Decomposing Polygons to Find Area worksheet for calculating total area by breaking shapes into smaller rectangles and triangles.

Let's solve each of the four polygon area problems by decomposing them into simpler shapes (like rectangles and triangles), calculating their areas, and then adding them together.

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🔷 Problem 1

Shape: An irregular shape that looks like a rectangle with a smaller rectangle removed from the top-left corner.

Given dimensions:
- Total width = 10 in
- Height on left side = 6 in
- The "notch" at the top has a height of 3 in and a width of 3 in (since it's missing from the top-left)

But actually, let's re-analyze:

Looking closely:
- The full bottom is 10 in.
- The right side is 3 in tall.
- The left side is 6 in tall.
- The top horizontal segment is only 3 in long (on the left), and then there's a gap before the rest.

So we can split this into two rectangles:

#### ✔ Option: Decompose into two rectangles

1. Bottom rectangle:
- Width = 10 in
- Height = 3 in
- Area = $10 \times 3 = 30$ in²

2. Top-left rectangle:
- Width = 3 in (from the left)
- Height = 6 in – 3 in = 3 in (since the total height is 6 in, and bottom part is 3 in)
- Wait — no, the total height on the left is 6 in, but the bottom rectangle is 3 in high, so the top rectangle must be 6 in high? That doesn’t fit.

Wait — let’s draw it mentally.

Actually, better to interpret as:

The figure has:
- A large rectangle on the bottom: 10 in wide × 3 in high → area = $10 \times 3 = 30$
- A smaller rectangle on top left: 3 in wide × 3 in high → because the total height is 6 in, and bottom is 3 in, so top is also 3 in?

But wait — the total height on the left is 6 in, and the bottom part is 3 in, so the top part must be 3 in tall. But the top extends only 3 in across.

So yes:
- Bottom rectangle: 10 in × 3 in = 30 in²
- Top rectangle: 3 in × 3 in = 9 in²

Total area = $30 + 9 = \boxed{39}$ in²

✔ Answer for #1: 39 in²

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🔷 Problem 2

Shape: A trapezoid or a rectangle with a triangle cut off.

Given:
- Left vertical side = 12 cm
- Right vertical side = 10 cm
- Base = 22 cm

This is a trapezoid, but we can decompose it.

Alternatively, think of it as:
- A rectangle of height 10 cm and width 22 cm
- Plus a triangle on top with base 22 cm and height = 12 - 10 = 2 cm

Wait — but the slanted side goes from 12 cm down to 10 cm.

Actually, the shape has:
- A rectangle of height 10 cm and width 22 cm
- On top of that, a triangle with:
- Base = 22 cm
- Height = 12 - 10 = 2 cm

Wait — but is that correct? Let's see:

If the left side is 12 cm tall and the right side is 10 cm tall, and the base is 22 cm, then the shape is a trapezoid.

We can use the trapezoid area formula:
$$
\text{Area} = \frac{1}{2} \times (b_1 + b_2) \times h
$$
But here, the two parallel sides are the top and bottom? Wait — no.

Actually, the vertical sides are not parallel unless it's a rectangle.

Wait — the shape appears to have:
- Bottom base = 22 cm
- Left side = 12 cm (vertical)
- Right side = 10 cm (vertical)
- Top is slanted

Wait — this seems inconsistent.

Let me reinterpret:

Looking at the diagram:
- It's a right trapezoid with:
- One vertical leg = 12 cm (left)
- Another vertical leg = 10 cm (right)
- Bottom base = 22 cm
- Top is slanted

So, the height of the trapezoid is the distance between the two vertical sides — but they're both vertical, so the horizontal distance is 22 cm.

Wait — no. Actually, the bases are the horizontal sides.

But the top is slanted, so perhaps the bottom is 22 cm, and the top is shorter?

Wait — maybe the height of the trapezoid is not given directly.

Wait — actually, looking at the labels:

- Bottom: 22 cm
- Left side: 12 cm
- Right side: 10 cm
- And the top is slanted from the top of the 12 cm side to the top of the 10 cm side

So this is a trapezoid with:
- Two parallel vertical sides? No — they’re both vertical, so they are parallel.

Wait — if both left and right sides are vertical, then the shape has two vertical sides: 12 cm and 10 cm, and the bottom is 22 cm, and the top connects the tops of these two verticals.

So the height of the trapezoid is the horizontal distance between the two vertical sides — which is 22 cm.

But then the bases are the vertical sides? No — bases are the parallel sides.

In a trapezoid, the bases are the two parallel sides.

Here, the left and right sides are both vertical, so they are parallel.

So the two bases are:
- Left base = 12 cm
- Right base = 10 cm
- Distance between them (the "height" of the trapezoid) = 22 cm

Wait — but that would mean the height is horizontal, and the bases are vertical.

That’s unusual, but mathematically possible.

But usually, we think of the bases as the horizontal ones.

Wait — actually, the bottom is labeled 22 cm — so that's a horizontal base.

And the top is slanted.

So the vertical sides are not the bases.

Let me reconsider.

Perhaps it's a quadrilateral with:
- Bottom: 22 cm
- Left side: 12 cm (vertical)
- Right side: 10 cm (vertical)
- Top: slanted from top of left to top of right

So this is a trapezoid with:
- Two parallel sides: the left and right sides are both vertical → so they are parallel.
- So the bases are the vertical sides: 12 cm and 10 cm
- The distance between them (the horizontal length) is 22 cm

So area of a trapezoid:
$$
A = \frac{1}{2} \times (b_1 + b_2) \times h
$$
where $b_1 = 12$, $b_2 = 10$, $h = 22$

So:
$$
A = \frac{1}{2} \times (12 + 10) \times 22 = \frac{1}{2} \times 22 \times 22 = 11 \times 22 = 242 \text{ cm}^2
$$

But wait — that seems large. Is that correct?

Wait — no! If the bases are the vertical sides, then the height of the trapezoid is the horizontal distance between them — which is 22 cm.

Yes — so the area is:
$$
\frac{1}{2} \times (12 + 10) \times 22 = 11 \times 22 = 242 \text{ cm}^2
$$

But let’s check with decomposition.

Alternative approach: decompose into a rectangle and a triangle.

Imagine:
- Draw a horizontal line from the top of the 10 cm side to the left, making a rectangle of:
- Width = 22 cm
- Height = 10 cm
- Area = $22 \times 10 = 220$ cm²

Then, above that, there's a triangle on the left:
- Base = 22 cm (same width)
- Height = 12 - 10 = 2 cm
- Area = $\frac{1}{2} \times 22 \times 2 = 22$ cm²

Wait — but that triangle would extend across the whole width, but the left side is only 12 cm, so the extra 2 cm is only on the left?

No — the top is slanted, so the difference in height is 2 cm over 22 cm.

So the shape is a trapezoid with:
- Two parallel vertical sides: 12 cm and 10 cm
- Horizontal distance between them: 22 cm

So area = average of the two vertical sides × horizontal width
= $\frac{12 + 10}{2} \times 22 = 11 \times 22 = 242$ cm²

✔ Answer for #2: 242 cm²

But let’s double-check: is the height of the trapezoid the horizontal distance? Yes — since the two bases are vertical, the perpendicular distance between them is horizontal.

So yes, area = $\frac{1}{2}(b_1 + b_2) \times h = \frac{1}{2}(12+10)\times22 = 242$

✔ Answer: 242 cm²

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🔷 Problem 3

Shape: A complex shape with multiple steps.

Dimensions:
- Overall width = 20 ft
- Heights: 6 ft, 11 ft, 6 ft, 5 ft

It looks like a step-like shape.

Let’s break it into rectangles.

From the bottom up:

We can split it into three rectangles:

1. Bottom rectangle:
- Width = 20 ft
- Height = 5 ft
- Area = $20 \times 5 = 100$ ft²

2. Middle rectangle:
- Width = 20 ft
- Height = 6 ft (but wait — the middle section is only 11 ft high, and the bottom is 5 ft, so the next layer is 6 ft? Wait — look at labels.)

Wait — labels:
- From bottom: 5 ft
- Then up: 6 ft
- Then up: 11 ft
- Then up: 6 ft

Wait — no — it's likely:
- The shape has:
- A bottom rectangle: 20 ft wide × 5 ft high
- Then a middle rectangle: 20 ft wide × 6 ft high
- Then a top rectangle: 11 ft wide × 6 ft high

Wait — but the total height on the left is 6 + 5 = 11? Or is it 6 ft + 11 ft?

Wait — the label says:
- On the left: 6 ft, then 11 ft, then 6 ft
- On the right: 5 ft, then 6 ft

So likely:

- The shape has three horizontal levels:
1. Bottom level: height = 5 ft, width = 20 ft
2. Middle level: height = 6 ft, width = 20 ft
3. Top level: height = 6 ft, width = 11 ft

Wait — but the 11 ft is the width of the top rectangle.

Yes — the top rectangle is 11 ft wide and 6 ft high.

So:
- Rectangle 1 (bottom): $20 \times 5 = 100$ ft²
- Rectangle 2 (middle): $20 \times 6 = 120$ ft²
- Rectangle 3 (top): $11 \times 6 = 66$ ft²

Total area = $100 + 120 + 66 = \boxed{286}$ ft²

✔ Answer for #3: 286 ft²

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🔷 Problem 4

Shape: A pentagon-like shape with a right triangle on the right.

Dimensions:
- Left side: 13 cm
- Right side: 8 cm
- Bottom: 15 cm
- Small rectangle on right: 4 cm wide and 8 cm high

Wait — the figure shows:
- A large rectangle on the left: 13 cm high, but width unknown
- A small rectangle on the right: 4 cm wide, 8 cm high
- The total bottom is 15 cm

So:
- The total width = 15 cm
- The right part is 4 cm wide
- So the left part is $15 - 4 = 11$ cm wide

Now, the height of the left part is 13 cm, but the right part is only 8 cm high.

So the top is slanted — forming a triangle on top of the right rectangle.

Wait — or is the left part a rectangle of 11 cm wide × 13 cm high?

And the right part is a rectangle of 4 cm wide × 8 cm high?

But then the top is slanted — so the top edge goes from the top of the left (13 cm) to the top of the right (8 cm).

So the shape is:
- A rectangle on the left: 11 cm × 13 cm
- A rectangle on the right: 4 cm × 8 cm
- But the top is slanted — so actually, the right side is only 8 cm high, while the left is 13 cm.

Wait — but the total height on the left is 13 cm, and on the right is 8 cm.

So the shape is a trapezoid or a polygon that can be split into:
- A rectangle of 11 cm × 8 cm (common height)
- A triangle on top of that, with base 11 cm and height 13 - 8 = 5 cm
- And a rectangle on the right: 4 cm × 8 cm

Wait — no — the right side is only 8 cm high, and the left is 13 cm high.

So better:

Split the shape into:
1. Rectangle on the right: 4 cm wide × 8 cm high → area = $4 \times 8 = 32$ cm²
2. Left part: 11 cm wide × 8 cm high → rectangle = $11 \times 8 = 88$ cm²
3. Triangle on top of the left part: base = 11 cm, height = 13 - 8 = 5 cm → area = $\frac{1}{2} \times 11 \times 5 = 27.5$ cm²

Total area = $32 + 88 + 27.5 = \boxed{147.5}$ cm²

Alternatively, think of the whole shape as:
- A large rectangle of 15 cm × 8 cm = 120 cm²
- Plus a triangle on top of the left 11 cm portion: base 11 cm, height 5 cm → 27.5 cm²
- Plus a rectangle on the right: already included?

Wait — no: the right side is only 8 cm high, so the 15 cm × 8 cm rectangle includes the right 4 cm and the left 11 cm.

Then the extra is the triangle on top of the left 11 cm portion.

So:
- Base rectangle: $15 \times 8 = 120$ cm²
- Top triangle: base = 11 cm, height = 5 cm → $\frac{1}{2} \times 11 \times 5 = 27.5$ cm²
- Total = $120 + 27.5 = 147.5$ cm²

✔ Answer for #4: 147.5 cm²

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✔ Final Answers:

1. 39 in²
2. 242 cm²
3. 286 ft²
4. 147.5 cm²

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Let me know if you'd like diagrams or further explanation!