Area of Compound Shapes Textbook Exercise - Corbettmaths - Free Printable

Let's solve Question 1 from the image: Work out the area of each of these compound shapes (a) to (f).

We'll break each shape into simpler rectangles, calculate their areas, and then add or subtract as needed.

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🔷 (a)

This shape looks like a rectangle with a smaller rectangle missing from the bottom right.

But actually, it’s better to split it into two rectangles:

- Top rectangle:
Width = 9 cm, Height = 8 cm → Area = 9 × 8 = 72 cm²

- Bottom rectangle:
Width = 7 cm, Height = 5 cm → Area = 7 × 5 = 35 cm²

Wait! But the bottom rectangle is only 7 cm wide, but the top is 9 cm. So the bottom part must be aligned to the right? Let's re-express.

Actually, the total width at the bottom is 7 cm, and the top is 9 cm — so the top overhangs by 1 cm on each side?

No — let’s look carefully:

The figure has:
- Top part: 9 cm wide, 8 cm tall
- Bottom part: 7 cm wide, 5 cm tall
- The vertical gap between them suggests that the bottom rectangle is aligned to the right, and the top is wider.

So we can split the shape into two parts:

✔ Left part:
- A vertical strip: height = 8 + 5 = 13 cm? No — wait, the heights are stacked vertically.

Better approach: Split into two rectangles:

1. Left rectangle:
- Width = 5 cm (from the left edge to where the step is)
- Height = 8 cm (top part) + 5 cm (bottom part) = 13 cm
- Area = 5 × 13 = 65 cm²

2. Right rectangle:
- Width = 7 cm (bottom), but only 5 cm high
- Wait — no. Actually, the right rectangle is only the bottom part: 7 cm wide, 5 cm high → 7 × 5 = 35 cm²

But now we have overlap? No — better to use horizontal split.

Alternative method: Divide into two rectangles:

- Top rectangle: 9 cm × 8 cm = 72 cm²
- Bottom rectangle: 7 cm × 5 cm = 35 cm²

But they don’t overlap — the bottom one is directly below the right portion.

Wait — the total width at the bottom is 7 cm, and the top is 9 cm, so the bottom is shorter, and the top extends beyond.

So the total shape consists of:
- A large rectangle on the top: 9 cm × 8 cm = 72 cm²
- A smaller rectangle attached below the right side: 7 cm × 5 cm = 35 cm²

But is there a gap? Yes — the bottom rectangle is only 7 cm wide, so the top rectangle sticks out 1 cm on each side? No — the bottom is 7 cm wide, and the top is 9 cm wide — so the bottom is centered? Not necessarily.

But looking at the diagram: the bottom rectangle is 7 cm wide, and the top rectangle is 9 cm wide, but the vertical drop is at the left side, suggesting that the bottom rectangle is shifted right.

Let’s reconstruct:

From the diagram:
- The top is 9 cm wide, 8 cm high.
- Then it drops down 5 cm, and the bottom is 7 cm wide.
- The step occurs on the left side.

So the shape is:
- Left side: full height = 8 + 5 = 13 cm, width = 5 cm (since the bottom is 7 cm, and the top is 9 cm, so the overhang is 1 cm on the right?)

Wait — this is ambiguous.

Let’s assume the horizontal lines are aligned.

Better idea: Split the shape into two rectangles:

1. Left rectangle:
- Width = 5 cm (from left edge to the step)
- Height = 8 cm (top) + 5 cm (bottom) = 13 cm
- Area = 5 × 13 = 65 cm²

2. Right rectangle:
- Width = 7 cm (bottom width)
- Height = 5 cm (only the bottom part)
- But wait — the top part is 9 cm wide, so the right rectangle should be 4 cm wide (since 9 − 5 = 4)? No.

Wait — the bottom is 7 cm wide, and the top is 9 cm wide. The step is on the left.

So the bottom starts 2 cm in from the left? Let's think.

Total width of top: 9 cm
Total width of bottom: 7 cm
So if the bottom is centered, it would be offset by 1 cm on each side.

But the diagram shows a step on the left, meaning the bottom rectangle is shifted right.

So:
- The top rectangle is 9 cm wide, 8 cm high
- The bottom rectangle is 7 cm wide, 5 cm high, and it sits under the right 7 cm of the top rectangle

Therefore, the overlap is 7 cm wide.

So the total area = area of top + area of bottom = 9×8 + 7×5 = 72 + 35 = 107 cm²

✔ Answer for (a): 107 cm²

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🔷 (b)

This is a T-shape.

We can split it into:
- Top rectangle: 25 cm × 12 cm → but wait, the height is 12 cm total, and the stem is 10 cm high.

Actually:
- The top bar is 25 cm wide, and the height from top to the start of the stem is 12 cm? No — the total height is 12 cm, and the stem is 10 cm high? That doesn't make sense.

Wait — the total height is 12 cm, and the stem is 10 cm high? Then the top bar is 12 − 10 = 2 cm high?

Yes — likely.

So:
- Top bar: 25 cm × 2 cm = 50 cm²
- Stem: 6 cm wide, 10 cm high → 6 × 10 = 60 cm²

Total area = 50 + 60 = 110 cm²

✔ Answer for (b): 110 cm²

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🔷 (c)

This is an L-shaped figure.

We can split it into two rectangles:

Option 1: Vertical split

- Left rectangle: width = 13 − 4 = 9 cm? Wait — total width is 13 cm, top part is 4 cm wide, so bottom is 13 cm wide, but height is only 3 cm.

Actually:
- Bottom rectangle: 13 cm wide, 3 cm high → area = 13 × 3 = 39 cm²
- Top rectangle: 4 cm wide, height = 10 − 3 = 7 cm → area = 4 × 7 = 28 cm²

Total area = 39 + 28 = 67 cm²

✔ Answer for (c): 67 cm²

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🔷 (d)

This is a step-up shape.

We can split it into three rectangles, or two.

Better: split into:
- Bottom rectangle: 9 cm wide, 3 cm high → 9 × 3 = 27 cm²
- Middle rectangle: 4 cm wide, 5 cm high → 4 × 5 = 20 cm²
- Top rectangle: 2 cm wide, 5 cm high → 2 × 5 = 10 cm²

Wait — but the middle and top are stacked.

Actually:
- The bottom is 9 cm × 3 cm = 27 cm²
- On top of it, there's a middle rectangle: 4 cm wide, 5 cm high → but wait — the height is 5 cm, and the bottom is already 3 cm high, so total height is 8 cm? But the figure shows the top is 5 cm high above the bottom.

Wait — the total height is not given — the middle is 5 cm high, and the top is 2 cm wide, 5 cm high.

But the middle is 4 cm wide, and the top is 2 cm wide, both sitting on the same level.

So:
- Bottom rectangle: 9 cm × 3 cm = 27 cm²
- Middle rectangle: 4 cm × 5 cm = 20 cm²
- Top rectangle: 2 cm × 5 cm = 10 cm²

But are they all connected? Yes.

Total area = 27 + 20 + 10 = 57 cm²

Wait — but the middle and top are both 5 cm high — so they are stacked? No — the middle is 4 cm wide, and the top is 2 cm wide, both on top of the bottom.

But the middle is 4 cm wide, and the top is 2 cm wide — and they are placed on top of the bottom.

But the top is narrower than the middle.

So:
- Bottom: 9 cm × 3 cm = 27 cm²
- Middle: 4 cm × 5 cm = 20 cm²
- Top: 2 cm × 5 cm = 10 cm²

But the middle and top are on top of the bottom, so total height is 3 + 5 = 8 cm.

Is the top sitting on the middle? Yes — it’s a step.

So the top is on top of the middle, which is on top of the bottom.

So the middle is 4 cm wide, 5 cm high → area = 20 cm²
The top is 2 cm wide, 5 cm high → area = 10 cm²
But the top is only 2 cm wide, so it sits on the right side of the middle?

But the total width is 9 cm, and the bottom is 9 cm wide.

So the middle is 4 cm wide, probably centered? Or left-aligned?

But the top is 2 cm wide — and it’s shown above the middle.

But the middle is 4 cm wide, and the top is 2 cm wide — so the top is only half the width.

But the bottom is 9 cm wide, so the middle must be 4 cm wide, and the top is 2 cm wide, sitting on top of it.

So total area = bottom + middle + top = 27 + 20 + 10 = 57 cm²

✔ Answer for (d): 57 cm²

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🔷 (e)

This is a rectangle with two notches.

We can compute:
- Full rectangle: 24 cm × 20 cm = 480 cm²
- Subtract the two notches

Each notch:
- Width = 6 cm, depth = 8 cm → area = 6 × 8 = 48 cm²
- There are two such notches → 2 × 48 = 96 cm²

But wait — the last notch is only 2 cm wide? Let’s check.

The bottom has:
- Left notch: 6 cm
- Middle notch: 6 cm
- Right notch: 2 cm

And the depth of each notch is 8 cm.

So:
- First notch: 6 × 8 = 48 cm²
- Second notch: 6 × 8 = 48 cm²
- Third notch: 2 × 8 = 16 cm²

Total removed = 48 + 48 + 16 = 112 cm²

So area = 480 − 112 = 368 cm²

✔ Answer for (e): 368 cm²

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🔷 (f)

This is a rectangle with a rectangular hole in the middle.

We can:
- Compute full rectangle: 8 cm × 7 cm = 56 cm²
- Subtract the cut-out part

The cut-out:
- Width = 3 cm
- Height = 3 cm (since it goes up 3 cm from the bottom)

So area of cut-out = 3 × 3 = 9 cm²

But wait — the cut-out is not a full rectangle — it's inset.

Looking at the diagram:
- The main rectangle is 8 cm wide, 7 cm high
- Inside, there is a rectangular cut-out that is 3 cm wide and 3 cm high
- It is positioned with 2 cm on the left, then 3 cm cut-out, then 3 cm on the right? Let’s see.

The base:
- Left: 2 cm
- Cut-out: 3 cm
- Right: 3 cm → total = 2 + 3 + 3 = 8 cm → correct.

So the cut-out is 3 cm wide and 3 cm high.

So area of cut-out = 3 × 3 = 9 cm²

But the height of the cut-out is 3 cm, and the total height is 7 cm — so it's only from the bottom.

So the area of the shape = area of big rectangle − area of cut-out = 8×7 − 3×3 = 56 − 9 = 47 cm²

✔ Answer for (f): 47 cm²

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✔ Final Answers:

(a) 107 cm²
(b) 110 cm²
(c) 67 cm²
(d) 57 cm²
(e) 368 cm²
(f) 47 cm²

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Let me know if you'd like diagrams or step-by-step drawings!