Breaking Shapes Apart to Find Area - 3rd Grade Math

Section 1

Where Did This Idea Come From?

People have needed to measure the size of flat surfaces for thousands of years. Farmers wanted to know how big their fields were. Builders needed to know how much stone to cut for a floor. Even kids sharing a pizza want to know who got the bigger piece! The idea of area — how much space a flat shape covers — has been important for a very long time.

~3000 B.C.

Ancient Egyptians measured the area of farm fields along the Nile River. They used ropes to mark off rectangles!

~300 B.C.

A Greek teacher named Euclid wrote a famous math book. He showed that you can break big shapes into smaller ones to find their area.

~800 A.D.

Builders in many countries used rectangles to plan castles and temples. They added the areas of each room to find the total.

Today

You use the same idea! When a floor or wall has an odd shape, you split it into rectangles and add the areas. That is what this lesson is all about.

Here is the big question we will answer: What do you do when a shape is NOT a simple rectangle? You break it into pieces that ARE rectangles, find each area, and add them together. Let's learn how!

Section 2

The Big Ideas

Before we start splitting shapes, let's make sure we know four important ideas. Read each card below carefully.

Area = Space Inside

Area tells us how much flat space is inside a shape. We measure it in square units, like square centimeters (cm²) or square inches (in²).

Area Is Additive

Additive means you can add pieces together. If you break a shape into parts, adding the areas of the parts gives you the total area.

Rectilinear Shapes

A rectilinear shape is made only of straight lines that meet at right angles (like the corners of a book). It looks like rectangles stuck together.

Non-Overlapping

Non-overlapping means the rectangles do NOT sit on top of each other. Every square unit is counted exactly once — no more, no less.

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Key Takeaway

Think of a rectilinear shape like an L-shaped swimming pool. You can't just use one length × width to find how much water fills it. But if you picture a wall splitting the pool into two rectangles, you can find the area of each rectangle and add them together. That's the whole trick!

Section 3

See It: Splitting a Shape

Look at the L-shaped figure below. It is a rectilinear shape because every corner is a right angle. We draw one line to split it into Rectangle A and Rectangle B. Then we find each area and add.

L-shaped figure decomposed into two non-overlapping rectangles, A and B, with dimensions labeled.

See the dashed yellow line? It splits the L-shape into two rectangles that do not overlap. Rectangle A is 7 cm wide and 3 cm tall. Rectangle B is 8 cm wide and 4 cm tall. To find the total area, we find each area and add them together.

Finding the Total Area

Area of A = 7 × 3 = 21 cm² Area of B = 8 × 4 = 32 cm² Total = 21 + 32 = 53 cm²

Section 4

The Math: Step by Step

Here are the three steps you will use every time you see a rectilinear shape. Follow these and you'll always get the right answer!

Step 1 — Rectangle Area Formula

Area = length × width

Length is how long one side is. Width is how long the other side is. Multiply them to get the area in square units.

Step 2 — Decompose (Break Apart)

Draw lines to split the shape into rectangles.

Look for corners that stick out. Draw a straight line from one edge to another to make rectangles. Make sure the rectangles do NOT overlap.

Step 3 — Add the Areas

Total Area = Area₁ + Area₂ + …

Find the area of each rectangle. Then add all the areas together. The sum is the total area of the whole shape!

You can sometimes split the same shape in different ways. That's okay! No matter how you split it, the total area will be the same. That is what "area is additive" means.

Section 5

More Shapes You Might See

Not every rectilinear shape is an L. Some look like a T, a U, or even a plus sign (+). The good news is: the same trick works every time. Let's look at a T-shape.

T-shaped rectilinear figure decomposed into two rectangles with dimensions labeled.

The T-shape above has two parts. Rectangle C (the top bar) is 10 cm × 2 cm = 20 cm². Rectangle D (the stem) is 2 cm × 5 cm = 10 cm². The total area is 20 + 10 = 30 cm².

Here is a handy table showing common rectilinear shapes and how many rectangles you might need.

Shape	Looks Like	Rectangles Needed
L-shape	A book with a piece cut off	2
T-shape	A letter T or a hammer	2
U-shape	A horseshoe or goal post	3
Plus sign (+)	A first-aid cross	3 (or 2 with overlap thinking)
Staircase	Steps going up	2, 3, or more

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Key Takeaway

Imagine you have a chocolate bar that is shaped like the letter T. You can't eat it all in one bite! But you can snap it into two smaller rectangles and eat each one. Finding area works the same way — snap the shape apart, find each area, and add.

Section 6

Worked Example

Let's solve a full problem together. Read each step slowly.

Worked Example — U-Shaped Garden

1
Problem

Maria's garden is shaped like the letter U. The outside of the shape is 8 m wide and 6 m tall. The notch in the top middle is 4 m wide and 3 m tall. What is the area of the garden?
2
Step 1 — Picture the Shape

Think of a big 8 m × 6 m rectangle. Now imagine cutting out a 4 m × 3 m rectangle from the top middle. What's left is the U-shape.
3
Step 2 — Split Into Rectangles

We can split the U into three rectangles: a left column, a bottom bar, and a right column. Or we can think of it as a big rectangle minus the cut-out piece. Let's use the subtraction way because it's quicker here!
4
Step 3 — Find the Big Rectangle Area

Area of the big rectangle = 8 × 6 = 48 m².
5
Step 4 — Find the Cut-Out Area

Area of the cut-out notch = 4 × 3 = 12 m².
6
Step 5 — Subtract

Area of the U-shape = 48 − 12 = 36 m².

Maria's garden covers 36 square meters. Great job — we're done!

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Tip

Subtracting a cut-out piece works too. The area is still additive — you just think of it as the big area minus the missing part.

Section 7

Splitting vs. Subtracting

You've now seen two ways to find the area of a rectilinear shape. Both work! Here's a quick comparison so you know when each method is handy.

	Splitting (Decomposing)	Subtracting
What You Do	Draw lines to break the shape into rectangles. Find each area and add.	Picture a big rectangle around the whole shape. Find its area, then subtract the missing part.
Best For	L-shapes and T-shapes where the split is easy to see.	U-shapes and shapes with a notch cut out of one side.
Number of Calculations	2 or 3 multiplications, then add.	2 multiplications, then subtract.
Common Mistake	Accidentally overlapping two rectangles (counting some area twice).	Forgetting to subtract, so the area is too big.

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Key Takeaway

Both methods give the same answer. It's like building a LEGO wall: you can snap small bricks together (adding), or start with a big flat piece and pop off the parts you don't need (subtracting). Either way, you end up with the same shape!

Section 8

What Comes Next?

Right now you are working with shapes that have only right angles (rectilinear shapes). As you move through math, you'll start finding the area of triangles, trapezoids, and even circles. The big idea stays the same: you can almost always break a tricky shape into simpler shapes you already know how to measure.

	What You Learn Now	What You'll Learn Later
Shapes	Rectangles and rectilinear figures	Triangles, parallelograms, circles
Strategy	Split into rectangles, add areas	Split into triangles and rectangles, use new formulas
Core Idea	Area is additive	Area is still additive! (This never changes.)

So everything you practice today is building a super-strong foundation. The idea that area is additive will be your friend all through school!

Section 9

Practice Problems

Try these five problems on your own. Click "Show Answer" when you're ready to check your work. Good luck!

Question 1: Problem 1 — Thinking Question

Sam says: "I can find the area of any rectilinear shape if I know how to find the area of a rectangle." Is Sam right? Why or why not?

Show answer

Yes, Sam is right! Every rectilinear shape can be split into rectangles because all its sides are straight and all its corners are right angles. You find the area of each rectangle (length × width) and add the areas together. That gives the total area. Nice thinking, Sam!

Question 2: Problem 2 — Basic

An L-shaped rug is made of two rectangles. Rectangle 1 is 5 feet long and 3 feet wide. Rectangle 2 is 4 feet long and 2 feet wide. What is the total area of the rug?

Show answer

Area of Rectangle 1 = 5 × 3 = 15 square feet. Area of Rectangle 2 = 4 × 2 = 8 square feet. Total area = 15 + 8 = 23 square feet. Great job if you got 23!

Question 3: Problem 3 — Intermediate

A T-shaped stage has a top bar that is 10 meters wide and 2 meters tall. The stem below is 3 meters wide and 4 meters tall. What is the total area of the stage?

Show answer

Top bar area = 10 × 2 = 20 m². Stem area = 3 × 4 = 12 m². Total area = 20 + 12 = 32 m². You split the T into two rectangles and added — perfect!

Question 4: Problem 4 — Word Problem

Lily is painting a wall that is shaped like a big rectangle with a window cut out. The wall is 9 feet wide and 7 feet tall. The window is 3 feet wide and 2 feet tall. How many square feet does Lily need to paint?

Show answer

Big wall area = 9 × 7 = 63 square feet. Window area = 3 × 2 = 6 square feet. Lily does NOT paint the window, so we subtract: 63 − 6 = 57 square feet. Remember, subtracting a missing part still uses the additive idea — the total is just the big area minus the hole!

Question 5: Problem 5 — Challenge

Jake splits a rectilinear shape into three rectangles. He finds the areas are 12 cm², 8 cm², and 15 cm². His friend Emma splits the same shape into two rectangles and gets areas of 20 cm² and 15 cm². They both say the total area is 35 cm². Who is correct — Jake, Emma, both, or neither? Explain.

Show answer

Let's check the math. Jake's total = 12 + 8 + 15 = 35 cm². Emma's total = 20 + 15 = 35 cm². They both get 35 cm², so both are correct! This makes sense because area is additive. No matter how you split the shape — into 2 pieces or 3 pieces — the total area stays the same, as long as the pieces don't overlap and nothing is left out. Way to go if you figured this out!

Summary

What We Learned

Summary

Area is the amount of flat space inside a shape, measured in square units. A rectilinear shape is a figure made only of straight sides and right-angle corners — like rectangles stuck together to form an L, T, U, or other step-like shape. The big idea is that area is additive: you can decompose (break apart) any rectilinear figure into non-overlapping rectangles, find the area of each rectangle using length × width, and then add all the areas together to get the total.

You can also find the area by picturing a big rectangle around the whole shape and subtracting the missing parts. Both strategies work because area is additive — no matter how you split or subtract, you get the same total. This idea will help you all through math, even when you meet triangles, circles, and other shapes later on!