Area of Polygons

📐 Area of Polygons

Part 1 of 5 — What Is Area? Rectangles & Squares

Topics in This Part

🔑 Key Concept: Area is the amount of flat space a shape covers, measured in square units. Every area formula in this lesson is really just a clever way to count those squares without drawing them all.

What Is Area?

Imagine covering a shape with $1 \times 1$ unit squares, with no gaps and no overlaps. The number of squares that fit is the area.

If a rectangle is $5$ units wide and $3$ units tall, you can fit $3$ rows of $5$ squares:

$5 + 5 + 5 = 5 \times 3 = 15 \text{ square units}$

That is exactly why the rectangle formula is length times width — it counts the rows of squares for you.

Units Matter

Area is always measured in square units, written with a small $2$:

⚠️ Don't mix it up: perimeter is the distance around a shape (just units, like cm). Area is the space inside (square units, like $\text{cm}^2$).

Area of a Rectangle

$A = \text{length} \times \text{width} = l \times w$

Worked Example

A rectangle is $8\text{ cm}$ long and $5\text{ cm}$ wide.

$A = l \times w = 8 \times 5 = 40 \text{ cm}^2$

💡 It doesn't matter which side you call "length" and which you call "width" — $8 \times 5$ and $5 \times 8$ both give $40$. Multiplication can be done in any order.

Concept Check 🎯

Area of a Square

A square is just a rectangle whose length and width are equal, so both sides are the same value $s$:

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

Worked Example

A square has sides of $9\text{ in}$.

$A = s^2 = 9 \times 9 = 81 \text{ in}^2$

🔑 Key Idea: $s^2$ is read "$s$ squared." The word squared literally comes from finding the area of a square.

Compute the Area 🧮

Enter the area as a number only (the unit is already shown).

1) Rectangle, $15\text{ ft}$ by $4\text{ ft}$ $\;\Rightarrow\; ? \text{ ft}^2$ 2) Square with side $6\text{ cm}$ $\;\Rightarrow\; ? \text{ cm}^2$ 3) Square with side $12\text{ m}$ $\;\Rightarrow\; ? \text{ m}^2$

Working Backwards: Finding a Missing Side

If you know the area and one side of a rectangle, you can find the other side by dividing.

Worked Example

A rectangle has area $72\text{ cm}^2$ and a width of $8\text{ cm}$. How long is it?

$l = \frac{A}{w} = \frac{72}{8} = 9 \text{ cm}$

💡 Multiplication and division are opposites. Since $A = l \times w$, you can always undo the multiplication by dividing. We'll use this "work backwards" trick for every shape in this lesson.

Find the Missing Side 🧮

1) A rectangle has area $40\text{ m}^2$ and length $8\text{ m}$. Its width is $? \text{ m}$. 2) A rectangle has area $54\text{ in}^2$ and width $6\text{ in}$. Its length is $? \text{ in}$.

📐 Area of Polygons

Part 2 of 5 — Parallelograms

🔑 The Big Idea: A parallelogram can be cut and rearranged into a rectangle with the same area. That's why its formula looks so much like the rectangle's: $A = b \times h$.

Turning a Parallelogram into a Rectangle

Slice a right triangle off one end of a parallelogram and slide it to the other side. You get a rectangle — same area, just rearranged.

• The base $b$ becomes the rectangle's length.
• The height $h$ becomes the rectangle's width.

$A = b \times h$

⚠️ Height Is NOT the Slanted Side

The height is the straight-up (perpendicular) distance between the two bases — measured at a right angle, not along the tilted edge.

⚠️ The most common mistake is multiplying the base by the slanted side. Always use the perpendicular height.

Worked Example

A parallelogram has base $10\text{ cm}$ and height $6\text{ cm}$. (Its slanted side is $7\text{ cm}$, but we ignore that.)

$A = b \times h = 10 \times 6 = 60 \text{ cm}^2$

A Second Example

Base $12\text{ m}$, height $5\text{ m}$:

$A = 12 \times 5 = 60 \text{ m}^2$

💡 Notice this is the exact same multiplication you used for rectangles. A rectangle is really just a parallelogram whose height equals its side.

Concept Check 🎯

Pick the Right Numbers 🔽

A parallelogram has base $9$, height $8$, and a slanted side of $10$. Choose correctly at each step.

Parallelogram Practice 🧮

1) Base $15$, height $3$ $\;\Rightarrow\; A = ?$ 2) Base $6$, height $11$ $\;\Rightarrow\; A = ?$ 3) A parallelogram has area $48$ and base $8$. Its height is $?$

📐 Area of Polygons

Part 3 of 5 — Triangles

🔑 The Big Idea: Two identical triangles snap together to make a parallelogram. So one triangle is exactly half of that parallelogram: $A = \dfrac{1}{2} \, b \, h$.

Why Triangles Use One-Half

Take any triangle and make a copy of it. Flip the copy and slide it next to the original — together they form a parallelogram with the same base $b$ and height $h$.

Since the parallelogram's area is $b \times h$ and the triangle is half of it:

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

Same Height Rule as Parallelograms

The height is again the perpendicular distance from the base straight up to the opposite vertex (corner) — not a slanted side.

⚠️ For a right triangle, the two short sides (legs) meet at a right angle, so they can serve as the base and height directly. For other triangles, look for the dashed perpendicular line marking the height.

Worked Examples

Example 1. Base $8\text{ cm}$, height $5\text{ cm}$:

$A = \frac{1}{2} \times 8 \times 5 = \frac{1}{2} \times 40 = 20 \text{ cm}^2$

Example 2. Base $12\text{ m}$, height $7\text{ m}$:

$A = \frac{1}{2} \times 12 \times 7 = \frac{1}{2} \times 84 = 42 \text{ m}^2$

💡 Smart shortcut: multiply the two lengths first, then take half. Or, if one of the numbers is even, halve it first: $\frac{1}{2} \times 12 = 6$, then $6 \times 7 = 42$. Same answer, easier mental math.

Concept Check 🎯

Triangle Practice 🧮

1) Base $9$, height $4$ $\;\Rightarrow\; A = ?$ 2) Base $14$, height $5$ $\;\Rightarrow\; A = ?$ 3) Base $6$, height $9$ $\;\Rightarrow\; A = ?$ (decimal is fine)

Work Backwards 🔽

A triangle has area $24\text{ cm}^2$ and base $8\text{ cm}$. Find its height.

📐 Area of Polygons

Part 4 of 5 — Trapezoids

🔑 The Big Idea: A trapezoid has two parallel bases of different lengths. Average those two bases, then multiply by the height: $A = \dfrac{1}{2}\,(b_1 + b_2)\,h$.

The Trapezoid Formula

A trapezoid has exactly one pair of parallel sides — the two bases, called $b_1$ ("base one") and $b_2$ ("base two"). The height $h$ is the perpendicular distance between them.

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

Read It as "Average of the Bases Times Height"

The piece $\dfrac{b_1 + b_2}{2}$ is just the average of the two bases — a typical width halfway up the shape. Multiply that average width by the height and you have the area.

💡 Order of operations: always add the two bases first (inside the parentheses), then multiply by the height, then take half.

Worked Examples

Example 1. Bases $b_1 = 6$, $b_2 = 10$, height $h = 4$:

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

Example 2. Bases $b_1 = 8$, $b_2 = 12$, height $h = 5$:

$A = \frac{1}{2}(8 + 12)(5) = \frac{1}{2}(20)(5) = \frac{1}{2}(100) = 50 \text{ units}^2$

⚠️ Watch the steps: $6 + 10 = 16$ comes before any multiplying. A common error is computing $\frac{1}{2} \times 6$ first and forgetting to include $b_2$.

Concept Check 🎯

Build the Answer Step by Step 🔽

Trapezoid with bases $b_1 = 7$, $b_2 = 11$, and height $h = 8$.

Trapezoid Practice 🧮

1) Bases $10$ and $14$, height $3$ $\;\Rightarrow\; A = ?$ 2) Bases $4$ and $8$, height $10$ $\;\Rightarrow\; A = ?$ 3) Bases $3$ and $7$, height $5$ $\;\Rightarrow\; A = ?$ (decimal is fine)

📐 Area of Polygons

Part 5 of 5 — Composite Figures & Mastery Check

🔑 The Big Idea: Any complicated polygon can be broken into rectangles, triangles, and trapezoids you already know. Find each piece's area, then add them (or subtract a missing chunk).

Two Strategies for Composite Shapes

Strategy 1 — Add the pieces. Split the figure into simple shapes, find each area, and add.

Example — a "house" pentagon: a rectangle ($8 \times 6$) with a triangle roof (base $8$, height $3$) on top.

$A = \underbrace{8 \times 6}_{\text{rectangle}} + \underbrace{\tfrac{1}{2} \times 8 \times 3}_{\text{triangle}} = 48 + 12 = 60 \text{ units}^2$

Strategy 2 — Subtract a missing piece. Find the area of a big rectangle that encloses the shape, then subtract the corner that isn't there.

Example — an L-shape: a full $10 \times 8$ rectangle with a $4 \times 3$ rectangle cut out of one corner.

$A = \underbrace{10 \times 8}_{\text{big}} - \underbrace{4 \times 3}_{\text{cut-out}} = 80 - 12 = 68 \text{ units}^2$

💡 Either strategy works — choose whichever splits the figure into the fewest, friendliest pieces.

Concept Check 🎯

Quick Reference — All Five Formulas

⚠️ Top three traps: (1) using a slanted side instead of the perpendicular height, (2) forgetting the $\frac{1}{2}$ for triangles and trapezoids, and (3) reporting area in plain units instead of square units.

Mixed Practice 🧮

Pick the right formula for each shape, then compute.

1) Triangle, base $7$, height $6$ $\;\Rightarrow\; A = ?$ 2) Trapezoid, bases $9$ and $15$, height $6$ $\;\Rightarrow\; A = ?$ 3) Parallelogram, base $12$, height $5$ $\;\Rightarrow\; A = ?$ 4) L-shape: a $10 \times 8$ rectangle with a $4 \times 3$ corner removed $\;\Rightarrow\; A = ?$

Exit Quiz ✅

Answer all three to finish the lesson.