ m}$, height $5\text{ m}$: $A = \,?\ \text{m}^2$

📐 Perimeter and Area

Part 4 of 5 — Circles: Circumference & Area

🔑 The star of circles is $\pi$ (pi), about $3.14$. It's the number you get when you divide any circle's distance-around by its diameter — it never changes.

Radius, Diameter & $\pi$

• The radius $r$ reaches from the center to the edge.
• The diameter $d$ stretches all the way across, through the center.
• They're related by $d = 2r$ (the diameter is two radii).

The distance around a circle has a special name — circumference — and two equivalent formulas:

$C = 2\pi r \qquad\text{or}\qquad C = \pi d$

Example: radius $5\text{ cm}$ (use $\pi \approx 3.14$)

$C = 2\pi r = 2(3.14)(5) = 31.4\text{ cm}$

💡 If you're given the diameter, $C = \pi d$ is faster. If you're given the radius, use $C = 2\pi r$.

Concept Check 🎯

Area of a Circle

The space inside a circle uses the radius squared:

$A = \pi r^2$

Example: radius $5\text{ cm}$ (use $\pi \approx 3.14$)

$A = \pi r^2 = 3.14 \times 5^2 = 3.14 \times 25 = 78.5\text{ cm}^2$

⚠️ Square the radius first, then multiply by $\pi$. $r^2$ means $r \times r$, so $5^2 = 25$ — not $5 \times 2 = 10$. And if you're given the diameter, cut it in half to get $r$ before using this formula.

Circle Quantities 🔽

A circle has diameter $12\text{ in}$. Use $\pi \approx 3.14$.

Circle Practice 🧮

Use $\pi \approx 3.14$. Enter each value as a decimal.

1) Circumference of a circle with radius $3\text{ cm}$ ($C = 2\pi r$): $\,?\ \text{cm}$ 2) Area of a circle with radius $3\text{ cm}$ ($A = \pi r^2$): $\,?\ \text{cm}^2$ 3) Area of a circle with radius $10\text{ m}$: $\,?\ \text{m}^2$

📐 Perimeter and Area

Part 5 of 5 — Composite Figures, Real-World Problems & Mastery Check

You now have every basic formula. Real problems often combine shapes or ask "how much fencing / how much paint?" — let's put it all together.

Quick Reference

⚠️ Remember the units: perimeter/circumference use plain units; area uses square units. And in $\frac{1}{2}bh$ or $\pi r^2$, always do the multiplication/squaring before the $\frac{1}{2}$ or before adding $\pi$.

Composite Figures: Split, Then Add

A composite figure is made of simpler shapes glued together. To find its area, break it into pieces, find each area, and add.

Example: An "L" shape

Picture a $10\times 6$ rectangle with a $4\times 3$ rectangle removed from one corner. You can either subtract, or split into two rectangles. Subtracting:

$A = (10 \times 6) - (4 \times 3) = 60 - 12 = 48\text{ units}^2$

💡 Two strategies: Split-and-add (cut the figure into shapes you know) or subtract (start with a big rectangle and remove the missing chunk). Both give the same area — pick whichever is easier to see.

Real-World Problems 🎯

Composite & Applications 🧮

1) An L-shape: a $10\text{ m} \times 7\text{ m}$ rectangle with a $3\text{ m} \times 2\text{ m}$ rectangle cut out. Total area $= \,?\ \text{m}^2$ 2) A room floor is $15\text{ ft} \times 10\text{ ft}$. How many square feet of tile cover it? $\,?\ \text{ft}^2$ 3) A square photo has side $6\text{ in}$. How much frame trim goes around it (perimeter)? $\,?\ \text{in}$

Exit Quiz ✅

Answer all three to finish the lesson.