Grade 6 Area Worksheets | Find the Area of Compound Shapes - Free Printable

Let's solve each of the three area problems step by step. These are composite shapes (made up of rectangles), so we'll break them down into simpler rectangles, find their areas, and then add or subtract as needed.

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Problem 1

Shape: L-shaped figure
Given dimensions:
- Total width = 12 m
- Height on left = 8 m
- Right section height = 3 m
- The horizontal segment at the top is 8 m

We can split this shape into two rectangles:

#### Rectangle A (left part):
- Width = 12 m - 8 m = 4 m (since the right rectangle is 8 m wide)
- Height = 8 m
- Area = 4 × 8 = 32 m²

Wait! Let’s re-analyze carefully.

Actually, better to divide it differently:

#### Option: Split into two rectangles:

1. Top rectangle (horizontal):
- Length = 8 m
- Height = 8 m - 3 m = 5 m
- Area = 8 × 5 = 40 m²

2. Bottom rectangle (right side):
- Length = 12 m
- Height = 3 m
- Area = 12 × 3 = 36 m²

But wait — if we do that, the top rectangle is only 8 m long, but the bottom is 12 m long. So the total width is 12 m, and the top rectangle is only 8 m across — meaning there’s a "step" down.

Better way:

Split vertically:
- Left rectangle: 4 m wide (12 - 8) and 8 m tall → 4 × 8 = 32 m²
- Right rectangle: 8 m wide and 3 m tall → 8 × 3 = 24 m²

Wait — no. The total width is 12 m, and the top part goes 8 m in from the left, then drops down. So:

Let’s draw mentally:
- From the left: a rectangle that is 4 m wide and 8 m tall (the full height).
- Then a rectangle to the right: 8 m wide and 3 m tall (bottom part).

So yes:
- Left rectangle: 4 m × 8 m = 32 m²
- Right rectangle: 8 m × 3 m = 24 m²

Total area = 32 + 24 = 56 m²

✔ Answer for Problem 1: 56 m²

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Problem 2

Shape: Rectangle with a smaller rectangle removed from the top-right corner.

Dimensions:
- Outer rectangle: 20 cm high, 18 cm wide
- Cut-out: 10 cm high, 5 cm wide

So:
- Full area without cut = 20 × 18 = 360 cm²
- Area of cut-out = 10 × 5 = 50 cm²
- Subtract: 360 - 50 = 310 cm²

✔ Answer for Problem 2: 310 cm²

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Problem 3

Shape: Large rectangle with a rectangular hole in the middle.

Dimensions:
- Outer rectangle: 14 m wide, 10 m high
- Inner cut-out: 5 m wide, 4 m high

But note: the cut-out is not centered — it's from the top, and the sides are different heights.

Let’s analyze:

The shape has:
- Left side: 10 m high
- Right side: 8 m high
- Bottom: 14 m wide
- There's a rectangular notch in the middle, 5 m wide and 4 m high

But actually, the cut-out is inside, so we need to find the area of the outer rectangle minus the area of the missing rectangle.

But wait — is it a single rectangle with a hole?

Looking at the diagram:
- The base is 14 m.
- Left side is 10 m high, right side is 8 m high.
- There is a rectangle removed from the center, 5 m wide and 4 m high.

But the height of the cut-out is 4 m, so it goes from the top down 4 m.

So the total area is:

We can compute as:
- Area of full rectangle if flat: 14 × 10 = 140 m²
- But the right side is only 8 m high, so the top-right portion is missing? Wait — no.

Actually, the figure is like a large rectangle with a rectangular notch taken out from the top center.

Let’s re-express:

From the diagram:
- The entire base is 14 m.
- The height on the left is 10 m.
- The height on the right is 8 m.
- There is a rectangular gap of 5 m width and 4 m depth in the center.

Wait — perhaps better to split the shape into parts.

Alternative approach: Divide into three rectangles

Let’s divide the shape into:
1. Left rectangle
2. Middle rectangle (below the notch)
3. Right rectangle

But the notch is 5 m wide and 4 m deep, so it removes a 5×4 rectangle from the top.

But the total height varies.

Wait — let's look at the vertical lines:

- The total width is 14 m.
- There is a notch in the center: 5 m wide, and 4 m high (so it's missing from the top).
- The left and right parts go all the way up to 10 m and 8 m respectively?

Wait — actually, looking at the diagram:

- The top of the shape is flat except for a dip in the center.
- The left side is 10 m high.
- The right side is 8 m high.
- The notch is 5 m wide and 4 m high (from the top down).

But the total height on the left is 10 m, on the right is 8 m.

So the shape is like a rectangle with a rectangular hole in the center, but the hole is only 4 m high.

But the hole is only 4 m high, so it's not going all the way down.

So the shape is:

- A large rectangle of 14 m × 10 m, but the right side is only 8 m high, so we have to adjust.

Wait — no. The base is 14 m, and the height on the left is 10 m, on the right is 8 m.

But the notch is in the middle, 5 m wide and 4 m high, so it cuts into the top.

But since the left is 10 m high and right is 8 m high, the top edge is not straight.

Actually, the best way is to divide the shape into three parts:

Let’s assume the notch is centered and removes a 5 m × 4 m rectangle from the top.

But the total height is 10 m on the left, 8 m on the right.

Wait — perhaps the outer boundary is:

- The bottom is 14 m.
- The left side goes up 10 m.
- The right side goes up 8 m.
- The top has a notch in the center: a 5 m wide section missing from the top, 4 m deep.

So the top surface is like a shelf: 14 m wide, but with a 5 m wide dip of 4 m.

But the heights are measured from the bottom.

So the full height is 10 m on the left, 8 m on the right.

But the notch is 5 m wide and 4 m high — so it's a rectangular hole in the top.

So the shape is:

- A large rectangle: 14 m wide × 10 m high → area = 140 m²
- But the right side is only 8 m high, so we subtract a rectangle of 14 m × (10 - 8) = 14 × 2 = 28 m²? No — that would be wrong because the notch is only in the center.

Wait — the diagram shows:

- The notch is 5 m wide, and 4 m high — so it's a rectangle of 5 m × 4 m missing from the top.

But also, the right side is shorter — only 8 m, while left is 10 m.

So the shape has:
- A left part: 5 m wide? Not necessarily.

Wait — let's use coordinates.

Let’s label the points:

Assume the bottom-left is (0,0). Then:

- Bottom: from (0,0) to (14,0)
- Left side: up to (0,10)
- Top-left: (0,10)
- Then the top goes to some point, dips down, then up.

But the notch is 5 m wide, and 4 m high, so it's a rectangle of 5×4 missing from the top.

Also, the right side is only 8 m high, so from (14,0) to (14,8)

And the top is flat except for a 5 m wide dip of 4 m.

So the top edge is:
- From (0,10) to (x,10), then down to (x+5,6), then up to (y,10), then to (14,8)? This is getting messy.

Alternatively, the notch is 5 m wide and 4 m high, so it's a rectangle of 5×4 removed from the top.

But the overall height on the left is 10 m, on the right is 8 m.

So the shape consists of:

1. A rectangle on the left: 5 m wide, 10 m high → area = 5 × 10 = 50 m²
2. A rectangle on the right: 5 m wide, 8 m high → area = 5 × 8 = 40 m²
3. A middle rectangle: 4 m wide, 6 m high? Wait — no.

Wait — total width is 14 m.

Let’s see: the notch is 5 m wide, so the remaining width is 14 - 5 = 9 m.

But how is it distributed?

From the diagram:
- The left side is 5 m wide? No — the notch is 5 m wide, but where?

Actually, the notch is centered? Or not?

Look at the labels:
- On the left: 5 m — probably the width of the left rectangle?
- Then the notch is 5 m wide
- Then the right side is 4 m? But the total is 14 m.

Wait — the bottom is labeled 14 m, and the notch is 5 m wide, and the left and right parts are shown.

From the diagram:
- Left rectangle: 5 m wide, 10 m high
- Middle notch: 5 m wide, 4 m high (missing)
- Right rectangle: 4 m wide, 8 m high

Wait — 5 + 5 + 4 = 14 m → yes!

So:
- Left rectangle: 5 m × 10 m = 50 m²
- Right rectangle: 4 m × 8 m = 32 m²
- But the middle is not a rectangle — it's a gap of 5 m wide and 4 m high.

But the middle area between left and right is 5 m wide, but only 6 m high (since the notch is 4 m deep, and total height on left is 10 m, so the middle is 10 m high? No.

Wait — confusion.

Actually, the notch is a rectangular hole of 5 m × 4 m cut out from the top.

But the bottom of the notch is at 6 m above ground (since it's 4 m high), and the top is at 10 m.

But the right side is only 8 m high.

So the shape is:

- The left part: 5 m wide, 10 m high → area = 50 m²
- The middle part: 5 m wide, but only 6 m high (because the notch is 4 m deep, so from 0 to 6 m) → area = 5 × 6 = 30 m²
- The right part: 4 m wide, 8 m high → area = 4 × 8 = 32 m²

Wait — but the notch is only 5 m wide and 4 m high, so the middle is not a solid rectangle — it's missing the top 4 m.

So the middle section is only 6 m high, not 10 m.

Yes.

So:
- Left: 5 m × 10 m = 50 m²
- Middle: 5 m × 6 m = 30 m²
- Right: 4 m × 8 m = 32 m²

Total area = 50 + 30 + 32 = 112 m²

But wait — is the middle connected? Yes.

Alternatively, think of it as:

- A large rectangle: 14 m × 10 m = 140 m²
- Subtract the notch: 5 m × 4 m = 20 m²
- But also, the right side is only 8 m high, so we must subtract an extra rectangle: 4 m × 2 m = 8 m² (from x=10 to 14, y=8 to 10)

Wait — this is better.

Let’s try subtraction method:

Start with a rectangle of 14 m × 10 m = 140 m²

Now, remove:
1. The notch: 5 m × 4 m = 20 m²
2. The right-top corner: from x=9 to 14 (5 m wide), y=8 to 10 (2 m high) → but the notch is already in the middle.

Wait — the notch is 5 m wide, so likely from x=5 to 10? But the right side is only 8 m high.

Actually, the notch is centered? Or not?

From the diagram:
- The notch is 5 m wide, and it's in the center.
- The left rectangle is 5 m wide (labeled 5m), then the notch is 5 m wide, then the right rectangle is 4 m wide.

So the widths:
- Left: 5 m
- Notch: 5 m
- Right: 4 m
- Total: 14 m

So the notch is from x=5 to 10.

But the right side is only 8 m high, so from x=10 to 14, the height is only 8 m, not 10 m.

So the full rectangle of 14×10 includes:
- Left: 5×10 = 50
- Middle: 5×10 = 50
- Right: 4×10 = 40

But in reality:
- Left: 5×10 = 50 ✔
- Middle: only 5×6 = 30 (because the notch removes 4 m from the top)
- Right: 4×8 = 32

So total = 50 + 30 + 32 = 112 m²

Alternatively:
- Start with 14×10 = 140
- Subtract the notch: 5×4 = 20
- Subtract the right-top strip: 4×2 = 8 (from x=10 to 14, y=8 to 10)

But wait — the notch is from x=5 to 10, y=6 to 10 → 5×4 = 20
- The right-top strip is from x=10 to 14, y=8 to 10 → 4×2 = 8
- But these are adjacent, so total subtract = 20 + 8 = 28
- 140 - 28 = 112 m² ✔

Yes!

So both methods give 112 m²

✔ Answer for Problem 3: 112 m²

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✔ Final Answers:

1. Area = 56 m²
2. Area = 310 cm²
3. Area = 112 m²

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