Area of Composite Figures

What Is a Composite Figure? 🧩

A composite figure (sometimes called a complex figure) is a shape made by putting together two or more basic shapes. Think of it like a puzzle: the big shape is built from smaller pieces you already know how to handle.

The basic building blocks are usually:

• Rectangles and squares
• Triangles
• Circles and semicircles
• Trapezoids and parallelograms

You see composite figures everywhere in real life:

• The floor plan of a house
• A garden split into different sections
• A swimming pool shaped like a rectangle with a rounded end
• Letters and logos like the giant "L" on a sign

The big idea is simple: break the messy shape into simple shapes, find each area, then combine them.

Review: The Area Formulas You Need 📐

Before we combine shapes, let's review the area formula for each basic piece. Keep this table handy!

Shape	Formula	Meaning
Rectangle	$A = l \times w$	length times width
Square	$A = s^2$	side times side
Triangle	$A = \frac{1}{2}bh$	half of base times height
Parallelogram	$A = bh$	base times height
Trapezoid	$A = \frac{1}{2}(b_1 + b_2)h$	half the sum of the two bases, times height
Circle	$A = \pi r^2$	pi times radius squared
Semicircle	$A = \frac{1}{2}\pi r^2$	half of a circle

Remember: the radius $r$ is half the diameter. If a circle's diameter is $6$ cm, then $r = 3$ cm.

We'll use $\pi \approx 3.14$ when we need a number for circle problems.

The 3-Step Strategy ✅

Every composite-figure problem follows the same plan:

Step 1 — Break it apart. Look for rectangles, triangles, and circles hiding inside. Draw a line to split the figure into simple pieces.

Step 2 — Find each area. Use the right formula for each piece and label your dimensions carefully.

Step 3 — Combine. Usually you add the pieces together:

$A_{total} = A_1 + A_2 + A_3 + \dots$

Worked example — an L-shaped figure. Split the "L" into two rectangles:

• Top rectangle: $8$ ft by $3$ ft → $A_1 = 8 \times 3 = 24$ $ft^{2}$
• Bottom rectangle: $5$ ft by $4$ ft → $A_2 = 5 \times 4 = 20$ $ft^{2}$
• Total: $A = 24 + 20 = 44$ $ft^{2}$ ✓

That's it — split, solve each piece, then add!

Concept Check ✅

Let's make sure the main idea is clear before moving on.

Worked Example: A "House" Shape 🏠

A house-shaped figure is a rectangle with a triangle roof on top.

• Rectangle (the walls): $12$ m wide and $8$ m tall
• Triangle (the roof): base $= 12$ m, height $= 5$ m

Step 1 — Break it apart. It's already two pieces: one rectangle + one triangle.

Step 2 — Find each area.

• Rectangle: $A_1 = l \times w = 12 \times 8 = 96$ $m^{2}$
• Triangle: $A_2 = \frac{1}{2}bh = \frac{1}{2} \times 12 \times 5 = \frac{1}{2} \times 60 = 30$ $m^{2}$

Step 3 — Add them. $A = 96 + 30 = 126 \text{ m}^2 ✓$

Tip: For the triangle, multiply base times height first ($12 \times 5 = 60$), then take half. The numbers stay friendly!

Worked Example: Rectangle + Semicircle 🏊

A swimming pool is a rectangle with a semicircle on one end.

• Rectangle: $10$ cm long and $6$ cm wide
• Semicircle: it sits on the $6$ cm end, so the diameter is $6$ cm and the radius is $r = 3$ cm

Step 1 — Break it apart. One rectangle + one semicircle.

Step 2 — Find each area (use $\pi \approx 3.14$).

• Rectangle: $A_1 = 10 \times 6 = 60$ $cm^{2}$
• Semicircle: $A_2 = \frac{1}{2}\pi r^2 = \frac{1}{2} \times 3.14 \times 3^2 = \frac{1}{2} \times 3.14 \times 9 = \frac{1}{2} \times 28.26 = 14.13$ $cm^{2}$

Step 3 — Add them. $A = 60 + 14.13 = 74.13 \text{ cm}^2 ✓$

Notice we found the radius by cutting the diameter in half. That's a super common step — watch for it!

Your Turn — Fill in the Boxes ✍️

A figure is a rectangle with a triangle on top (a "house" shape).

• Rectangle: $9$ ft wide and $6$ ft tall
• Triangle on top: base $= 9$ ft, height $= 4$ ft

Box 1: Area of the rectangle in $ft^{2}$ (use $A = l \times w$) Box 2: Area of the triangle in $ft^{2}$ (use $A = \frac{1}{2}bh$) Box 3: The total area in $ft^{2}$

Guided Practice: Choose the Answer 🎯

Break each figure into basic shapes, then combine the areas.

Match the Piece to Its Formula 🔽

A composite figure is made of a rectangle and a semicircle. Choose the correct formula for each piece.

Real-World Composite Figures 🌍

Composite-figure math shows up whenever a real object isn't a perfect single shape.

Example — a garden. Maya's vegetable garden is shaped like a rectangle with a triangular flower bed attached to one end. To buy the right amount of soil, she needs the total area.

• Rectangle part: $15$ ft by $8$ ft → $A_1 = 15 \times 8 = 120$ $ft^{2}$
• Triangle part: base $8$ ft, height $6$ ft → $A_2 = \frac{1}{2} \times 8 \times 6 = 24$ $ft^{2}$
• Total area: $120 + 24 = 144$ $ft^{2}$

So Maya needs enough soil to cover $144$ $ft^{2}$. Breaking the garden into a rectangle and a triangle made the problem easy. Now try a couple yourself.

Word Problem — Fill in the Boxes ✍️

A classroom rug is shaped like a rectangle with a semicircle on one end.

• Rectangle: $8$ ft long and $4$ ft wide
• Semicircle on the $4$ ft end: diameter $= 4$ ft, so radius $r = 2$ ft

Use $\pi \approx 3.14$.

Box 1: The radius of the semicircle in ft Box 2: Area of the rectangle in $ft^{2}$ Box 3: Area of the semicircle in $ft^{2}$ (use $A = \frac{1}{2}\pi r^2$, rounded to two decimals)

Application Question 🧮

Use the subtraction method for this one.

Summary: Putting It All Together 🎓

You now know how to find the area of any composite figure! Here's the whole plan in one place:

Step	What you do
1. Break apart	Split the figure into rectangles, triangles, and circles
2. Find each area	Use the right formula for each piece
3. Combine	Add the pieces — or subtract a missing hole

The two methods:

• Addition method: when pieces are joined together, find each area and add: $A = A_1 + A_2 + \dots$
• Subtraction method: when a piece is cut out, find the big area and subtract the missing piece: $A = A_{big} - A_{hole}$

Key reminders:

• The radius is half the diameter.
• Area is always measured in square units ($cm^{2}$, $ft^{2}$, $m^{2}$).
• A semicircle is half a circle: $A = \frac{1}{2}\pi r^2$.

Now take on the challenge questions below!

Challenge Questions 🏆

These mix the addition and subtraction methods. Take your time and work step by step.