Why Do We Need to Find Area?
People have been figuring out the area of shapes for thousands of years. Ancient farmers needed to know the size of their fields. Builders needed to know how much stone to cut. Even today, you use area when you figure out how much paint covers a wall or how much carpet fits a room.
The word area means the amount of flat space inside a shape. It is measured in square units like square feet (ft²) or square centimeters (cm²). Finding area for rectangles is pretty simple — just multiply length times width. But what about triangles, trapezoids, or weird-looking polygons?
That is exactly the problem mathematicians solved long ago. They figured out that you can take any tricky shape and either break it into simpler pieces (decompose) or combine pieces to form a rectangle (compose). Let's see how this idea developed over time.
The big question this lesson answers is: How can you find the area of any polygon, even if it is not a simple rectangle? The answer: compose or decompose!
Core Principles & Definitions
Before we dive into examples, let's nail down four key ideas. These are the building blocks for every area problem you will meet in this lesson.
Area
Decompose
Compose
Base & Height
See It: Decomposing a Polygon
The diagram below shows an L-shaped polygon. On the left, you see the original shape. On the right, a dashed line splits it into two rectangles. You find the area of each rectangle and add them together.
Notice how the dashed line turns one hard shape into two easy ones. The trick is choosing where to draw that line. Look for corners that stick out — those are great places to cut.
Formulas You Need
You only need a handful of area formulas to handle any polygon in this lesson. Let's list them out and explain what each letter means.
Shape-by-Shape Breakdown
Let's look more closely at how composing and decomposing works for each shape. The diagram below shows four common shapes and the strategy for each one.
| Shape | Strategy | Formula |
|---|---|---|
| Right Triangle | Half of a rectangle | A = ½ × b × h |
| Any Triangle | Drop a height line to the base | A = ½ × b × h |
| Parallelogram | Slice off triangle, slide to form rectangle | A = b × h |
| Trapezoid | Split into rectangle + two triangles, or use formula | A = ½ × (b₁ + b₂) × h |
| Irregular Polygon | Decompose into known shapes OR compose into a rectangle and subtract | Sum of pieces (or rect − extras) |
Worked Example: Area of a Trapezoid-Shaped Garden
Maya wants to put grass seed on her trapezoid-shaped garden. The two parallel sides measure 8 ft and 14 ft. The height (perpendicular distance between them) is 6 ft. How many square feet of seed does she need?
Composing vs. Decomposing — When to Use Each
Both methods always give the same answer. But sometimes one is much easier than the other. Here's a quick guide to help you choose.
| Feature | Decomposing (Break Apart) | Composing (Build Up) |
|---|---|---|
| Best for | L-shapes, T-shapes, shapes with straight interior cuts | Irregular polygons with slanted edges on a grid |
| How it works | Cut the shape into rectangles and/or triangles. Add all the areas. | Enclose the shape in a rectangle. Subtract the extra corner areas. |
| Strengths | Simple; fewer subtraction steps; easy to visualize | Works for any polygon; great on coordinate grids |
| Watch out for | Missing a piece or counting a piece twice | Forgetting to subtract one of the extra corner pieces |
Looking Ahead — Surface Area & Beyond
The skills you are learning now are the foundation for harder topics later. In 6th and 7th grade, you will use decomposing to find the surface area of 3-D shapes like rectangular prisms and pyramids. A surface area is just the total area of all the flat faces — and each face is a 2-D shape you already know how to handle!
| This Lesson (6.G.1) | What Comes Next |
|---|---|
| Area of triangles and rectangles | Surface area of prisms (6.G.4) |
| Decomposing polygons on flat paper | Unfolding 3-D nets into 2-D pieces |
| Composing shapes into rectangles | Finding area of circles (7.G.4) — using composing ideas with wedge slices |
| Real-world word problems (gardens, rooms) | Volume and real-world design projects |
In high school, you will even use these same decomposing ideas in calculus to find the area under a curved line. So the strategy of slicing a hard shape into easy pieces never goes away — it just gets more powerful!
Practice Problems
Try these five problems. They start easy and get harder. For each one, think about whether decomposing or composing is the better strategy.
Lesson Summary
You can find the area of any polygon by using two powerful strategies. Decomposing means cutting a shape into simpler pieces — usually rectangles and triangles — finding each piece's area, and adding them together. Composing means surrounding the shape with a rectangle and subtracting the extra parts. Both methods always give the same answer.
The key formulas are A = b × h for rectangles and parallelograms, A = ½ × b × h for triangles, and A = ½ × (b₁ + b₂) × h for trapezoids. Always use the perpendicular height, not a slanted side. These skills connect directly to surface area of 3-D shapes and many real-world problems like painting walls, seeding gardens, and designing rooms.