Math worksheet for calculating the area of composite shapes.

Explanation:
Let’s solve each problem one by one. We need to find the white space (the area not shaded) in each composite shape.

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Problem 1:
We have a rectangle that is 12 m wide and 12 m tall (since both horizontal and vertical sides are labeled 12 m). Inside it, there’s a shaded triangle with base = 12 m and height = 8 m.

- Area of rectangle = length × width = 12 m × 12 m = 144 m²
- Area of shaded triangle = (base × height) ÷ 2 = (12 × 8) ÷ 2 = 96 ÷ 2 = 48 m²
- White space = rectangle − triangle = 144 − 48 = 96 m²

✔ Check: Makes sense — triangle is less than half the rectangle (half would be 72), but since height is only 8/12 = 2/3 of full height, area is (1/2)×12×8 = 48, yes.

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Problem 2:
This is an L-shaped figure. We’re asked for the area of the whole shape (since no part is shaded — it's all white). So we find total area of the composite shape.

One easy way: break it into two rectangles.

Option A:
- Left big rectangle: width = 10 cm, height = 15 cm → area = 10 × 15 = 150 cm²
- Top right small rectangle: width = 5 cm, height = (15 − 12) = 3 cm → because total height is 15 cm, and the lower part is 12 cm tall, so the “step” is 3 cm high.
So area = 5 × 3 = 15 cm²

Total area = 150 + 15 = 165 cm²

Alternative check: Full outer rectangle would be 15 cm tall × (10 + 5) = 15 × 15 = 225 cm², but missing a rectangle of size 10 cm × 3 cm = 30 cm² (the bottom-right empty part). So 225 − 30 = 195? Wait—that contradicts!

Hold on — let’s re-express the shape carefully.

Looking again:
The shape has:
- Left side: height = 15 cm
- Bottom: width = 10 cm
- Right side: from bottom up, a vertical segment of 12 cm, then a horizontal top segment of 5 cm extending right.

So the shape looks like:
- Bottom rectangle: 10 cm (width) × 12 cm (height) = 120 cm²
- Top right rectangle: 5 cm (width) × (15 − 12) = 5 × 3 = 15 cm²
→ Total = 120 + 15 = 135 cm²

Wait — earlier I used left rectangle 10×15 = 150, but that assumes the full left side goes all the way up — but the top right part sticks out, meaning the left part is only 12 cm tall at the bottom, and above that (from y=12 to y=15), only the left 10 cm is present? No — actually, the left side is full height 15 cm — the shape is like an “L” where the vertical arm is on the left (10 cm wide, 15 cm tall), and the horizontal arm sticks out to the right from the top — but the diagram shows the horizontal part is only 5 cm wide and sits on top of the right side of the lower part.

Let me reconstruct coordinates:

Assume bottom-left corner is (0,0).
- Bottom edge: from (0,0) to (10,0)
- Left edge: from (0,0) to (0,15)
- Then from (0,15) to (10,15) — wait, no! The top of the left side is at y=15, but the right side only goes up to y=12 at x=10, then goes right 5 cm to x=15 at y=12, then up to y=15? No — the diagram shows a step: from (10,0) up to (10,12), then right to (15,12), then up to (15,15)? But no vertical line shown from (15,12) to (15,15). Actually, the shape is likely:

Points in order:
(0,0) → (10,0) → (10,12) → (15,12) → (15,15) → (0,15) → back to (0,0)

Yes! That makes an L-shape: full height on left (15 cm), bottom width 10 cm, then a 5 cm extension to the right at height 12–15 cm.

So area = area of big rectangle 15 cm × 15 cm? No — width is not 15 everywhere.

Better: split into:
1. Rectangle A: from x=0 to x=10, y=0 to y=15 → 10 × 15 = 150
2. Rectangle B: from x=10 to x=15, y=12 to y=15 → 5 × 3 = 15
Total = 150 + 15 = 165 cm²

But does rectangle A include the region from y=12 to y=15 at x=0–10? Yes — and rectangle B adds the top-right part. No overlap. So 165 is correct.

Alternative: full bounding box is 15 (wide) × 15 (tall) = 225. Missing part is rectangle from x=10 to 15, y=0 to 12 → 5 × 12 = 60. So 225 − 60 = 165 cm² ✔ Confirmed.

So Problem 2 area = 165 cm²

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Problem 3:
Another L-shaped figure, in meters.

Given:
- Total width = 11 m
- Left height = 9 m
- Top right extension: width = 2 m, and the vertical drop from top to the step is 3 m — meaning the lower part is 9 − 3 = 6 m tall? Wait.

Let’s assign coordinates again:
Bottom-left = (0,0)
Bottom-right = (11,0)
Left-top = (0,9)
Then from (0,9) to (11,9) is top edge? No — the shape has a step: from left side up to 9 m, then across 11 m, but then down 3 m, then left 2 m? Actually, the label says:
- Top horizontal segment = 11 m
- Right vertical segment (down from top) = 3 m
- Then left horizontal segment = 2 m
- Then down to bottom at height 9 m? Wait, left side is labeled 9 m — so total height is 9 m.

So points likely:
(0,0) → (11,0) → (11,9) ??? No — if right side is only 3 m tall from top, then top is at y=9, and the step is at y = 9 − 3 = 6 m.

So:
- Bottom: (0,0) to (11,0)
- Up right side: (11,0) to (11,6) — height 6 m
- Left: (11,6) to (11−2=9,6) → 2 m left
- Up: (9,6) to (9,9) → 3 m up
- Left: (9,9) to (0,9) → 9 m
- Down: (0,9) to (0,0)

Yes — that matches labels: total top width = 11 m (from x=0 to x=11 at y=9), the “notch” is 2 m wide and 3 m deep (so missing rectangle is 2 m wide × 3 m tall? Wait — actually, the shape includes the top bar and the left bar).

Easier: split into two rectangles:
1. Left rectangle: width = 9 m (from x=0 to x=9), height = 9 m → 9 × 9 = 81 m²
2. Right-top rectangle: width = 2 m (x=9 to 11), height = 6 m (y=0 to 6)? No — from diagram, the right part only goes up to y=6, but the top is at y=9, so the right part is from y=6 to y=9? Let's visualize:

From description:
- Full top length = 11 m
- Left side height = 9 m
- There’s a step: from the top right, go down 3 m, then left 2 m.

So the “missing” part (if it were a full rectangle 11×9) is a rectangle of width 2 m and height 3 m at the bottom-right? No — actually, the shape is the full 11×9 rectangle minus a rectangle of size 2 m (width) × 3 m (height) at the bottom-right corner.

Check:
- Full rectangle: 11 m × 9 m = 99 m²
- Missing piece: the bottom-right corner is cut out — width = 2 m (since top is 11 m, and the lower part is only 11 − 2 = 9 m wide), height = 3 m (since the right side rises only 3 m from bottom before stepping left). So missing area = 2 × 3 = 6 m²
→ Shape area = 99 − 6 = 93 m²

Verify by splitting:
- Bottom rectangle: width = 9 m, height = 9 m? No — bottom part is only 9 m wide and full height? Wait, the left part is 9 m wide and 9 m tall — yes. Then on top of the right side, there’s a 2 m × 3 m rectangle (from y=6 to y=9, x=9 to 11). So:
- Left big rectangle: 9 × 9 = 81
- Top-right small rectangle: 2 × 3 = 6
Total = 87? That contradicts.

Wait — if left part is 9 m wide and 9 m tall, and top-right is 2 m wide and 3 m tall, then total width at top is 9 + 2 = 11 m (good), but the height of the right part is only 3 m, sitting on top of the left part — meaning the left part must be only 6 m tall? No, the left side is labeled 9 m — so the left edge goes full 9 m.

The correct interpretation (standard for such diagrams):
The shape consists of:
- A rectangle 11 m wide and 6 m tall (bottom part), plus
- A rectangle 2 m wide and 3 m tall on top of the right side.

Why 6 m tall? Because total height is 9 m, and the top part is 3 m tall, so bottom part is 9 − 3 = 6 m tall.

And the bottom part’s width is 11 m? But then the top part sits on top of the rightmost 2 m of the bottom part — so bottom rectangle: 11 m × 6 m = 66
Top rectangle: 2 m × 3 m = 6
Total = 72 m²

But left side is labeled 9 m — that’s total height, okay. And the vertical segment labeled “3m” is the height of the top step, and “2m” is its width. The bottom width is 11 m, and the left side goes full 9 m — yes, that fits: from (0,0) to (0,9), then to (11,9), then down 3 m to (11,6), then left 2 m to (9,6), then down to (9,0), then left to (0,0). Wait — that would make the bottom part from x=0 to x=9 at y=0 to y=6, and x=9 to 11 at y=0 to y=6? No, (9,6) to (9,0) is vertical, so from x=0 to 9, height = 9; from x=9 to 11, height = 6.

So better split:
- Rectangle A: x=0 to 9, y=0 to 9 → 9 × 9 = 81
- Rectangle B: x=9 to 11, y=0 to 6 → 2 × 6 = 12
Total = 81 + 12 = 93 m²

Yes! That matches the “full rectangle minus missing corner” method:
Full 11×9 = 99
Missing part: x=9 to 11, y=6 to 9 → 2 × 3 = 6
99 − 6 = 93 ✔

So Problem 3 area = 93 m²

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Final answers:
1. 96 m²
2. 165 cm²
3. 93 m²

Final Answer:
96 m², 165 cm², 93 m²