Area of Polygons Worksheets - Math Monks - Free Printable

Let's solve each of the irregular polygon area problems step by step. The strategy is to break each shape into simpler rectangles, calculate their areas, and then add or subtract as needed.

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Problem 1

Dimensions:
- Left rectangle: 16 m × 10 m
- Right rectangle: 8 m × (19 - 10) = 8 m × 9 m? Wait — let's analyze carefully.

Actually, total width is 19 m.
Left part height = 16 m, right part height = 8 m.
So we can split it into:
- A large rectangle: 16 m × 19 m? No — that would include extra.

Better way:
Split into two rectangles:
1. Left rectangle: 16 m (height) × 10 m (width) → area = 16 × 10 = 160 m²
2. Right rectangle: 8 m (height) × (19 - 10) = 8 × 9 = 72 m²

But wait: Is the 10 m the width of the left part?

Yes — from diagram:
- Total base = 19 m
- Left part has width 10 m, height 16 m
- Right part has width = 19 - 10 = 9 m, height = 8 m

But the top of the left part is at 16 m, and the right part is only 8 m high, so there's a "step".

So:
- Rectangle A: 10 m × 16 m = 160 m²
- Rectangle B: 9 m × 8 m = 72 m²

Total area = 160 + 72 = 232 m²

✔ Answer: 232 m²

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Problem 2

- Left rectangle: 12 cm × 10 cm = 120 cm²
- Right rectangle: 5 cm × 22 cm = 110 cm²

Wait — but the left rectangle is 12 cm tall and 10 cm wide. Then the bottom extends 22 cm horizontally.

So:
- Top part: 10 cm wide, 12 cm high → 10 × 12 = 120 cm²
- Bottom part: 22 cm wide, 5 cm high → 22 × 5 = 110 cm²

But they overlap in width? Actually, the total horizontal span is 10 + 22 = 32 cm? But no — the 22 cm is the width of the lower rectangle, which starts where the upper one ends?

Wait — look again:

The shape has:
- A vertical rectangle on the left: 12 cm tall, 10 cm wide
- A horizontal extension to the right: 22 cm long, 5 cm tall

But the 22 cm includes the 10 cm? Or not?

No — the figure shows:
- Left side: 12 cm tall, 10 cm wide
- Then a rectangle extending to the right: 22 cm long, 5 cm tall — this means the total width is 10 + 22 = 32 cm?

But the height of the bottom rectangle is 5 cm, and the top one is 12 cm — so the bottom rectangle is under the top one?

Wait — better interpretation: The shape has:
- A tall rectangle: 12 cm × 10 cm (left)
- A short rectangle attached to its right: 5 cm high, 22 cm wide — but this must be placed such that the bottom part extends beyond.

But the total width is 10 cm (left) + 22 cm (right)? That doesn’t make sense.

Wait — actually, the 22 cm is the horizontal length of the bottom part. So:

- The top rectangle: 10 cm (width) × 12 cm (height)
- The bottom rectangle: 22 cm (width) × 5 cm (height)

But how are they connected?

From the diagram: The top rectangle is on the left, and the bottom rectangle extends to the right.

So total width of the shape = 10 cm (top) + ? But the bottom is 22 cm long — so likely the bottom rectangle goes from x=0 to x=22, and the top rectangle is from x=0 to x=10, y=12 down to y=5.

Wait — the total height is 12 cm (top), and bottom is 5 cm, so the overlap is 5 cm?

Ah! So we can break it into:
- Rectangle A: 10 cm × 12 cm = 120 cm² (left side)
- Rectangle B: 22 cm × 5 cm = 110 cm² (bottom)

But now we see that the bottom rectangle extends beyond the left one — so total area is sum of both.

But do they overlap? Yes — the overlapping region is 10 cm × 5 cm.

So if we add both, we double-count the overlap.

So total area = Area A + Area B – Overlap

But wait — actually, the bottom rectangle is below the top one, so the top rectangle sits on top of the bottom one for 10 cm width.

So the bottom rectangle is 22 cm wide, 5 cm high → 110 cm²
The top rectangle is 10 cm wide, 12 cm high — but the lower 5 cm is already included in the bottom rectangle.

So we should compute:
- Bottom part: 22 cm × 5 cm = 110 cm²
- Top part: only the extra 7 cm height (12 - 5 = 7 cm) over the 10 cm width → 10 × 7 = 70 cm²

Total area = 110 + 70 = 180 cm²

✔ Answer: 180 cm²

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Problem 3

This is a U-shape.

We can think of it as:
- A big rectangle minus a smaller rectangle (the middle gap)

Or add three rectangles.

Let’s use addition method.

It has:
- Left vertical rectangle: 1 yd (top) + 5 yd (middle) = 6 yd high? No — the sides are 9 yd high, and the middle is 5 yd.

Wait — labels:
- Left side: 1 yd (top), then 5 yd (middle), then bottom?
- Total height: 9 yd (on the right side)
- Middle section: 5 yd high
- Width: 10 yd across the bottom

So the shape is:
- Two vertical rectangles on left and right: each 1 yd (top) + 5 yd (middle) = 6 yd? But the total height is 9 yd.

Wait — look at the numbers:
- Left column: 1 yd (top), then 5 yd (middle), then what? It says 9 yd on the right side.

Actually:
- The entire height is 9 yd.
- The middle gap is 5 yd high.
- The top is 1 yd high, and the bottom is 1 yd high? Wait — the label says "1 yd" on top and bottom of the U.

Wait — the diagram shows:
- On the left: 1 yd (top), then 5 yd (middle), then 3 yd (bottom)? But it says 9 yd total height.

Wait — maybe:
- The vertical parts are 9 yd high (left and right)
- The middle horizontal bar is 5 yd high
- The gap is 1 yd high on top and 1 yd on bottom?

No — the labels are:
- Left: 1 yd (top), then 5 yd (middle), and total height 9 yd → so bottom part is 9 - 1 - 5 = 3 yd?

But the right side says 9 yd total.

Wait — the figure shows:
- Top: 1 yd
- Middle: 5 yd
- Bottom: 3 yd? But not labeled.

Actually, the total height is 9 yd, and the middle section is 5 yd high, with 1 yd above and 3 yd below? But the label says "5 yd" in the middle, and "1 yd" on top and bottom.

Wait — perhaps:
- The top is 1 yd high
- The middle is 5 yd high
- The bottom is 3 yd high
- Total = 1 + 5 + 3 = 9 yd — yes!

But the middle horizontal bar is 5 yd high, and the gap is 1 yd on top and 1 yd on bottom?

No — the U-shape has:
- Top bar: 1 yd high, 10 yd wide
- Bottom bar: 3 yd high, 10 yd wide
- Side bars: 5 yd high, 1 yd wide

Wait — no — the side bars are 1 yd wide, and go from top to bottom.

Let me re-analyze:

The shape has:
- Two vertical rectangles: left and right, each 1 yd wide, 9 yd high → area = 1×9 = 9 yd² each → total 18 yd²
- One horizontal rectangle in the middle: 10 yd wide, 5 yd high → 10×5 = 50 yd²

But wait — the middle rectangle is not 10 yd wide — because the sides are 1 yd wide, so the middle bar is 10 - 1 - 1 = 8 yd wide?

Yes! Because the total width is 10 yd, and the two side bars are 1 yd each.

So:
- Left vertical bar: 1 yd × 9 yd = 9 yd²
- Right vertical bar: 1 yd × 9 yd = 9 yd²
- Middle horizontal bar: 8 yd × 5 yd = 40 yd²

Total area = 9 + 9 + 40 = 58 yd²

✔ Answer: 58 yd²

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Problem 4

A rectangle with a notch cut out.

Dimensions:
- Outer rectangle: 14 mm × 25 mm → 14 × 25 = 350 mm²
- Cut-out: 2 mm × 3 mm → 6 mm²
- But there are two notches? No — only one.

Wait — the figure shows:
- A rectangle of width 14 mm, height 25 mm
- But on the left, a 2 mm × 3 mm rectangle is missing
- On the right, a 2 mm × 2 mm rectangle is missing?

Wait — labels:
- Left: 2 mm depth, 3 mm height
- Right: 2 mm depth, 2 mm height

So two small rectangles removed.

But are they inside?

Yes — the shape has:
- Main rectangle: 14 mm wide, 25 mm high → 14 × 25 = 350 mm²
- Subtract left notch: 2 mm × 3 mm = 6 mm²
- Subtract right notch: 2 mm × 2 mm = 4 mm²

Total area = 350 - 6 - 4 = 340 mm²

✔ Answer: 340 mm²

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Problem 5

A rectangle with a rectangular hole in the middle.

Outer dimensions:
- Width: 7 yd
- Height: 3 yd (bottom) + 3 yd (top) = 6 yd? But the middle is indented.

Wait — the shape has:
- Overall height: 3 yd (bottom) + 3 yd (top) = 6 yd
- But the middle is indented: a 4 yd × 2 yd rectangle missing

So:
- Full rectangle: 7 yd × 6 yd = 42 yd²
- Subtract inner rectangle: 4 yd × 2 yd = 8 yd²

Total area = 42 - 8 = 34 yd²

✔ Answer: 34 yd²

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Problem 6

L-shaped figure.

Break into two rectangles:
- Bottom rectangle: 5 ft × 2.5 ft = 12.5 ft²
- Top rectangle: width = 5 ft, height = 8.8 - 2.5 = 6.3 ft? Wait — no.

Wait — the vertical segment is 8.8 ft high, but the horizontal part is 3.8 ft long.

So:
- Bottom rectangle: 5 ft × 2.5 ft = 12.5 ft²
- Top rectangle: 3.8 ft × (8.8 - 2.5) = 3.8 × 6.3 = ?

Calculate: 3.8 × 6.3

= (3 + 0.8)(6 + 0.3) = 3×6 + 3×0.3 + 0.8×6 + 0.8×0.3 = 18 + 0.9 + 4.8 + 0.24 = 23.94 ft²

Total area = 12.5 + 23.94 = 36.44 ft²

Alternatively:
- Bottom: 5 × 2.5 = 12.5
- Top: 3.8 × 6.3 = 23.94
- Total: 36.44 ft²

✔ Answer: 36.44 ft²

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Problem 7

Irregular shape — like a rectangle with a bite taken out.

Break into two rectangles:
- Bottom rectangle: 20 km × 5 km = 100 km²
- Top rectangle: width = 16 km, height = 10 - 5 = 5 km → 16 × 5 = 80 km²

Total area = 100 + 80 = 180 km²

✔ Answer: 180 km²

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Problem 8

T-shaped figure.

Break into:
- Top rectangle: 22 mm × 2.5 mm = 55 mm²
- Bottom left leg: 6 mm × 4 mm = 24 mm²
- Bottom right leg: 6 mm × 4 mm = 24 mm²

But wait — the legs are 4 mm wide, and the top is 22 mm wide, but the legs are inset.

Wait — the top is 22 mm wide, 2.5 mm high.

Then two legs below: each 6 mm wide, 4 mm high.

But the legs are centered? The total width is 22 mm, and the legs are 4 mm each, so distance between them?

Wait — the figure shows:
- Top: 22 mm wide, 2.5 mm high
- Below, two rectangles: each 6 mm wide, 4 mm high
- Distance between them: 22 - 6 - 6 = 10 mm? But not labeled.

But the legs are separate — so total area is:
- Top: 22 × 2.5 = 55 mm²
- Left leg: 6 × 4 = 24 mm²
- Right leg: 6 × 4 = 24 mm²

Total = 55 + 24 + 24 = 103 mm²

✔ Answer: 103 mm²

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✔ Final Answers:

| Problem | Area |
|--------|------|
| 1 | 232 m² |
| 2 | 180 cm² |
| 3 | 58 yd² |
| 4 | 340 mm² |
| 5 | 34 yd² |
| 6 | 36.44 ft² |
| 7 | 180 km² |
| 8 | 103 mm² |

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✔ Summary of Methods Used:

- Decomposition: Break into rectangles.
- Addition: Add areas of sub-rectangles.
- Subtraction: Subtract missing parts from larger rectangles.
- Careful attention to dimensions and overlaps.

Let me know if you'd like a visual breakdown of any shape!