Area of Polygons Worksheets - Math Monks - Free Printable

Let’s solve each problem one by one. We’ll break each irregular shape into rectangles, calculate the area of each rectangle, and then add them up.

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Problem 1:

Shape is made of two rectangles:

- Left rectangle: height = 16 m, width = ?
Total bottom = 19 m, right part = 10 m → left width = 19 - 10 = 9 m
Area = 16 × 9 = 144 m²

- Right rectangle: height = 8 m, width = 10 m
Area = 8 × 10 = 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 rectangle: height = 12 cm, width = 10 cm → Area = 12 × 10 = 120 cm²

- Right rectangle: height = 5 cm, width = 22 cm → Area = 5 × 22 = 110 cm²

Wait — but look at the diagram: the total height on the left is 12 cm, and the right part is only 5 cm tall. That means the top part of the left rectangle sticks up above the right rectangle. So we’re not double-counting — it’s fine to add both.

But actually, let’s check if they overlap? No — the dashed line shows they are side by side vertically aligned at the bottom. So yes, just add.

Total = 120 + 110 = 230 cm²

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

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Problem 3:

This is a U-shape. Think of it as a big rectangle minus the middle missing part.

Big rectangle: width = 10 yd, height = 9 yd → Area = 10 × 9 = 90 yd²

Missing middle part: width = 10 - 1 - 1 = 8 yd, height = 5 yd → Area = 8 × 5 = 40 yd²

So actual area = 90 - 40 = 50 yd²

Alternatively, think of three parts:

- Left column: 1 yd wide × 9 yd high = 9 yd²

- Right column: same = 9 yd²

- Bottom bar: between them, width = 8 yd, height = (9 - 5) = 4 yd → 8 × 4 = 32 yd²

Total = 9 + 9 + 32 = 50 yd²

✔ Both methods give 50 → Correct.

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Problem 4:

Break into three rectangles:

- Top horizontal: 14 mm long, 3 mm high → Area = 14 × 3 = 42 mm²

- Bottom left small square: 2 mm × 2 mm = 4 mm²

- Bottom right small rectangle: 2.5 mm high, 2 mm wide → Area = 2.5 × 2 = 5 mm²

Wait — let’s check alignment.

Actually, looking at the diagram:

The whole shape has a top part that’s 14 mm wide and 3 mm tall.

Then below it, on the left, there’s a 2x2 square sticking down.

On the right, there’s a 2 mm wide piece that goes down 2.5 mm.

But note: the 2.5 mm includes the 3 mm top? No — the 2.5 mm is labeled on the right side, meaning from bottom to top of that leg.

Actually, better to split vertically or horizontally.

Alternative approach:

Split into:

1. Large top rectangle: 14 mm × 3 mm = 42 mm²

2. Left bottom rectangle: 2 mm wide × (total height - 3 mm). But total height isn’t given directly.

Wait — the right side says “2.5 mm” for the vertical part. And the left side has “2 mm” for the small square.

Actually, the diagram shows:

- From left: a 2 mm wide column that goes down 2 mm below the top.

- Then the main body is 14 mm wide, 3 mm tall.

- On the right, a 2 mm wide column that goes down 2.5 mm total — so below the top 3 mm, it extends 2.5 - 3? That can’t be.

I think I misread.

Looking again:

Label “3 mm” is on the left side — height of the top part.

Label “2 mm” is under the left protrusion — so that little square is 2 mm x 2 mm.

Label “2.5 mm” is on the right side — height of the right leg.

And “14 mm” is the top width.

Also, the right leg is 2 mm wide (since it matches the left?).

Actually, perhaps the shape is:

- A central rectangle: 14 mm wide, 3 mm high.

- Attached below on the left: a 2 mm x 2 mm square.

- Attached below on the right: a 2 mm wide rectangle that is 2.5 mm high — but since the top is already 3 mm, this must mean the right leg extends 2.5 mm downward from the bottom of the top part? That would make total height on right = 3 + 2.5 = 5.5 mm, but label says “2.5 mm” on the side — probably meaning the entire right leg is 2.5 mm tall, including the top? Confusing.

Wait — re-examining the diagram description:

It says:

“14 mm” on top.

Left side: “3 mm” vertical.

Below left: “2 mm” horizontal and “2 mm” vertical — so a 2x2 square attached below left end.

Right side: “2.5 mm” vertical — and since it's on the outer edge, likely the full height of the right leg is 2.5 mm.

But the top is 3 mm — contradiction unless the right leg doesn't include the top.

Perhaps the 3 mm is only the left part, and the right part is shorter? But diagram shows top is flat.

Better interpretation:

The shape consists of:

- A large rectangle on top: 14 mm wide, 3 mm high.

- Below it, on the far left, a 2 mm x 2 mm square attached (so extending down).

- Below it, on the far right, a rectangle that is 2 mm wide and 2.5 mm high — but since the top is already 3 mm, this 2.5 mm must be the additional height below the top? But 2.5 > 3? No.

I think there's a mistake in my reading.

Let me assume based on standard such problems:

Typically, in such diagrams, the labels indicate dimensions of segments.

So:

- The top horizontal segment is 14 mm.

- The left vertical segment is 3 mm (height of left part).

- Then below that, on the left, there's a 2 mm horizontal and 2 mm vertical — so a notch or extension.

Actually, looking at the way it's drawn (from your description), it's like a rectangle with two "feet" at the bottom left and right.

But the right foot is labeled 2.5 mm high, left foot 2 mm high.

And the top is 14 mm wide.

Also, the feet are each 2 mm wide (implied by the 2 mm labels).

So:

Area = area of top rectangle + area of left foot + area of right foot.

Top rectangle: width = 14 mm, height = 3 mm → 42 mm²

Left foot: 2 mm wide × 2 mm high = 4 mm²

Right foot: 2 mm wide × 2.5 mm high = 5 mm²

Total = 42 + 4 + 5 = 51 mm²

But is the top rectangle covering the entire 14 mm, including where the feet are attached? Yes, because the feet are below it.

So no overlap — correct.

✔ Answer: 51 mm²

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Problem 5:

This is a C-shape or frame.

Outer dimensions: width = 7 yd, height = ? Let's see.

From diagram:

Top: 7 yd

Right side: 3 yd (top arm), then gap, then 3 yd (bottom arm)

Inner cutout: 4 yd wide, 2 yd high

So, think of it as a big rectangle minus the inner hole.

Big rectangle: width = 7 yd, height = 3 + 2 + 3 = 8 yd? Wait, no.

Actually, the total height should be the sum of the arms and the gap.

The right side shows: top arm 3 yd, then the inner part is 2 yd high, then bottom arm 3 yd — so total height = 3 + 2 + 3 = 8 yd.

Width = 7 yd.

So big rectangle area = 7 × 8 = 56 yd²

Inner hole: width = 4 yd, height = 2 yd → area = 4 × 2 = 8 yd²

So actual area = 56 - 8 = 48 yd²

Alternatively, calculate the three parts:

- Top bar: 7 yd wide × 3 yd high = 21 yd²

- Bottom bar: same = 21 yd²

- Middle connector: on the left, width = 7 - 4 = 3 yd? Wait, no.

Actually, the sides are connected on the left.

The left side is solid: height = 8 yd, width = ? The total width is 7 yd, inner width is 4 yd, so the left strip is 7 - 4 = 3 yd wide? But that would be if the hole is on the right.

In the diagram, the hole is in the middle-right, so the left part is a vertical strip.

Width of left strip: since total width 7 yd, and the hole starts after some point.

Actually, from the labels: the inner rectangle is 4 yd wide and 2 yd high, positioned such that there's space on left and right? But the top and bottom arms are full width.

Standard way: the shape has:

- Left vertical rectangle: width = let's say W, height = 8 yd

But we don't have W directly.

Note that the top arm is 7 yd wide, which includes the left part and the part over the hole.

Similarly, the bottom arm is 7 yd wide.

The inner hole is 4 yd wide, so the material on the right of the hole is 7 - 4 = 3 yd? But that doesn't match.

Perhaps the left side is thicker.

Another way: the area can be calculated as:

Top rectangle: 7 × 3 = 21

Bottom rectangle: 7 × 3 = 21

Middle left rectangle: connecting them, width = ? Height = 2 yd (the gap height), and width = total width minus the hole width? But the hole is 4 yd wide, and if it's centered or not.

From the diagram description, it's likely that the left side is solid, and the hole is on the right.

So, the left part is a rectangle of width = 7 - 4 = 3 yd? But 7 - 4 = 3, and height = 8 yd, but then the top and bottom arms would be included in that, causing double-counting.

Better to use subtraction method as before.

Big rectangle: 7 yd wide × 8 yd high = 56 yd²

Inner hole: 4 yd wide × 2 yd high = 8 yd²

Area = 56 - 8 = 48 yd²

To verify: the shape consists of:

- Left vertical strip: width = let's calculate. Since the hole is 4 yd wide, and total width 7 yd, if the hole is flush right, then left strip is 3 yd wide, height 8 yd → 24 yd²

- Top horizontal strip on right: but wait, if left strip is 3 yd, then the remaining width is 4 yd, but the top arm is supposed to be full 7 yd — inconsistency.

I think I have it wrong.

Let me define coordinates.

Assume bottom-left corner is (0,0).

The shape goes:

- From (0,0) to (7,3) — bottom arm? No.

Typically for such C-shapes:

- The outer boundary is 7 yd wide and 8 yd high.

- There is a rectangular hole starting at x=3, y=3 to x=7, y=5? Let's see.

From the labels:

The inner rectangle is labeled "4 yd" wide and "2 yd" high.

And it's indented from the right.

Also, the top and bottom arms are each 3 yd high.

So, likely:

- The hole is located such that from the right, it's inset.

Total width 7 yd.

Hole width 4 yd, so if it's against the right, then left margin is 7 - 4 = 3 yd.

Height: total height 8 yd (3 top + 2 gap + 3 bottom).

Hole height 2 yd, positioned in the middle vertically, so from y=3 to y=5 if bottom is y=0.

So hole from x=3 to x=7, y=3 to y=5.

Then the area is big rectangle minus hole: 7*8 - 4*2 = 56 - 8 = 48 yd².

The material is:

- Left strip: x=0 to 3, y=0 to 8 → 3*8=24

- Top strip: x=3 to 7, y=5 to 8 → 4*3=12

- Bottom strip: x=3 to 7, y=0 to 3 → 4*3=12

Total = 24+12+12=48 yd² — perfect.

✔ So answer is 48 yd².

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Problem 6:

L-shaped figure.

Can be split into two rectangles.

Option 1:

- Vertical part: width = ? , height = 8.8 ft

- Horizontal part: width = 5 ft, height = 2.5 ft

But they overlap at the corner.

Better:

Split into:

- Bottom rectangle: 5 ft wide × 2.5 ft high = 12.5 ft²

- Right vertical rectangle: but the vertical part is 8.8 ft high, but the bottom 2.5 ft is already counted, so additional height = 8.8 - 2.5 = 6.3 ft

Width of vertical part: total width is 5 ft, but the horizontal part is 5 ft, and the vertical part extends up from the right end.

The label "3.8 ft" is on the top of the horizontal part, meaning the length of the horizontal arm is 3.8 ft? But it says "5 ft" at the bottom.

Diagram: bottom is 5 ft, left side is 2.5 ft, then it turns right for 3.8 ft, then up for 8.8 ft.

So, the shape is:

- Start at bottom-left, go right 5 ft, up 2.5 ft, left 3.8 ft? No, that would be inward.

Standard L-shape:

From bottom-left:

- Right 5 ft (bottom)

- Up 2.5 ft (left side of bottom part)

- Then right? No, typically it turns up.

Actually, from the labels:

"5 ft" at bottom — total width.

"2.5 ft" on left — height of left part.

"3.8 ft" on the top of the lower part — so the horizontal segment at the top of the bottom rectangle is 3.8 ft, meaning the vertical leg is inset.

"8.8 ft" on the right — height of the vertical leg.

So, the shape has:

- A bottom rectangle: width = 5 ft, height = 2.5 ft → area = 5 × 2.5 = 12.5 ft²

- A vertical rectangle on the right: width = ? , height = 8.8 ft

But the vertical rectangle sits on top of the bottom rectangle, and its width is the difference: total width 5 ft, and the bottom rectangle extends full width, but the vertical part is only on the right part.

The "3.8 ft" is the length of the top edge of the bottom part, which means that from the left, after 5 - 3.8 = 1.2 ft, it turns up? Let's think.

Actually, the horizontal distance from the left to the start of the vertical leg is 5 - 3.8 = 1.2 ft? But that doesn't make sense.

Better: the vertical leg has width = total width minus the overhang.

From the diagram, the bottom is 5 ft wide.

The top of the bottom part has a segment of 3.8 ft — this is likely the part that is covered by the vertical leg or something.

Standard interpretation:

The L-shape can be divided into:

1. A large rectangle on the bottom: 5 ft wide × 2.5 ft high = 12.5 ft²

2. A rectangle on the right side, standing on top of the bottom one: its width is the amount that sticks out, but since the bottom is full width, the vertical leg must be narrower.

The label "3.8 ft" is probably the width of the vertical leg? But it's labeled on the horizontal part.

Looking back: "3.8 ft" is written on the top edge of the lower horizontal part, between the left turn and the right turn.

So, from left to right on the top of the bottom rectangle: first a segment, then it turns up, but the 3.8 ft is the length before turning up? That would mean the vertical leg starts at 3.8 ft from left.

Then the width of the vertical leg is 5 - 3.8 = 1.2 ft.

Height of vertical leg is 8.8 ft.

But then the area of vertical leg is 1.2 × 8.8 = 10.56 ft²

Total area = bottom + vertical = 12.5 + 10.56 = 23.06 ft²

But is there overlap? The vertical leg is sitting on top of the bottom rectangle, so no overlap in area calculation since we're adding separate regions.

However, the bottom rectangle includes the area under the vertical leg, which is fine because the vertical leg is additional height.

In this case, the bottom rectangle is 5x2.5, and the vertical rectangle is 1.2x8.8, placed on top of the right part of the bottom rectangle.

So total area = 12.5 + 10.56 = 23.06 ft²

We can also think of it as:

- Full rectangle if it were complete: but it's L-shaped.

Another way: the shape is composed of:

- Left part: width = 3.8 ft, height = 2.5 ft → area = 3.8 × 2.5 = 9.5 ft²

- Right part: width = 1.2 ft, height = 2.5 + 8.8 = 11.3 ft → area = 1.2 × 11.3 = 13.56 ft²

Total = 9.5 + 13.56 = 23.06 ft² — same.

✔ So answer is 23.06 ft²

But let's confirm the width: total width 5 ft, and the top horizontal segment is 3.8 ft, which is the part not under the vertical leg, so the vertical leg width is 5 - 3.8 = 1.2 ft — yes.

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Problem 7:

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

Dimensions:

Bottom: 20 km

Left side: 10 km

Top: 16 km

Right side: 5 km

So, it's like a big rectangle minus a smaller rectangle on the top-right.

Big rectangle: 20 km wide × 10 km high = 200 km²

Missing part: width = 20 - 16 = 4 km, height = 10 - 5 = 5 km → area = 4 × 5 = 20 km²

So actual area = 200 - 20 = 180 km²

Alternatively, split into two rectangles:

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

- Right part: 4 km wide × 5 km high = 20 km²

Total = 160 + 20 = 180 km² — same.

✔ Correct.

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Problem 8:

T-shaped or H-shaped? From description: top bar 22 mm, then two legs down, each 6 mm wide, and between them a gap, and the legs are 4 mm apart at the bottom? Labels: "6 mm" on the sides of the legs, "4 mm" at the bottom between legs, and "2.5 mm" on the right side — probably the height of the top bar.

Diagram:

- Top horizontal bar: 22 mm wide, 2.5 mm high

- Below it, two vertical legs: each 6 mm wide, and they extend down.

- The distance between the legs at the bottom is 4 mm, but since the legs are 6 mm wide each, and total width is 22 mm, the gap between them should be 22 - 6 - 6 = 10 mm, but it says "4 mm" at the bottom — perhaps the 4 mm is the width of the gap at the bottom, but that doesn't make sense.

Labels:

"22 mm" on top — width of top bar.

"2.5 mm" on right — height of top bar.

"6 mm" on the left and right sides of the legs — probably the width of each leg.

"4 mm" at the bottom between the legs — so the gap between the two legs is 4 mm.

But if each leg is 6 mm wide, and gap is 4 mm, then total width occupied by legs and gap is 6 + 4 + 6 = 16 mm, but the top bar is 22 mm, so there must be overhangs.

Indeed, the top bar extends beyond the legs on both sides.

So, the legs are centered or positioned such that the overhang on each side is (22 - 16)/2 = 3 mm.

Now, to find area:

- Top bar: 22 mm × 2.5 mm = 55 mm²

- Two legs: each is 6 mm wide, and height? Not given directly.

The height of the legs is not specified. Problem.

In the diagram, probably the height of the legs is implied or standard.

Looking back at user input: for problem 8, it says "6 mm" on the sides, "4 mm" at bottom, "2.5 mm" on right, "22 mm" on top.

But no height for legs. Perhaps the "6 mm" is the height? But it's labeled on the side, which might mean width.

In many such problems, the vertical dimension is given.

Perhaps the "6 mm" is the height of the legs.

Let me assume that.

Commonly, in T-shapes, the stem height is given.

Here, since it's symmetric, and "6 mm" is labeled on the vertical sides of the legs, it likely means the height of each leg is 6 mm.

Yes, that makes sense.

So:

- Top bar: 22 mm × 2.5 mm = 55 mm²

- Left leg: 6 mm wide × 6 mm high = 36 mm²

- Right leg: same = 36 mm²

But do they overlap with the top bar? Yes, the legs are attached below the top bar, so no overlap in area — we add them.

Total area = 55 + 36 + 36 = 127 mm²

Is that correct? The legs are separate from the top bar, so yes.

To confirm: the total shape has the top bar, and two rectangles hanging down from it.

No overlapping areas.

✔ So 55 + 36 + 36 = 127 mm²

But let's check the gap: between legs is 4 mm, and each leg 6 mm, so total width of legs and gap is 16 mm, top bar 22 mm, so overhang 3 mm on each side — fine, doesn't affect area.

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Now, compiling all answers:

1. 224 m²

2. 230 cm²

3. 50 yd²

4. 51 mm²

5. 48 yd²

6. 23.06 ft²

7. 180 km²

8. 127 mm²

For problem 6, 23.06 is exact? 1.2 * 8.8 = 10.56, plus 12.5 = 23.06 — yes.

Sometimes they want fractions, but decimal is fine.

Now, final answers per problem.

Final Answer:
1. 224
2. 230
3. 50
4. 51
5. 48
6. 23.06
7. 180
8. 127