Find Area by Composing and Decomposing - 6th Grade Math

SECTION 1

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.

~2000 BCE • Ancient Egypt

Egyptian surveyors measured farm fields along the Nile River. They used ropes to split odd-shaped land into triangles and rectangles to figure out their area.

~300 BCE • Euclid's Elements

The Greek mathematician Euclid proved that a triangle's area equals half of a rectangle with the same base and height. This is the formula you still use today!

~250 BCE • Archimedes

Archimedes found the area of curved shapes by filling them with many tiny triangles. This was an early version of decomposing a shape.

Today • Modern Applications

Architects, game designers, and engineers use composing and decomposing every day. Computer programs split complex 3-D models into millions of tiny triangles.

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!

SECTION 2

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

The number of square units that cover a flat shape with no gaps or overlaps. Think of it as how many 1 × 1 tiles fit inside.

Decompose

To break a shape into smaller, simpler pieces (like triangles and rectangles). You find the area of each piece, then add them up.

Compose

To put shapes together to form a bigger, simpler shape (usually a rectangle). Then subtract any extra area you added.

Base & Height

The base (b) is any side you pick. The height (h) is the perpendicular (straight up-and-down) distance from the base to the opposite side or vertex.

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KEY TAKEAWAY

Think of finding area like solving a jigsaw puzzle in reverse. If you have a weird puzzle piece, you can either cut it into shapes you already know (decompose) or tape extra pieces onto it to make a rectangle and then subtract the tape parts (compose). Either way, you end up with the same area!

SECTION 3

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.

The L-shape is split into Rectangle A (6 m × 10 m = 60 m²) and Rectangle B (7 m × 4 m = 28 m²). Adding them gives 88 m².

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.

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Pro Tip

There is often more than one way to decompose a shape. Any correct split will give you the same total area. Pick the one that looks easiest!

SECTION 4

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.

RECTANGLE

A = b × h

A = area, b = base (length of one side), h = height (length of the other side). This is your go-to formula.

TRIANGLE

A = ½ × b × h

b = base, h = height measured perpendicular to the base. A triangle is exactly half of a rectangle — that's why you multiply by ½.

PARALLELOGRAM

A = b × h

Same formula as a rectangle! The height must be perpendicular to the base, not the slanted side length.

TRAPEZOID

A = ½ × (b₁ + b₂) × h

b₁ and b₂ are the two parallel sides (the top and bottom), and h is the perpendicular distance between them.

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REMEMBER THIS

Every formula above comes back to the rectangle formula, A = b × h. A triangle is half a rectangle. A parallelogram is a rectangle in disguise. A trapezoid is the average of two rectangles. If you know the rectangle formula, you can figure out everything else.

SECTION 5

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.

Top row: strategies for four common shapes. Bottom: the composing strategy — enclose an irregular polygon in a rectangle and subtract the extra pieces.

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)

Summary of decompose/compose strategies by shape

SECTION 6

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?

Finding the Area of Maya's Garden

1
Step 1 — Identify the Shape and Given Values

The garden is a trapezoid. The two bases are b₁ = 8 ft and b₂ = 14 ft. The height is h = 6 ft.
2
Step 2 — Write the Formula

The area of a trapezoid is A = ½ × (b₁ + b₂) × h.
3
Step 3 — Substitute the Numbers

A = ½ × (8 + 14) × 6. First add the bases: 8 + 14 = 22.

A = ½ × 22 × 6
4
Step 4 — Multiply

Multiply 22 × 6 = 132. Then take half: 132 ÷ 2 = 66.

A = 66 ft²
5
Step 5 — Check with the Average-Base Interpretation

The trapezoid formula can be rewritten as A = average of the two bases × height. The average of 8 ft and 14 ft is (8 + 14) ÷ 2 = 11 ft. Think of this as the width of a rectangle that has the same area as the trapezoid. Then A = 11 × 6 = 66 ft². This works for any trapezoid — no matter how the legs are slanted — because the formula only depends on the two parallel sides and the height between them. ✓ The answer matches!

66 ft² confirmed

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Did You Notice?

We verified the answer two different ways — using the trapezoid formula AND by thinking of it as a rectangle with the average width. Checking with a second method is a great habit. It catches mistakes!

SECTION 7

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

Comparison of two main area strategies

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KEY TAKEAWAY

Imagine you have an oddly shaped cookie. You can either break it into smaller pieces that are easy to measure (decompose), or you can press it into a rectangular cookie cutter and then figure out how much dough you trimmed off (compose). Both let you figure out how big the cookie really is.

SECTION 8

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

How 6.G.1 connects to future standards

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!

SECTION 9

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.

Question 1: CONCEPTUAL

Explain in your own words: why is the area of a triangle exactly half the area of a rectangle with the same base and height?

Show answer

If you draw a rectangle and then draw a diagonal from one corner to the opposite corner, you create two triangles that are the same size and shape. Each triangle takes up exactly half of the rectangle. That means Area of triangle = ½ × base × height, which is half of the rectangle's area (base × height). This works for right triangles directly, and any other triangle can be rearranged to show the same relationship.

Question 2: BASIC CALCULATION

Find the area of a right triangle with a base of 10 cm and a height of 7 cm.

Show answer

Use the triangle formula: A = ½ × b × h. Substitute: A = ½ × 10 × 7. Multiply 10 × 7 = 70, then take half: 70 ÷ 2 = 35. The area is 35 cm². You can check by imagining a 10 × 7 rectangle (70 cm²) and noting the triangle fills exactly half.

Question 3: INTERMEDIATE

A trapezoid has bases of 5 in. and 11 in. and a height of 8 in. Find its area using the formula AND by decomposing it into a rectangle and a triangle. Show that both give the same answer.

Show answer

Formula method: A = ½ × (5 + 11) × 8 = ½ × 16 × 8 = ½ × 128 = 64 in². Decompose method: place the shorter base (5 in.) on top. The rectangle in the middle is 5 × 8 = 40 in². The leftover part on the side is a triangle with base = 11 − 5 = 6 in. and height = 8 in. Triangle area = ½ × 6 × 8 = 24 in². Total = 40 + 24 = 64 in². Both methods give 64 in², confirming the answer.

Question 4: APPLIED

Carlos wants to paint one wall of his room. The wall is a rectangle that is 12 ft wide and 9 ft tall, but it has a triangular window cutout with a base of 4 ft and a height of 3 ft. One can of paint covers 50 ft². How many cans does Carlos need?

Show answer

First, find the total wall area: A = 12 × 9 = 108 ft². Next, find the window area: A = ½ × 4 × 3 = 6 ft². Subtract the window: 108 − 6 = 102 ft² to paint. Since one can covers 50 ft², Carlos needs 102 ÷ 50 = 2.04 cans. Because you can't buy part of a can, Carlos needs 3 cans to have enough paint. (Two cans would only cover 100 ft², which is not quite enough.)

Question 5: CRITICAL THINKING

A pentagon has vertices at (0, 0), (6, 0), (6, 4), (3, 7), and (0, 4) on a coordinate grid. Find its area. Hint: try enclosing it in a rectangle and subtracting corner triangles, or decompose it a different way.

Show answer

Compose method: the smallest rectangle that encloses the pentagon goes from (0, 0) to (6, 7), so its area is 6 × 7 = 42 square units. Two corner triangles are not part of the pentagon: upper-left triangle with vertices (0, 4), (0, 7), (3, 7) has base 3 and height 3 → area = ½ × 3 × 3 = 4.5. Upper-right triangle with vertices (3, 7), (6, 7), (6, 4) has base 3 and height 3 → area = ½ × 3 × 3 = 4.5. Subtract: 42 − 4.5 − 4.5 = 33 square units. Alternatively, decompose into a rectangle (0,0)-(6,0)-(6,4)-(0,4) with area 24, plus a triangle (0,4)-(6,4)-(3,7) with base 6 and height 3, area = ½ × 6 × 3 = 9. Total = 24 + 9 = 33 square units. Both methods confirm the area is 33 square units.

SUMMARY

Lesson Summary

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.