Area of Composite Figures Worksheets - Math Monks - Free Printable

To find the area of composite rectangles, we break each shape into simpler rectangles (or squares), calculate the area of each, and then add or subtract them as needed. The formula for the area of a rectangle is:

> Area = Length × Width

Let’s solve each problem one by one.

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Problem 1

Shape: L-shape
Dimensions:
- Left vertical part: 20 ft tall, 5 ft wide → Area = 20 × 5 = 100 ft²
- Top horizontal part: 30 ft long, but we must subtract the overlapping 5 ft → so 25 ft long, 15 ft tall? Wait — let’s re-express.

Actually, better to split it as:
- Bottom rectangle: 30 ft (length) × 5 ft (height) = 150 ft²
- Left vertical rectangle on top: height = 20 - 5 = 15 ft, width = 5 ft → 15 × 5 = 75 ft²

✔ Total Area = 150 + 75 = 225 ft²

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Problem 2

Shape: U-shape or notch in middle
Outer dimensions: 25 cm wide, 30 cm tall
Notch: 5 cm wide, 10 cm tall (centered)

We can compute:
- Full rectangle: 25 × 30 = 750 cm²
- Subtract the notch: 5 × 10 = 50 cm²

✔ Total Area = 750 - 50 = 700 cm²

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Problem 3

Shape: Rectangle with a smaller rectangle cut out from bottom center
Large rectangle: 36 m × 24 m → 36 × 24 = 864 m²
Cutout: 10 m wide, 14 m tall → 10 × 14 = 140 m²

✔ Total Area = 864 - 140 = 724 m²

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Problem 4

T-shape
Top bar: 12 ft × 6 ft = 72 ft²
Stem: 8 ft tall, 3 ft wide → 8 × 3 = 24 ft²

✔ Total Area = 72 + 24 = 96 ft²

*(Note: The stem is centered under the top bar — no overlap since it’s attached at bottom edge.)*

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Problem 5

H-shape or double notch
We can split into 3 rectangles:

- Top rectangle: 14 yd × 8 yd = 112 yd²
- Bottom rectangle: 14 yd × 8 yd = 112 yd²
- Middle connecting rectangle: 8 yd (height) × 3 yd (width) = 24 yd²
*(The middle part is 8 yd tall and 3 yd wide — since total width is 14, and each side has 5 yd, so middle is 14 - 5 - 5 = 4? Wait — let’s check.*

Wait — looking again: The middle section is between two notches. Each notch is 5 yd wide, so the middle bar is 14 - 5 - 5 = 4 yd wide, and 8 yd tall? But the diagram says “8 yd” vertically for the middle bar — actually, the middle bar connects the top and bottom, and its height is 8 yd? No — let’s read carefully.

Actually, the entire shape is symmetric. Total width = 14 yd. Each side “arm” is 5 yd wide, so the central bar is 14 - 5 - 5 = 4 yd wide. The height of the central bar is 8 yd (as labeled). So:

- Top bar: 14 × 8 = 112
- Bottom bar: 14 × 8 = 112
- Middle bar: 4 × 8 = 32? But that would be double-counting.

Actually, this is an H-shape: better to think as:

- Two vertical bars: each is 8 yd wide? No — wait, labels say “8 yd” for height of top and bottom, and “8 yd” for height of middle bar? Confusing.

Looking at diagram:
- Top rectangle: 14 yd wide, 8 yd tall → 112
- Bottom rectangle: 14 yd wide, 8 yd tall → 112
- Middle rectangle: 3 yd tall? Wait — the label says “8 yd” next to the middle bar — that’s likely the *height* of the middle bar.

Actually, standard way: The whole figure is 14 yd wide, 8 + 8 + 8 = 24 yd tall? But no — the middle bar is only 8 yd tall, and top and bottom are each 8 yd tall? That would make total height 24 yd, but the diagram doesn’t show that.

Better approach: The shape consists of:

- A top rectangle: 14 yd × 8 yd
- A bottom rectangle: 14 yd × 8 yd
- And a middle connecting rectangle: 8 yd (height) × (14 - 5 - 5) = 8 × 4 = 32 yd²? But then we’re triple-counting.

Actually, correct decomposition: The shape is made of three rectangles stacked vertically:

- Top: 14 yd × 8 yd = 112
- Middle: 4 yd × 8 yd = 32 (since 14 - 5 - 5 = 4)
- Bottom: 14 yd × 8 yd = 112

But wait — if you stack them, the total height is 8 + 8 + 8 = 24 yd, but the diagram shows the middle bar is only 8 yd tall, and the top and bottom are also 8 yd tall? That seems inconsistent.

Alternatively, perhaps the "8 yd" labeled on the side is the total height of the top and bottom sections combined? No.

Let me reinterpret: The diagram likely intends:

- The top horizontal bar: 14 yd wide, 8 yd tall
- The bottom horizontal bar: 14 yd wide, 8 yd tall
- The vertical bar in the middle: 8 yd tall (same as each horizontal bar), and 3 yd wide? But 14 - 5 - 5 = 4 yd.

Actually, looking at the diagram: there are two notches of 5 yd each on the sides, so the central vertical bar is 14 - 5 - 5 = 4 yd wide. The height of the central bar is given as 8 yd (the same as the top and bottom bars).

So total area = top + bottom + middle = 14×8 + 14×8 + 4×8 = 112 + 112 + 32 = 256 yd²

But that counts the middle bar separately — and since it’s connected, no overlap. Yes.

✔ Total Area = 256 yd²

*(Alternatively, you could see it as a big rectangle 14 yd × 24 yd minus two notches: each notch is 5 yd × 8 yd, so 2×5×8=80, and 14×24=336, 336-80=256 — same answer.)*

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Problem 6

Step-like shape
We can break into rectangles:

- Bottom rectangle: 28 m × 6 m = 168 m²
- Middle rectangle: 10 m × (18 - 6) = 10 × 12 = 120 m²
- Top rectangle: 12 m × (18 - 6 - ?) — wait, let’s see.

Actually, better:

From left to right:

- Leftmost: 6 m wide, 18 m tall → 6 × 18 = 108
- Middle: 10 m wide, 12 m tall (since 18 - 6 = 12) → 10 × 12 = 120
- Rightmost: 12 m wide, 6 m tall → 12 × 6 = 72

Total: 108 + 120 + 72 = 300 m²

Alternatively, decompose as:

- Bottom full width: 28 m × 6 m = 168
- Above that, from x=6 to x=16 (10 m wide), height 12 m → 10×12=120
- Above that, from x=16 to x=28 (12 m wide), height 6 m → 12×6=72

Same: 168+120+72=300

✔ Total Area = 300 m²

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Problem 7

F-shape
Break into rectangles:

- Vertical stem: 8 mm tall, 2 mm wide → 8 × 2 = 16 mm²
- Top horizontal bar: 6 mm long, 2 mm tall → 6 × 2 = 12 mm²
- Middle horizontal bar: 3 mm long, 1 mm tall → 3 × 1 = 3 mm²
- Bottom horizontal bar: 3 mm long, 1 mm tall → 3 × 1 = 3 mm²

Total: 16 + 12 + 3 + 3 = 34 mm²

Check: The vertical stem is 8 mm tall. The top bar is at top, 2 mm tall, so from y=6 to y=8. Then a gap, then middle bar at y=3 to y=4 (1 mm tall), bottom bar at y=0 to y=1 (1 mm tall). The widths: top bar extends 6 mm right, middle and bottom extend 3 mm right. All attached to left stem.

Yes, no overlap.

✔ Total Area = 34 mm²

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Problem 8

U-shape or frame with a cutout in the middle
Outer rectangle: 16 cm wide, 28 cm tall → 16 × 28 = 448 cm²
Inner cutout: 6 cm wide, 18 cm tall → 6 × 18 = 108 cm²

✔ Total Area = 448 - 108 = 340 cm²

*(Note: The cutout is centered, and the side walls are 8 cm and 6 cm — 8 + 6 + 6 = 20? Wait, 8 + 6 = 14, plus 6 is 20? No — 16 cm total width, cutout is 6 cm, so side margins: (16 - 6)/2 = 5 cm each? But diagram says left is 8 cm, right is 6 cm — that doesn't add up. 8 + 6 + 6 = 20 > 16. Mistake?*

Wait — looking at diagram:
- Total width: 16 cm
- Left side: 8 cm
- Right side: 6 cm
- Cutout width: ? 16 - 8 - 6 = 2 cm? But it says 6 cm for cutout.

Inconsistency. Probably, the cutout is 6 cm wide, and the left and right are each (16 - 6)/2 = 5 cm, but diagram labels 8 cm and 6 cm — that might be a typo, or perhaps the 8 cm and 6 cm are heights? No, they are widths.

Actually, re-examining: The diagram shows the outer width is 16 cm. The left part is labeled 8 cm (width), the right part 6 cm (width), and the cutout is 6 cm wide? 8 + 6 + 6 = 20 ≠ 16. Impossible.

Perhaps the 8 cm and 6 cm are the *heights* of the side walls? But no, they are labeled horizontally.

Another interpretation: The shape is a rectangle 16 cm wide, 28 cm tall, with a rectangular hole 6 cm wide and 18 cm tall, centered? But then side margins should be equal.

Given the labels, perhaps the cutout is not centered. The left wall is 8 cm wide, the right wall is 6 cm wide, so the cutout width is 16 - 8 - 6 = 2 cm? But the diagram says “6 cm” for the cutout width — contradiction.

Looking closely at the image: In problem 8, the cutout is labeled “6 cm” for width, and the left side is “8 cm”, right side “6 cm”. 8 + 6 + 6 = 20, which is more than 16. This is an error in the diagram or my reading.

Perhaps the “8 cm” and “6 cm” are the *heights*? But no, they are horizontal dimensions.

Alternative: Maybe the total width is not 16 cm for the outer, but the 16 cm is the top width, and the cutout is inset.

Actually, standard way: Ignore the side labels for now. The outer rectangle is 16 cm × 28 cm = 448 cm². The inner rectangle (cutout) is 6 cm × 18 cm = 108 cm². So area = 448 - 108 = 340 cm². The side widths (8 cm and 6 cm) might be mislabeled or refer to something else — perhaps the height of the side walls? But the cutout is 18 cm tall, so side walls are 28 - 18 = 10 cm tall? Not matching.

Given the ambiguity, and since the cutout is clearly 6 cm wide and 18 cm tall, and outer is 16 cm wide and 28 cm tall, we proceed with:

✔ Total Area = 16×28 - 6×18 = 448 - 108 = 340 cm²

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## ✔ Final Answers:

1. 225 ft²
2. 700 cm²
3. 724 m²
4. 96 ft²
5. 256 yd²
6. 300 m²
7. 34 mm²
8. 340 cm²

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Summary of Strategy:
- Break complex shapes into non-overlapping rectangles.
- Calculate area of each rectangle.
- Add areas if pieces are separate; subtract if one is cut out from another.
- Always check units and ensure dimensions match.

Let me know if you’d like diagrams or step-by-step visuals!