Area of Compound Shapes Textbook Exercise - Corbettmaths - Free Printable

Let's solve Question 1 from the image:
We are asked to find the area of each compound shape (a) through (f). Each shape is made up of rectangles, so we can break them down into simpler rectangles, calculate their areas, and then add or subtract as needed.

---

🔷 Strategy:

Break each compound shape into rectangles, calculate the area of each rectangle using:
> Area = length × width
Then sum the areas.

---

✔ (a)

This shape looks like a "step" or "L" shape. We can 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

But wait — the bottom rectangle overlaps with the top one? Let’s be careful.

Actually, the total height is 8 cm, and the bottom part is 5 cm high, so the top rectangle is only 8 cm high, but its width is 9 cm, and the bottom rectangle is 7 cm wide, sitting below.

But the bottom rectangle is only 7 cm wide, and the top rectangle is 9 cm wide, so there's a gap on the right side?

Wait — let's look carefully:

From the diagram:
- The left side has a vertical drop from 8 cm to 5 cm.
- The bottom rectangle is 7 cm wide and 5 cm high.
- The top rectangle is 9 cm wide and (8 - 5) = 3 cm high? No — that doesn't match.

Wait — actually, the vertical line at the left goes from 0 to 8 cm, then drops to 5 cm at the bottom. So the height of the top part is 8 cm, and the bottom part is 5 cm high, but they are aligned horizontally?

Let’s reconstruct:

Actually, the shape is like this:

- A large rectangle on the left:
Width = 5 cm, Height = 8 cm → Area = 5 × 8 = 40 cm²

- A smaller rectangle on the right:
Width = 7 cm, Height = 5 cm → Area = 7 × 5 = 35 cm²

But wait — the total width is 9 cm, and the left part is 5 cm, so the right part must be 9 − 5 = 4 cm? But it says 7 cm.

Wait — the bottom part is 7 cm wide, and the top part is 9 cm wide. That means the top rectangle extends beyond the bottom one.

So better way:

Split into:
1. Top rectangle: 9 cm wide × (8 − 5) = 3 cm high → Area = 9 × 3 = 27 cm²
2. Bottom rectangle: 7 cm wide × 5 cm high → Area = 7 × 5 = 35 cm²

Total area = 27 + 35 = 62 cm²

✔ Answer (a): 62 cm²

---

✔ (b)

This is a T-shaped figure.

We can split it into:
- Top rectangle: 25 cm wide, 12 cm high → Area = 25 × 12 = 300 cm²
- Bottom rectangle: 6 cm wide, (12 − 10) = 2 cm high? Wait — no.

Wait: The total height is 12 cm, and the middle section is 10 cm high, so the top rectangle is 2 cm tall?

No — let’s read carefully:

- The top part is 12 cm high, but the bottom stem is 10 cm high? No — the total height is 12 cm, and the stem is 10 cm high? That would make the top 2 cm.

But the width of the stem is 6 cm, and the top is 25 cm.

So:
- Top rectangle: 25 cm × (12 − 10) = 25 × 2 = 50 cm²
- Stem rectangle: 6 cm × 10 cm = 60 cm²

Total area = 50 + 60 = 110 cm²

✔ Answer (b): 110 cm²

---

✔ (c)

This is an L-shape with:
- Total height = 10 cm
- Right side: 4 cm wide, 10 cm high → Rectangle: 4 × 10 = 40 cm²
- Bottom part: 13 cm wide, 3 cm high → But wait, the total width is 13 cm, and the right part is 4 cm wide, so the bottom left part is 13 − 4 = 9 cm wide?

But the bottom rectangle is only 3 cm high, and the right rectangle is 10 cm high, so the bottom rectangle is 13 cm wide × 3 cm high = 39 cm²

But does it overlap? Yes — the right rectangle includes the top 7 cm of the right side (since 10 − 3 = 7), and the bottom rectangle covers the bottom 3 cm across the full width.

But the bottom rectangle is 13 cm wide and 3 cm high → Area = 13 × 3 = 39 cm²

The right rectangle is 4 cm wide and 10 cm high → Area = 4 × 10 = 40 cm²

But they overlap in a 4 cm × 3 cm region → Overlap area = 4 × 3 = 12 cm²

So total area = 39 + 40 − 12 = 67 cm²

Alternatively, split differently:

- Left bottom: 9 cm wide (13 − 4), 3 cm high → 9 × 3 = 27 cm²
- Right full: 4 cm × 10 cm = 40 cm²
- Total = 27 + 40 = 67 cm²

✔ Answer (c): 67 cm²

---

✔ (d)

This is a step-like shape, like a "reverse L".

We can split it into:
- Bottom rectangle: 9 cm wide, 3 cm high → Area = 9 × 3 = 27 cm²
- Middle rectangle: 4 cm wide, 5 cm high → But wait — the height is 5 cm above the base, and the bottom is 3 cm, so the middle is 5 cm high?
Actually, the total height is 5 cm (from base to top), but the bottom is 3 cm, so the top rectangle is 5 cm high?

Wait — the top rectangle is 2 cm wide and 5 cm high? But it's sitting on top of a 4 cm wide rectangle?

Wait — let's read:

- Bottom: 9 cm wide, 3 cm high → Area = 9 × 3 = 27 cm²
- Middle: 4 cm wide, 5 cm high? But the total height is 5 cm, and the bottom is 3 cm, so the middle part is 5 cm high? That would make total height 3 + 5 = 8 cm? But the label says 5 cm.

Wait — the vertical height from base to top is 5 cm, and the bottom rectangle is 3 cm high, so the top rectangle is 5 − 3 = 2 cm high?

But the top rectangle is labeled 2 cm wide and 5 cm high? That can’t be.

Wait — the label says:
- Top rectangle: 2 cm wide, and the vertical line shows 5 cm from base to top.

But the bottom rectangle is 3 cm high, so the top rectangle must be 2 cm high (5 − 3).

But the width of the top rectangle is 2 cm.

And the middle rectangle is 4 cm wide, but how high?

Wait — perhaps:

- Bottom rectangle: 9 cm wide, 3 cm high → Area = 9 × 3 = 27 cm²
- Middle rectangle: 4 cm wide, 2 cm high → Because the top is 5 cm high, bottom is 3 cm, so the middle step is 2 cm high → Area = 4 × 2 = 8 cm²
- Top rectangle: 2 cm wide, 2 cm high? But the top is only 2 cm wide, and height is 2 cm → Area = 2 × 2 = 4 cm²

Wait — but the top rectangle is only 2 cm wide, and sits on top of the 4 cm rectangle?

But the total width at the top is 2 cm, and the middle is 4 cm, and the bottom is 9 cm.

But the heights:
- From base to bottom of top: 3 cm
- Then from 3 cm to 5 cm: 2 cm

So:
- Bottom rectangle: 9 cm × 3 cm = 27 cm²
- Middle rectangle: 4 cm × 2 cm = 8 cm²
- Top rectangle: 2 cm × 2 cm = 4 cm²

Total = 27 + 8 + 4 = 39 cm²

But wait — the top rectangle is 2 cm wide, but where is it placed? It’s centered? Or on the right?

Looking at the diagram: The top is 2 cm wide, and the middle is 4 cm wide, and the bottom is 9 cm wide.

But the top sits on top of the middle, so the top rectangle is 2 cm wide and 2 cm high.

So yes.

✔ Answer (d): 39 cm²

---

✔ (e)

This is a rectangle with three rectangular holes cut out.

We can compute:
- Large rectangle: 24 cm wide, 20 cm high → Area = 24 × 20 = 480 cm²
- Cut-out sections: Three identical rectangles? Let’s see.

Each cut-out has:
- Width: 6 cm, 6 cm, and 2 cm → Wait, the bottom has:
- Left: 6 cm
- Middle: 6 cm
- Right: 2 cm
- Total = 6 + 6 + 2 = 14 cm

But the total width is 24 cm, so the cut-outs are not spanning the whole width.

Each cut-out is 8 cm high, because the height of the rectangle is 20 cm, and the cut-out depth is 8 cm.

So each cut-out is 8 cm high, and widths:
- First: 6 cm → Area = 6 × 8 = 48 cm²
- Second: 6 cm → 6 × 8 = 48 cm²
- Third: 2 cm → 2 × 8 = 16 cm²

Total cut-out area = 48 + 48 + 16 = 112 cm²

So final area = 480 − 112 = 368 cm²

✔ Answer (e): 368 cm²

---

✔ (f)

This is a rectangle with a U-shaped notch.

- Outer rectangle: 8 cm wide, 7 cm high → Area = 8 × 7 = 56 cm²
- Notch: A rectangle missing from the center
- Width: 3 cm (labeled)
- Height: 3 cm (labeled)
- So area removed = 3 × 3 = 9 cm²

But wait — the notch is not a full rectangle. It's only 3 cm wide, and the depth is 3 cm, but the top of the notch is open.

Wait — the notch is a rectangle cut out from the bottom, but it's only 3 cm wide and 3 cm deep?

But the horizontal lines show:
- The notch has a width of 3 cm, and height of 3 cm, so area = 3 × 3 = 9 cm²

But is that correct?

Wait — the outer rectangle is 8 cm wide, 7 cm high.

The notch is in the middle, and the bottom of the notch is 3 cm high, and the width is 3 cm.

So yes — it’s a rectangle of 3 cm × 3 cm removed.

So area = 56 − 9 = 47 cm²

But wait — the notch has two sides? No — it’s a single rectangular hole.

Wait — the diagram shows:
- Two vertical lines on the sides of the notch: 2 cm and 3 cm, so total width = 2 + 3 + 3 = 8 cm? Wait.

Wait — the notch has:
- Left side: 2 cm
- Middle gap: 3 cm
- Right side: 3 cm
- Total width = 2 + 3 + 3 = 8 cm → matches outer width.

But the height of the notch is 3 cm, and it’s cut out, so the missing area is a rectangle of 3 cm wide × 3 cm high = 9 cm²

So total area = 8 × 7 = 56 − 9 = 47 cm²

✔ Answer (f): 47 cm²

---

✔ Final Answers:

| Shape | Area |
|-------|------|
| (a) | 62 cm² |
| (b) | 110 cm² |
| (c) | 67 cm² |
| (d) | 39 cm² |
| (e) | 368 cm² |
| (f) | 47 cm² |

---

📌 Summary:

To find the area of compound shapes:
1. Break the shape into simple rectangles.
2. Calculate area of each.
3. Add for combined shapes.
4. Subtract for holes or notches.

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