Math worksheet for calculating the area of irregular polygons with labeled dimensions.

Let’s solve each problem one by one. We’ll break each irregular shape into rectangles, find the area of each rectangle (length × width), and then add them up to get the total area.

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Problem 1:

Shape is made of two rectangles:
- Left rectangle: height = 16 m, width = ?
Total bottom = 19 m, right part is 10 m → left part = 19 - 10 = 9 m
So left rectangle: 16 m × 9 m = 144 m²
- Right rectangle: 8 m × 10 m = 80 m²

Total area = 144 + 80 = 224 m²

✔ Check: 16×9=144, 8×10=80, 144+80=224 → Correct.

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Problem 2:

Break into two rectangles:
- Left tall rectangle: 12 cm × 10 cm = 120 cm²
- Right short rectangle: 5 cm × 22 cm = 110 cm²

Wait — check if they overlap or share space? The diagram shows a dashed line splitting it vertically. Actually, looking again:

The full shape has a vertical split. Left side is 12 cm high, 10 cm wide. Right side is 5 cm high, 22 cm wide. But note: the top of the right part aligns with the bottom of the left part? No — actually, the left rectangle goes from top to bottom (12 cm), and the right rectangle starts at the same base but only goes up 5 cm. So no overlap.

But wait — the total width isn’t given directly. However, since the dashed line separates them, we assume they are adjacent horizontally.

So yes:
Left: 12 × 10 = 120
Right: 5 × 22 = 110
Total = 120 + 110 = 230 cm²

✔ Check: 12×10=120, 5×22=110, sum=230 → Correct.

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Problem 3:

This is like a U-shape. Break into three parts:
- Left rectangle: 9 yd high × 1 yd wide = 9 yd²
- Right rectangle: same = 9 yd²
- Bottom middle rectangle: connects them. Width = total bottom 10 yd minus left 1 yd minus right 1 yd = 8 yd. Height = ? The total height is 9 yd, and the inner gap is 5 yd, so the bottom strip height = 9 - 5 = 4 yd? Wait — let's think.

Actually, look: the outer height is 9 yd. The inner “cutout” is 5 yd deep from the top. So the bottom horizontal bar must be 9 - 5 = 4 yd tall? But that doesn't match the labels.

Wait — better way: The entire shape can be seen as a big rectangle minus the missing middle top part.

Big rectangle: 10 yd wide × 9 yd tall = 90 yd²
Missing part: in the middle, width = 10 - 1 - 1 = 8 yd, height = 5 yd → area = 8 × 5 = 40 yd²
So area = 90 - 40 = 50 yd²

Alternatively, add the three pieces:
- Left: 1 × 9 = 9
- Right: 1 × 9 = 9
- Bottom: 8 × (9 - 5) = 8 × 4 = 32? Wait, 9 - 5 = 4, yes. Then 9 + 9 + 32 = 50 → Same.

✔ So area = 50 yd²

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Problem 4:

Break into three rectangles:
- Top long rectangle: 14 mm × 3 mm = 42 mm²
- Bottom left small square: 2 mm × 2 mm = 4 mm²
- Bottom right rectangle: 2.5 mm × 2 mm = 5 mm²? Wait — label says 2.5 mm height and 2 mm width? Let me check.

Actually, the shape looks like:
Top: 14 mm wide, 3 mm high → area = 14×3=42
Then below, on the left: a 2mm x 2mm square → area=4
On the right: a rectangle that is 2mm wide and 2.5mm high → area=2×2.5=5
But wait — is there a gap? The total height on right is 2.5 mm, and on left is 2 mm? That might mean the bottom right sticks down more.

Actually, perhaps better to see it as:

The whole thing can be split vertically? Or maybe:

Another way: Imagine the full bounding box? Not necessary.

Looking at dimensions:

From left to right:
- First column: height = 3 + 2 = 5 mm? No — the 3 mm is top, then below it on left is 2 mm, so total left height = 5 mm? But right side is labeled 2.5 mm — inconsistency?

Wait — re-examining the image description:

It says:
Top: 14 mm (width), 3 mm (height)
Below left: 2 mm (width), 2 mm (height)
Below right: 2 mm (width), 2.5 mm (height)

And between them? There’s a gap? Actually, the 14 mm top spans over both lower parts? Probably not — likely the 14 mm is only the top part, and below it are two separate blocks.

But then how are they connected? Perhaps the 14 mm includes the space above both lower blocks.

Assume:
- Top rectangle: 14 mm wide × 3 mm high = 42 mm²
- Below it, on the left: a block 2 mm wide × 2 mm high = 4 mm²
- On the right: a block 2 mm wide × 2.5 mm high = 5 mm²
- And what about the middle? Between the two lower blocks, is there empty space? The total width is 14 mm, and lower blocks take 2 + 2 = 4 mm, so 10 mm gap? That seems odd.

Wait — perhaps the 14 mm is the total width of the top, and the lower parts are inset.

Alternative approach: Use subtraction.

Imagine a big rectangle: width = 14 mm, height = max(3+2, 3+2.5) = 5.5 mm? Too messy.

Better: Look at the dashed lines — probably indicating splits.

Actually, standard way for such shapes: divide into non-overlapping rectangles.

Let me define:

Rectangle A: top part — 14 mm × 3 mm = 42 mm²

Rectangle B: bottom left — 2 mm × 2 mm = 4 mm²

Rectangle C: bottom right — 2 mm × 2.5 mm = 5 mm²

But now, is there a rectangle connecting them? Or is the top sitting on both? If the top is 14 mm wide, and the two bottom blocks are only 2 mm each, then the top extends beyond, which is fine — no overlap.

So total area = 42 + 4 + 5 = 51 mm²

But wait — the bottom right is 2.5 mm high, while bottom left is 2 mm — so they don’t align at bottom. That’s okay.

Is there any overlapping? No, because they are in different positions.

So I think 42 + 4 + 5 = 51 mm² is correct.

✔ Double-check: 14×3=42, 2×2=4, 2×2.5=5, sum=51 → Okay.

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Problem 5:

This is a C-shape or maze-like. Break into rectangles.

We can do:

- Outer frame minus inner hole? Or add parts.

Labelled dimensions:
Top: 7 yd
Right side: 3 yd (top segment), then 4 yd (middle horizontal?), wait.

Actually, let's trace:

Start from top-left:
- Go right 7 yd
- Down 3 yd
- Left 4 yd (so now we're at x=3 yd from left)
- Down 2 yd
- Right 4 yd? Wait, no — after going left 4 yd, then down 2 yd, then right to make the bottom.

Actually, better to split vertically or horizontally.

Split into three horizontal strips:

Top strip: width 7 yd, height 3 yd → area = 21 yd²

Middle strip: this is tricky. After going down 3 yd, we go left 4 yd, so the middle section has a "notch". Actually, the middle part is only on the sides.

Perhaps:

Left column: full height? Total height = 3 + 2 + 3 = 8 yd? From top to bottom: 3 (top) + 2 (middle drop) + 3 (bottom) = 8 yd.

Width of left column: from left edge to where the notch starts. Since top is 7 yd, and we go left 4 yd in the middle, that means the left arm is 7 - 4 = 3 yd wide? Let's see.

Define:

- Left rectangle: width = 3 yd (since 7 - 4 = 3), height = 8 yd (full height) → area = 3 × 8 = 24 yd²

- Right rectangle: similarly, on the right, after the notch, we have a part. When we go down 3 yd, then left 4 yd, then down 2 yd, then right — probably to the right edge. So the right part should also be 3 yd wide? Because total width is 7 yd, left is 3 yd, middle gap is 4 yd? But 3 + 4 = 7, so right part would be zero? That can't be.

I think I messed up.

Look at the shape: it's like a rectangle with a bite taken out of the middle right.

Standard way: calculate as large rectangle minus the missing part.

Large rectangle: if we fill the notch, what would be the size?

The overall width is 7 yd.

Overall height: from top to bottom, we have segments: top 3 yd, then down 2 yd (the depth of the notch), then bottom 3 yd, so total height = 3 + 2 + 3 = 8 yd.

So large rectangle: 7 × 8 = 56 yd²

Now, the missing part: it's a rectangle in the middle right. Width = 4 yd (as labeled), height = 2 yd (the drop). So missing area = 4 × 2 = 8 yd²

Thus, actual area = 56 - 8 = 48 yd²

Verify by adding parts:

- Top rectangle: 7 × 3 = 21

- Bottom rectangle: 7 × 3 = 21

- Middle left part: between top and bottom, on the left, width = 7 - 4 = 3 yd, height = 2 yd → area = 3 × 2 = 6

Total = 21 + 21 + 6 = 48 → Yes!

✔ So area = 48 yd²

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Problem 6:

L-shaped figure.

Can split into two rectangles:

Option 1:
- Vertical part: width = ? Total width at bottom is 5 ft. The horizontal part sticks out 3.8 ft, so the vertical part width = 5 - 3.8 = 1.2 ft? Height = 8.8 ft → area = 1.2 × 8.8

But 1.2 × 8.8 = let's compute: 1.2 × 8 = 9.6, 1.2 × 0.8 = 0.96, total 10.56

- Horizontal part: 3.8 ft × 2.5 ft = 9.5 ft²

Total = 10.56 + 9.5 = 20.06 ft²

But is that correct? The horizontal part is at the bottom, height 2.5 ft, and it extends 3.8 ft to the right, but the vertical part is behind it? Actually, in L-shape, they share the corner.

Better to avoid double-counting.

Standard way:

Rectangle A: the tall part: width = 5 - 3.8 = 1.2 ft, height = 8.8 ft → area = 1.2 × 8.8 = 10.56 ft²

Rectangle B: the wide part at bottom: but this includes the part under the tall rectangle? No — if we take the bottom rectangle as 5 ft wide × 2.5 ft high, that would include the base of the tall part, so we'd be double-counting the overlap.

So instead, for the bottom part, only the extension: width = 3.8 ft, height = 2.5 ft, but this is already not overlapping if we define properly.

Actually, in the L-shape, the two rectangles are:

- One: 1.2 ft (width) × 8.8 ft (height) — the stem

- Two: 3.8 ft (width) × 2.5 ft (height) — the foot, attached to the bottom of the stem? But the stem is only 1.2 ft wide, and the foot is 3.8 ft wide, so they meet at the corner.

The total area should be stem plus foot, no overlap because the foot is to the right of the stem.

In coordinates: suppose bottom-left is origin.

Stem: from x=0 to x=1.2, y=0 to y=8.8

Foot: from x=1.2 to x=1.2+3.8=5.0, y=0 to y=2.5

Yes, no overlap. So areas add.

So area = (1.2 × 8.8) + (3.8 × 2.5)

Calculate:

1.2 × 8.8 = 1.2 × (8 + 0.8) = 9.6 + 0.96 = 10.56

3.8 × 2.5 = 3.8 × 2 + 3.8 × 0.5 = 7.6 + 1.9 = 9.5

Sum = 10.56 + 9.5 = 20.06 ft²

But let's write as fraction or decimal? Probably keep as decimal.

Note: 1.2 = 6/5, 8.8=44/5, etc., but decimal is fine.

✔ So area = 20.06 ft²

But perhaps simplify: 20.06 is exact? 1.2×8.8=10.56, 3.8×2.5=9.5, sum 20.06 — yes.

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Problem 7:

Irregular polygon, looks like a rectangle with a bite taken out of the top right.

Can use subtraction.

Full rectangle if no bite: width = 20 km, height = 10 km → area = 200 km²

Bite: the missing part. From the top, the full width is 20 km, but the top edge is only 16 km, so the bite width = 20 - 16 = 4 km

Height of bite: the right side is 5 km, while left is 10 km, so the bite height = 10 - 5 = 5 km

So missing rectangle: 4 km × 5 km = 20 km²

Thus, area = 200 - 20 = 180 km²

Verify by adding parts:

- Left rectangle: 16 km × 10 km = 160 km²? No, because the right part is shorter.

Better:

Split vertically at x=16 km.

Left part: 16 km wide × 10 km high = 160 km²

Right part: from x=16 to 20, so width 4 km, but height only 5 km (since it steps down) → area = 4 × 5 = 20 km²

Total = 160 + 20 = 180 km² → Same.

✔ So area = 180 km²

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Problem 8:

T-shape with legs. Looks like a T with two feet.

Dimensions:

Top bar: 22 mm wide, 2.5 mm high → area = 22 × 2.5 = 55 mm²

Then below, two legs: each is 4 mm wide, and height? The total height from top to bottom of legs is not given directly.

The legs hang down from the top bar. The distance from the bottom of the top bar to the bottom of the legs is not specified, but we have the height of the legs themselves? Wait, the diagram shows:

After the top bar, there are two downward extensions. Each leg is 4 mm wide, and the height of each leg is... wait, the label "6 mm" is on the side, but that might be the height of the leg.

Looking: it says "6 mm" next to each leg, and "4 mm" for the width of each leg.

Also, between the legs, there is a gap. The top bar is 22 mm wide. Each leg is 4 mm wide, and there is a space between them.

The distance from left edge to first leg: probably symmetric.

Total width 22 mm.

Each leg 4 mm wide.

Space between legs: let's say S.

Then 4 + S + 4 = 22? No, because the legs are not at the edges necessarily.

Actually, the "6 mm" is likely the height of each leg (how far they extend down).

And the top bar is 2.5 mm high.

So the two legs are rectangles: each 4 mm wide × 6 mm high → area per leg = 24 mm², so two legs = 48 mm²

Plus top bar: 22 × 2.5 = 55 mm²

Total = 55 + 48 = 103 mm²

But is there overlap? The legs are attached to the bottom of the top bar, so no overlap — they are separate regions.

So yes, total area = top + two legs = 55 + 24 + 24 = 103 mm²

✔ Confirm: 22×2.5=55, 4×6=24 each, 24×2=48, 55+48=103 → Correct.

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Now, compiling all answers:

1. 224 m²
2. 230 cm²
3. 50 yd²
4. 51 mm²
5. 48 yd²
6. 20.06 ft²
7. 180 km²
8. 103 mm²

For problem 6, 20.06 is exact, but sometimes written as fraction. 20.06 = 2006/100 = 1003/50, but decimal is acceptable.

In the context, probably fine.

Final Answer:

1. 224
2. 230
3. 50
4. 51
5. 48
6. 20.06
7. 180
8. 103

With units as given.

But since the question asks for "Area = ______", and in the worksheet, units are included in the diagram, so in final answer, we can just put the number, assuming units are understood.

To be precise, but the instruction says "final correct answer", and in such worksheets, usually just the number.

So:

Final Answer:
1. 224
2. 230
3. 50
4. 51
5. 48
6. 20.06
7. 180
8. 103