Irregular Polygons Worksheet: Calculate the total area of each irregular shape.

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 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 ✔️

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Problem 2:

Break into two rectangles:

- Left vertical: 12 cm × 10 cm = 120 cm²

- Right horizontal: 5 cm × 22 cm = 110 cm²

Wait — but they overlap? No, looking at diagram: it’s an L-shape. The total width on top is 10 cm, then extends 22 cm to the right. But the full height on left is 12 cm, and right part is only 5 cm tall.

Actually, better way: split vertically or horizontally.

Split horizontally:

Top part: 10 cm wide × (12 - 5) = 7 cm high → 10 × 7 = 70 cm²

Bottom part: full width = 10 + 22 = 32 cm? Wait no — look again.

Actually, from diagram: left side is 12 cm tall, 10 cm wide. Then attached to its right is a rectangle that is 22 cm long and 5 cm tall — so it sticks out to the right, starting from the bottom.

So total shape: left rectangle 12x10, plus right rectangle 5x22.

But do they overlap? The right rectangle starts at the bottom of the left one, so no overlap.

Area = (12 × 10) + (5 × 22) = 120 + 110 = 230 cm²

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

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Problem 3:

U-shaped figure.

Total outer rectangle would be 10 yd wide × 9 yd tall = 90 yd²

But there’s a missing middle part.

The U has two sides: each 1 yd wide, and center gap.

Height of the “legs” is 9 yd, but the bottom bar is 5 yd high? Wait — label says 5 yd inside the U.

Actually, the inner empty space is: width = 10 - 1 - 1 = 8 yd, height = 9 - 5 = 4 yd? Wait no.

Look: the vertical parts are 9 yd tall, and the horizontal base of the U is 5 yd tall? Actually, the label "5 yd" is inside the U, meaning the depth of the U is 5 yd? Or the height of the bottom bar?

Better approach: divide into three rectangles.

Left leg: 1 yd × 9 yd = 9 yd²

Right leg: 1 yd × 9 yd = 9 yd²

Bottom bar: connects them, width = 10 - 1 - 1 = 8 yd, height = ? The total height is 9 yd, and the legs go all the way up, so the bottom bar must be the part between the legs at the bottom. But if the legs are 9 yd tall, and the bottom bar is connecting them, then the bottom bar’s height is not given directly.

Wait — the label “5 yd” is written inside the U, vertically. That likely means the height of the open part is 5 yd, so the bottom bar is 9 - 5 = 4 yd tall? But that doesn’t match.

Alternative interpretation: the entire shape is 10 yd wide and 9 yd tall. The U has two arms of 1 yd width, and the bottom of the U is 5 yd above the bottom? No.

Actually, standard way: the U-shape can be seen as a big rectangle minus a smaller rectangle in the middle.

Big rectangle: 10 yd × 9 yd = 90 yd²

Inner rectangle (the hole): width = 10 - 1 - 1 = 8 yd, height = 9 - 5 = 4 yd? Why 5? The label “5 yd” is placed vertically inside the U — probably indicating the height of the inner void.

If the total height is 9 yd, and the bottom of the U is solid for some height, but the label “5 yd” is next to the inner vertical line — actually, looking at diagram: the 5 yd is labeled on the inner vertical segment, meaning the height of the inner rectangle is 5 yd? But that would mean the bottom bar is 9 - 5 = 4 yd? Confusing.

Wait — perhaps the 5 yd is the height of the bottom bar? Let me re-read.

In diagram 3: left arm 1 yd wide, right arm 1 yd wide, total width 10 yd. Height of whole thing is 9 yd. Inside, there's a horizontal line with label “5 yd” — this likely means the distance from the top of the bottom bar to the top of the shape is 5 yd? Or the height of the bottom bar is 5 yd?

Actually, common interpretation: the U has a base that is 5 yd tall, and the two sides extend up 9 yd total, so the sides are 9 yd tall, and the base is 5 yd tall, but that doesn't make sense because the base should be shorter.

I think I got it: the shape is composed of:

- Two vertical rectangles: each 1 yd wide × 9 yd tall → 9 yd² each → total 18 yd²

- One horizontal rectangle at the bottom: width = 10 - 1 - 1 = 8 yd, height = ? The total height is 9 yd, but the vertical parts already cover the full height, so the bottom bar must be overlapping? No.

Actually, in such diagrams, the U-shape usually has the bottom bar connecting the two legs at the bottom, and the legs go up. So the bottom bar has height h, and the legs have height H, but here both are given?

Label “5 yd” is inside the U, vertically aligned with the inner edge — likely it means the height of the inner cavity is 5 yd, so the bottom bar is 9 - 5 = 4 yd tall? But that seems arbitrary.

Another way: the total area can be calculated as the area of the outer rectangle minus the area of the inner rectangle.

Outer: 10 yd × 9 yd = 90 yd²

Inner rectangle: width = 10 - 1 - 1 = 8 yd, height = 9 - 5 = 4 yd? Why 5? If the 5 yd is the height from the bottom to the start of the opening, then the opening height is 9 - 5 = 4 yd.

But the label “5 yd” is placed next to the inner vertical line, which suggests that the vertical dimension of the inner part is 5 yd. In many worksheets, for a U-shape, the number inside indicates the height of the "cut-out" part.

Assume that the inner rectangle (the hole) has height 5 yd and width 8 yd (since 10 - 1 - 1 = 8).

Then area of hole = 8 × 5 = 40 yd²

Total area = outer - hole = 90 - 40 = 50 yd²

But let's verify with addition:

Left leg: 1 × 9 = 9

Right leg: 1 × 9 = 9

Bottom bar: 8 × (9 - 5) = 8 × 4 = 32? But 9 - 5 = 4, yes.

Then total = 9 + 9 + 32 = 50 yd² ✔️

Yes, that makes sense. The bottom bar is 4 yd tall (since the total height is 9 yd, and the opening is 5 yd, so the solid part at bottom is 4 yd).

So area = 50 yd²

✔ Confirmed.

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Problem 4:

This is like a rectangle with a bite taken out, or can be split.

Dimensions: overall width 14 mm, but with indentations.

From left: 2 mm wide section, height 3 mm

Then middle part: width = 14 - 2 - 2 = 10 mm? Wait, right side also has 2 mm.

Actually, the shape has:

- Left protrusion: 2 mm × 3 mm

- Middle main body: but it's indented.

Better to see as a large rectangle minus two small rectangles.

Large rectangle: width 14 mm, height = max height. From diagram, left side is 3 mm, right side is 2.5 mm, but there's a step.

Actually, the shape is symmetric? No.

List the parts:

We can divide into three rectangles:

1. Left small rectangle: 2 mm wide × 3 mm high = 6 mm²

2. Right small rectangle: 2 mm wide × 2.5 mm high = 5 mm²

3. Middle rectangle: width = 14 - 2 - 2 = 10 mm, height = ? What is the height of the middle part?

Looking at diagram: the middle part connects the left and right. The left is 3 mm tall, right is 2.5 mm tall, but the middle might be lower.

Actually, the label "2 mm" is on the right side, indicating the height of the right protrusion is 2.5 mm? Wait, labels: left has "3 mm" height, right has "2.5 mm" height, and there's a "2 mm" label on the right side vertically? No.

Re-examining: the diagram shows:

- Overall width 14 mm

- On left, a rectangle sticking up: width 2 mm, height 3 mm

- On right, a rectangle sticking down? No, it's all one shape.

Actually, it's a single polygon. From left to right:

Start at bottom left, go up 3 mm, right 2 mm, down to some level, then right, then up to 2.5 mm, etc.

Perhaps easier: the shape can be seen as a rectangle of 14 mm by 3 mm, but with a notch on the right bottom.

Note that the right part has height 2.5 mm, while left is 3 mm, so the difference is 0.5 mm.

But there's a label "2 mm" on the right side — wait, in the diagram, next to the right vertical segment, it says "2 mm", but that might be the width of the right protrusion.

Standard way: divide into rectangles.

Rectangle A: left part, 2 mm × 3 mm = 6 mm²

Rectangle B: middle part, width = 14 - 2 - 2 = 10 mm, height = min(3, 2.5) = 2.5 mm? But that might not be accurate.

Actually, the middle part has height equal to the lower of the two sides? No.

Let's think of the bounding box.

The highest point is 3 mm (left), lowest is 0, but the right side goes up to 2.5 mm.

The shape has a "step" on the right.

From the diagram description: it's like a rectangle with a rectangular cutout on the bottom right or something.

Another approach: use the fact that the total area can be calculated by adding the areas of the visible rectangles.

From left:

- Rectangle 1: 2 mm (w) × 3 mm (h) = 6 mm²

- Then, to the right of that, there is a rectangle that is 10 mm wide (since 14-2-2=10) and 2.5 mm high? But why 2.5? Because the right side is 2.5 mm tall, and assuming the middle is at that height.

But the left part is taller, so between x=2 to x=12, the height is 2.5 mm? Then from x=12 to x=14, it's 2.5 mm high, but that's already included.

Actually, if we take:

- Full width 14 mm, but height varies.

The shape consists of:

- A rectangle from x=0 to x=2, y=0 to y=3: area 6

- A rectangle from x=2 to x=12, y=0 to y=2.5: width 10, height 2.5 = 25

- A rectangle from x=12 to x=14, y=0 to y=2.5: width 2, height 2.5 = 5

But then the first rectangle overlaps in y from 0 to 2.5 with the second, but since it's the same region, we shouldn't double-count.

Mistake.

Correct division:

The shape can be divided as:

1. The left tower: 2 mm × 3 mm = 6 mm²

2. The main body: from x=2 to x=14, but height is 2.5 mm for most, but at x=12 to 14, it's still 2.5 mm, and from x=2 to 12, it's 2.5 mm, but the left tower is separate.

Actually, from x=0 to 2: height 3 mm

From x=2 to 14: height 2.5 mm? But at x=12 to 14, it's specified as 2.5 mm, and from 2 to 12, it should be the same.

But is there a drop? The diagram likely shows that from x=2 to x=12, the height is 2.5 mm, and from x=12 to 14, it's also 2.5 mm, but the left part from 0 to 2 is 3 mm.

So total area = area of left strip + area of right part.

Left strip: 2 mm × 3 mm = 6 mm²

Right part: from x=2 to x=14, width 12 mm, height 2.5 mm = 30 mm²

But is that correct? The right part includes from x=2 to 14, which is 12 mm wide, and if height is uniformly 2.5 mm, then yes.

But in the diagram, there is a label "2 mm" on the right side — wait, in the user's image description, for problem 4, it says "2 mm" next to the right vertical segment, but that might be the width of the right protrusion.

Upon second thought, in many such problems, the shape is composed of three rectangles:

- Left: 2mm x 3mm

- Middle: 10mm x 2.5mm (since 14-2-2=10)

- Right: 2mm x 2.5mm

And they are adjacent, no overlap.

So area = (2*3) + (10*2.5) + (2*2.5) = 6 + 25 + 5 = 36 mm²

But is the middle part really 2.5 mm high? Yes, because the right part is 2.5 mm, and the middle connects to it.

The left part is taller, but it's separate.

So total area = 6 + 25 + 5 = 36 mm²

✔ Check: 2*3=6, 10*2.5=25, 2*2.5=5, sum=36 ✔️

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Problem 5:

This is a C-shape or U-shape rotated.

Dimensions: outer width 7 yd, outer height: let's see.

From diagram: top horizontal 7 yd, then down 3 yd, then left 4 yd, then down 2 yd, then right 3 yd, then up to close.

Actually, it's like a rectangle with a bite taken out.

Can be seen as a large rectangle minus a smaller rectangle.

Large rectangle: width 7 yd, height = 3 + 2 + 3 = 8 yd? Let's calculate total height.

From top to bottom: the left side has segments: down 3 yd, then after the indentation, down another 2 yd? No.

Trace the perimeter:

Start at top-left, go right 7 yd, down 3 yd, left 4 yd, down 2 yd, right 3 yd, then up to meet the start? But up how much?

After going right 3 yd, we need to go up to the top level. The total height should be consistent.

From the moves: after going down 3 yd, then left 4 yd, then down 2 yd, then right 3 yd, now we are at a point that is 3 yd below the top on the right side? Let's coordinate.

Set origin at top-left.

Go right 7: to (7,0)

Down 3: to (7,-3)

Left 4: to (3,-3)

Down 2: to (3,-5)

Right 3: to (6,-5)

Now, to close the shape, we need to go up to (6,0)? But that would be up 5 yd, but then from (6,0) to (0,0) is left 6 yd, but we started at (0,0).

Actually, from (6,-5), we go up to (6,0)? But the top is at y=0, so up 5 yd to (6,0), then left to (0,0).

But the width at top is 7 yd, from x=0 to x=7.

At y=0, from x=0 to x=7.

At y=-3, from x=3 to x=7 (since we went left to x=3 at y=-3)

At y=-5, from x=3 to x=6 (since we went right to x=6 at y=-5)

Then from (6,-5) up to (6,0), then left to (0,0).

So the shape has:

- From x=0 to x=7, y=0 to y=-3? No, only where filled.

Actually, the filled regions:

- Rectangle A: x=0 to 7, y=0 to -3? But at y=-3, from x=3 to 7 is present, but from x=0 to 3 is not necessarily.

Better to divide into rectangles.

Rectangle 1: top part: x=0 to 7, y=0 to -3 → but is it full? From the path, when we go down from (7,0) to (7,-3), then left to (3,-3), so the top rectangle is from x=3 to 7, y=0 to -3? No.

Let's list the polygons.

The shape can be divided as:

- Left vertical rectangle: but it's not straight.

Notice that the shape has a "notch" on the bottom left.

Total area can be calculated as the area of the outer rectangle minus the missing part.

Outer rectangle: width 7 yd, height = total height. From top to bottom, the lowest point is y=-5, so height 5 yd? But at x=0, what is the height?

From the construction, at x=0, the shape goes from y=0 down to y= - ? When we go from (0,0) to (7,0) to (7,-3) to (3,-3) to (3,-5) to (6,-5) to (6,0) to (0,0).

From (6,0) to (0,0) is along y=0, so at x=0, it's only at y=0, no downward extension? That can't be.

I think I have a mistake.

When we go from (6,-5) up to (6,0), then left to (0,0), so the left side from x=0 to x=6 at y=0 is the top, but for x<6, at y<0, is there material?

For example, at x=0, the shape is only at y=0? That doesn't make sense for a closed shape.

Perhaps the last segment is from (6,-5) up to (6,0), then left to (0,0), but then the left side from (0,0) down is not defined.

Actually, in such diagrams, the shape is closed by returning to start, so from (6,0) to (0,0) is the top edge, and from (0,0) down to where? It must be that from (0,0) down to (0,-5) or something, but that's not indicated.

Let's read the labels: "7 yd" on top, "3 yd" on right top, "4 yd" on the indent, "2 yd" on the bottom indent, "3 yd" on the right bottom.

Probably, the total height is 3 + 2 = 5 yd, and the width is 7 yd.

The missing part is a rectangle of 4 yd by 2 yd or something.

Standard way for such C-shapes: area = area of large rectangle minus area of the cut-out.

Large rectangle: 7 yd wide × 5 yd high = 35 yd² (since 3+2=5)

Cut-out rectangle: the indentation is 4 yd wide and 2 yd high? From the diagram, the horizontal indent is 4 yd, and the vertical drop is 2 yd, so the cut-out is 4 yd × 2 yd = 8 yd²

Then area = 35 - 8 = 27 yd²

Verify with addition:

Divide into three rectangles:

1. Top rectangle: 7 yd × 3 yd = 21 yd²

2. Bottom left rectangle: but after the indent.

From the path: after going down 3 yd on right, left 4 yd, down 2 yd, right 3 yd, so the bottom part is from x=3 to x=6 at y=-5 to y=-3? Let's define.

Rectangle A: top: x=0 to 7, y=0 to -3 → area 7*3=21

But at y<-3, only from x=3 to x=6 is present? From the move: at y=-3, we are at x=3, then down to y=-5 at x=3, then right to x=6 at y=-5, then up to y=0 at x=6.

So the additional part is from x=3 to 6, y=-3 to -5 → that's a rectangle 3 yd wide (6-3=3) and 2 yd high, area 6 yd²

But is that correct? From x=3 to 6, y=-3 to -5, yes.

Then total area = top rectangle 21 + bottom rectangle 6 = 27 yd²

Yes.

The bottom rectangle is 3 yd wide (since 6-3=3) and 2 yd high, area 6.

Total 21+6=27.

✔ Confirmed.

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Problem 6:

L-shaped or corner shape.

Dimensions: total width 5 ft, total height 8.8 ft, but with a step.

From diagram: bottom part is 5 ft wide, 2.5 ft high.

Then on top of that, on the right, a vertical part that is 8.8 ft high, but since the bottom is 2.5 ft, the additional height is 8.8 - 2.5 = 6.3 ft, and width? The total width is 5 ft, and the bottom is 5 ft, but the vertical part on right has width? Label "3.8 ft" is on the horizontal part of the step.

Actually, the shape has:

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

- Right vertical rectangle: width = ? The total width is 5 ft, and the bottom is full width, but the vertical part on right extends up. The label "3.8 ft" is likely the width of the vertical part? Or the length of the horizontal segment.

From diagram: after the bottom 2.5 ft high, there is a horizontal segment of 3.8 ft, then up 8.8 ft.

But 8.8 ft is the total height, so from the top of the bottom part, up 8.8 - 2.5 = 6.3 ft.

And the width of the vertical part: since the total width is 5 ft, and the horizontal segment is 3.8 ft, which is probably the overhang or something.

Actually, the vertical part on the right has width = total width - the left part, but there is no left part specified.

Standard interpretation: the shape is composed of two rectangles:

1. Bottom: 5 ft × 2.5 ft = 12.5 ft²

2. Right vertical: this sits on top of the bottom, but only on the right part. The width of this vertical rectangle is the amount that sticks out, but in this case, since the bottom is full width, and the vertical part is attached to the right, its width should be such that it doesn't exceed.

The label "3.8 ft" is on the horizontal line between the bottom and the vertical part, likely indicating the length of that horizontal segment, which is the width of the vertical rectangle.

In other words, the vertical rectangle is 3.8 ft wide and (8.8 - 2.5) = 6.3 ft high.

Then area of vertical part = 3.8 × 6.3

Calculate: 3.8 × 6.3

3.8 × 6 = 22.8, 3.8 × 0.3 = 1.14, total 23.94 ft²

Then total area = bottom + vertical = 12.5 + 23.94 = 36.44 ft²

But is that correct? The bottom is 5 ft wide, and the vertical part is 3.8 ft wide, so if it's attached to the right, then the total width is still 5 ft, which matches, and the vertical part is within the width.

Yes.

We can think of it as a large rectangle 5 ft × 8.8 ft = 44 ft², minus the missing part on the top left.

Missing part: width = 5 - 3.8 = 1.2 ft, height = 8.8 - 2.5 = 6.3 ft, area = 1.2 × 6.3 = 7.56 ft²

Then area = 44 - 7.56 = 36.44 ft² same as before.

✔ So area = 36.44 ft²

But perhaps keep as fraction or decimal.

3.8 × 6.3 = let's compute exactly: 38/10 * 63/10 = (38*63)/(100)

38*60=2280, 38*3=114, total 2394, so 23.94

12.5 + 23.94 = 36.44

Yes.

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Problem 7:

Irregular pentagon or hexagon.

Dimensions: bottom 20 km, left side 10 km, top 16 km, right side 5 km.

Can be divided into a rectangle and a triangle or two rectangles.

Divide vertically or horizontally.

One way: draw a line from the top-right corner down to the bottom.

The shape has a "step" on the right.

From left: width 20 km at bottom, but top is 16 km, so the right part is indented.

Specifically, the right side has a vertical segment of 5 km, then horizontal to the left.

So, we can divide into:

- Left rectangle: width 16 km, height 10 km? But the left side is 10 km, but the top is 16 km, bottom is 20 km.

Actually, the difference in width is 20 - 16 = 4 km, which is on the right.

The right part has a rectangle that is 4 km wide and 5 km high? Let's see.

Total height on left is 10 km, on right is 5 km, so the right part is shorter.

So, divide into:

Rectangle A: left part, width 16 km, height 10 km = 160 km²

But then the right part: from x=16 to x=20, width 4 km, but height is only 5 km (since the right side is 5 km tall), and it's at the bottom.

So rectangle B: 4 km × 5 km = 20 km²

Total area = 160 + 20 = 180 km²

Is that correct? The left rectangle is 16x10, which covers from x=0 to 16, y=0 to 10.

The right rectangle is from x=16 to 20, y=0 to 5.

But in the shape, at x=16 to 20, y=5 to 10 is empty, which is fine, and the top is from x=0 to 16 at y=10, and from x=16 to 20 at y=5, so yes, the shape is covered.

Area = 16*10 + 4*5 = 160 + 20 = 180 km²

✔ Confirmed.

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Problem 8:

T-shaped or H-shaped? From diagram: top bar 22 mm wide, 2.5 mm high.

Then below, two legs: each 6 mm wide? Labels: "6 mm" on the sides, "4 mm" at the bottom of each leg.

Actually, it's like a T with a gap, or an H.

Top rectangle: 22 mm × 2.5 mm = 55 mm²

Then below, two vertical rectangles: each is 6 mm wide? But the label "6 mm" is on the side, likely the height of the legs.

And "4 mm" is the width of each leg at the bottom.

Also, the distance between the legs: total width 22 mm, each leg is 4 mm wide, so the gap between them is 22 - 4 - 4 = 14 mm, but that might not be needed.

The legs are attached to the bottom of the top bar.

Each leg has width 4 mm (as per "4 mm" at bottom), and height 6 mm (as per "6 mm" on side).

But are the legs 4 mm wide or 6 mm? The label "6 mm" is vertical, so likely height, and "4 mm" is horizontal, so width.

In the diagram, for each leg, it says "6 mm" on the side (height), and "4 mm" at the bottom (width).

Also, the top bar is 22 mm wide, 2.5 mm high.

The legs are positioned such that they are under the top bar.

Typically, for such shapes, the legs are centered or something, but here, since no specification, assume they are attached directly below.

The width of each leg is 4 mm, height 6 mm.

But are they separate? Yes.

So area of top bar: 22 × 2.5 = 55 mm²

Area of left leg: 4 × 6 = 24 mm²

Area of right leg: 4 × 6 = 24 mm²

Total area = 55 + 24 + 24 = 103 mm²

But is there overlap? The legs are attached to the bottom of the top bar, so no overlap in area calculation since we're adding distinct regions.

Yes.

To confirm, the total shape has no overlapping parts.

So area = 103 mm²

✔ Check: 22*2.5=55, 4*6=24, twice is 48, 55+48=103 ✔️

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Now, summarize all answers:

1. 224 m²

2. 230 cm²

3. 50 yd²

4. 36 mm²

5. 27 yd²

6. 36.44 ft²

7. 180 km²

8. 103 mm²

For problem 6, 36.44 is exact since 3.8 * 6.3 = 23.94, and 12.5 + 23.94 = 36.44.

We can write it as fraction: 3.8 = 38/10 = 19/5, 6.3 = 63/10, so (19/5)*(63/10) = (19*63)/(50) = 1197/50 = 23.94

12.5 = 25/2 = 625/50? Better: 12.5 + 23.94 = 36.44, or 3644/100 = 911/25, but decimal is fine.

Since the inputs are decimals, output decimal is acceptable.

Now, final answers.

Final Answer:
1. 224 m²
2. 230 cm²
3. 50 yd²
4. 36 mm²
5. 27 yd²
6. 36.44 ft²
7. 180 km²
8. 103 mm²