Determining Rectilinear Area Worksheet Download - Free Printable

Let’s solve each shape one by one. We’ll break each rectilinear shape into rectangles, find the area of each rectangle, and add them up.

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Shape 1:
It looks like a big rectangle with a small rectangle cut out from the bottom left.
- Big rectangle: width = 3 mm, height = 7 mm → Area = 3 × 7 = 21
- Cut-out part: width = (3 - 2) = 1? Wait — let’s look again.
Actually, better to split it as two rectangles:
Top part: 3 wide × 2 high = 6
Bottom right part: (7 - 2) = 5 high, and width = ? The total width is 3, but the left side has a step of 2 down and then 4 up? Hmm.

Wait — let’s use coordinates or just subtract.

Alternative method: Imagine full rectangle 3×7 = 21. Then subtract the missing piece on bottom left.
The missing piece: width = (3 - 2) = 1? No — actually, looking at the diagram:

Left side: from bottom, goes up 4, then right 2, then up 2 more to top? Total height is 7, so 4 + 2 = 6? That doesn’t match.

Wait — let me re-read the labels.

In Shape 1:
- Top horizontal: 3
- Right vertical: 7
- Left side: from bottom, up 4, then right 2, then up to top? So the “step” is 2 units right and (7 - 4) = 3 units up? But labeled 2 on the top-left horizontal.

Actually, standard way: Split into two rectangles.

Rectangle A (top): width 3, height 2 → area = 6
Rectangle B (bottom right): width = 3 - 2 = 1? No — wait, the bottom part extends full width? Let's think differently.

Better approach: Use the outer dimensions and subtract the notch.

Full rectangle if no notch: 3 wide × 7 tall = 21
Notch: it’s a rectangle on the bottom left. Its width is 2 (given), its height is (7 - 4) = 3? Because left side says 4 from bottom to the step, so the notch height is 7 - 4 = 3? But that would make the notch 2×3=6, so area = 21 - 6 = 15.

But let’s verify by adding parts:

Top rectangle: 3 × 2 = 6
Bottom rectangle: the remaining height is 7 - 2 = 5, but only the right part? Width of bottom part: since the step is 2 from left, the bottom rectangle is (3 - 2) = 1 wide? No — actually, the bottom part should be full width minus the step? I’m confusing myself.

Let me draw mentally:

From bottom-left corner:
- Go up 4 mm
- Then go right 2 mm
- Then go up 2 mm (to reach top, since 4+2=6, but total height is 7? Inconsistency?)

Wait — the label "7" is on the right side, meaning total height is 7.
Label "4" is on the left side from bottom to the inner corner.
Then from that inner corner, going right 2 mm (labeled), then up to top — that last segment must be 7 - 4 = 3 mm? But it’s not labeled. However, the top horizontal is labeled 3, which matches the full width.

So the shape can be seen as:
- A rectangle on the right: width = 3 - 2 = 1? No.

Actually, best way: divide vertically.

Left part: from x=0 to x=2, y=0 to y=4 → rectangle 2×4 = 8
Right part: from x=2 to x=3, y=0 to y=7 → rectangle 1×7 = 7
Total = 8 + 7 = 15

Yes! That works.

So Shape 1: 15 mm²

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Shape 2:
Looks like a cross or T-shape.

Split into three rectangles? Or two.

Top horizontal bar: width 3, height ? The right side has a label "2", and the vertical part on left is 6.

Actually, the shape has:
- A vertical rectangle on left: width ? Let's see.

From diagram:
- Left vertical side: 6
- Top horizontal: 3
- Right extension: 3 long, 2 high

So perhaps:
Main vertical rectangle: width = ? The total width at top is 3, but the right part sticks out.

Actually, the shape is symmetric? No.

Let me define:

The central part: from left, width = 3 - 3? Confusing.

Better: imagine the shape as a large rectangle minus nothing, but composed of parts.

Part A: the left vertical stem: width = let's say W, height = 6. But what is W?

Notice that the top horizontal segment is 3, and the right arm is also 3 long and 2 high.

Perhaps the main body is 3 wide and 6 high, but with an extra rectangle on the right middle.

Standard way: split horizontally.

Top part: a rectangle 3 wide × ? The height of the top part isn't given directly.

Look at the right side: there's a segment labeled 2, which is the height of the right arm.

And the left side is 6, which is the full height.

So the right arm is attached to the middle of the left rectangle.

So:
- Left rectangle: width = 3 - 3? No.

Actually, the total width at the top is 3, and the right arm extends 3 to the right, so the left part must be narrower.

I think I need to interpret the labels carefully.

In Shape 2:
- The top horizontal line is labeled 3 — this is the width of the top part.
- The right vertical segment of the arm is labeled 2 — height of the arm.
- The left vertical side is labeled 6 — full height of the left part.
- The horizontal segment of the arm is labeled 3 — length of the arm.

So, the shape consists of:
1. A vertical rectangle on the left: width = ? Let's call it W. Height = 6.
2. A horizontal rectangle on the right: width = 3, height = 2, attached to the left rectangle.

But where is it attached? Probably centered or at some point.

The top of the left rectangle is at the same level as the top of the arm? The label "3" on top suggests that the top width is 3, which might be the width of the left part.

Assume the left part is 3 wide and 6 high. Then the right arm is attached to the right side, starting from some height.

The arm is 3 long and 2 high. If it's attached to the middle, then the area would be left rectangle plus arm.

But if left is 3x6=18, arm is 3x2=6, total 24, but they overlap? No, if attached externally, no overlap.

But visually, in such diagrams, the arm is usually attached without overlapping, so total area = area of left rect + area of arm.

But is the left rect 3 wide? The top label is 3, and it's the top of the left part, so yes.

However, when you attach the arm to the right, the total width becomes 3 + 3 = 6, but that's fine.

But let's check the height: the arm is 2 high, and it's placed somewhere on the 6-high rectangle. Since no other labels, probably it's attached such that it doesn't extend beyond, so we can assume it's fully attached.

So area = (3 * 6) + (3 * 2) = 18 + 6 = 24

But is there overlap? Only if the arm is inside, but typically in these problems, the shapes are non-overlapping unions.

Another way: the shape can be seen as a large rectangle 6 high and (3+3)=6 wide, but with corners missing? No.

Let's calculate by dividing.

Divide into:
- Bottom part: from y=0 to y=h1, but complicated.

Notice that the right arm is 2 high, and the left is 6 high, so the arm is likely attached to the top or bottom or middle.

In many such problems, the arm is attached to the side, and the height is measured from the base.

Perhaps the 6 includes the arm's position.

Let's look for consistency.

I recall that in Shape 2, a common interpretation is:

The main rectangle is 3 wide and 6 high, area 18.

Then a rectangle 3 wide and 2 high is attached to the right side, but since the main is 6 high, and the arm is 2 high, it must be attached at a specific location, but for area, as long as no overlap, we add.

But if attached to the side, and the arm is 2 high, it could be attached at the top, bottom, or middle, but the area added is still 3*2=6, so total 24.

But let's verify with another method.

Imagine the bounding box: width = 3 (left) + 3 (arm) = 6, height = 6.

Area of bounding box = 6*6=36.

Now, what is missing? On the top-right and bottom-right, if the arm is in the middle, there are two rectangles missing.

If the arm is attached to the middle of the right side, then above and below the arm, there are empty spaces.

Height of arm is 2, total height 6, so space above and below: (6-2)/2 = 2 each, assuming centered.

Then missing areas: top-right rectangle: width 3, height 2 = 6

Bottom-right rectangle: width 3, height 2 = 6

So total missing = 12, so area = 36 - 12 = 24.

Same as before.

If not centered, but since no specification, and area is the same regardless of vertical position as long as no overlap, so 24 is correct.

So Shape 2: 24 mm²

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Shape 3:
L-shaped.

Can be split into two rectangles.

Option 1:
- Vertical part: width 2, height ? Total height is not given, but from diagram, the right side has a label 3, and bottom has 2.

Labels:
- Top horizontal: 6
- Right vertical: 3
- Bottom horizontal: 2
- Inner vertical: 2

So, the shape has:
- A top rectangle: width 6, height ?
- A bottom-left rectangle: width 2, height ?

From the inner corner: from bottom-left, go right 2, then up 2, then right to end, then down 3 to bottom? Total height should be consistent.

Let's define:

The full height on the right is 3, so from top to bottom on right is 3.

On the left, from bottom to the inner corner is 2 (labeled), then from inner corner to top is ? Since total height is 3, and the inner corner is at height 2 from bottom, so from there to top is 3 - 2 = 1? But not labeled.

Actually, the vertical segment from inner corner to top is not labeled, but we can infer.

Split into:
- Rectangle A (bottom-left): width 2, height 2 → area = 4
- Rectangle B (top): width 6, height ? The height of the top part: since on the right, the full height is 3, and the bottom part is 2 high, so the top part must be 3 - 2 = 1 high? But that seems small.

If the bottom-left rectangle is 2x2, and it's at the bottom, then above it, from y=2 to y=3, there is a rectangle spanning the full width 6, so height 1, area 6*1=6.

Total area = 4 + 6 = 10.

But is the top part really 1 high? Let's see the labels.

The right side is labeled 3, which is the total height.

The bottom horizontal is labeled 2, which is the width of the bottom part.

The inner vertical is labeled 2, which is the height of the bottom-left part.

So yes, the remaining height on the right is 3 - 2 = 1, and since the top part spans the full width 6, area = 6*1 = 6.

Plus bottom-left 2*2=4, total 10.

Another way: full rectangle 6x3=18, minus the missing part on bottom-right.

Missing part: width = 6 - 2 = 4, height = 2 (since bottom is 2 high, and missing from x=2 to x=6, y=0 to y=2), so area 4*2=8, so 18-8=10.

Same.

So Shape 3: 10 mm²

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Shape 4:
Similar to Shape 3.

Labels:
- Top horizontal: 8
- Right vertical: 2 (for the top part?)
- Bottom horizontal: 5
- Inner vertical: 5

Diagram: L-shape, with a step.

From bottom-left: go right 5, then up 5, then right to end, then down 2 to bottom? Total height on right is 2, but that can't be if inner is 5.

Let's read carefully.

The label "2" is on the right side, probably the height of the top-right part.

Label "5" on bottom, width of bottom part.

Label "5" on the inner vertical, height of the left part.

Label "8" on top, full width.

So, similar to Shape 3.

Split into:
- Bottom-left rectangle: width 5, height 5 → area = 25
- Top rectangle: width 8, height ? The height of the top part: on the right, the total height from top to bottom is not directly given, but the label "2" is on the right vertical segment, which is likely the height of the top part.

In the diagram, the right side has a short segment labeled 2, which is the height of the protruding part.

Since the inner vertical is 5, and it's on the left, the total height of the shape is 5 (from bottom to the step) plus the height of the top part.

The top part's height is given as 2 on the right.

So total height = 5 + 2 = 7? But not labeled.

For area, we don't need total height.

The top rectangle spans the full width 8, and has height 2, so area = 8*2 = 16.

The bottom-left rectangle is 5 wide and 5 high, area 25.

But do they overlap? The bottom-left is from x=0 to 5, y=0 to 5.

The top rectangle is from y=5 to y=7 (assuming), x=0 to 8.

So no overlap, total area = 25 + 16 = 41.

Is that correct? The bottom part only goes to x=5, but the top goes to x=8, so yes.

We can think of the missing part: if full rectangle 8x7=56, but what is missing? On the bottom-right, from x=5 to 8, y=0 to 5, a rectangle 3x5=15, so 56-15=41. Same.

But is the total height 7? From the labels, the inner vertical is 5, and the top-right vertical is 2, and they are adjacent, so yes, total height 7.

So Shape 4: 41 mm²

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Shape 5:
Rectangle with a notch on the left.

Labels:
- Top horizontal: 9
- Right vertical: 5
- Bottom horizontal: 8
- Left vertical: 3 (but it's indented)

So, the shape is almost a rectangle 9x5, but with a notch on the left bottom.

The bottom is labeled 8, which is less than 9, so the notch is on the left.

Specifically, from bottom-left, instead of going full width, it goes right 8, but the top is 9, so there's a step.

The left side has a label 3, which is the height of the notch or something.

Let's see: the vertical segment on the left is labeled 3, and it's the height from bottom to the inner corner.

Then from there, it goes right to the full width.

So, the shape can be seen as:
- A large rectangle 9x5 = 45
- Minus a small rectangle on the bottom-left: width = 9 - 8 = 1, height = 3? Because the bottom is 8, so the missing width is 1, and the height of the missing part is 3 (since the left side drops down 3).

In the diagram, the left side has a segment of 3 down, then it turns right, and the bottom is 8, while top is 9, so yes, the missing part is a rectangle 1 wide (9-8) and 3 high.

Area of missing part = 1 * 3 = 3

So area of shape = 45 - 3 = 42

Split into parts:
- Top part: full width 9, height = 5 - 3 = 2? Because the notch is 3 high, so above it, height 2, area 9*2=18
- Bottom part: width 8, height 3, area 24
Total = 18+24=42

Same.

So Shape 5: 42 mm²

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Shape 6:
L-shape.

Labels:
- Top horizontal: 6
- Right vertical: 4
- Bottom horizontal: 2
- Left vertical: 6

So, similar to previous.

Split into:
- Left rectangle: width ? Height 6
- Bottom rectangle: width 2, height ?

From diagram, the bottom is labeled 2, which is the width of the bottom part.

The left side is 6, full height.

The right side is 4, which is the height of the top part.

So, total height is 6, so the bottom part height is 6 - 4 = 2? But not labeled.

Assume:
- Top rectangle: width 6, height 4 → area = 24
- Bottom-left rectangle: width 2, height 2 (since 6-4=2) → area = 4
Total = 28

Check with subtraction: full rectangle 6x6=36, minus missing part on bottom-right: width 6-2=4, height 2, area 8, so 36-8=28.

Yes.

So Shape 6: 28 mm²

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Shape 7:
Almost a square, with a notch on the left.

Labels:
- Top horizontal: 7
- Right vertical: 7
- Bottom horizontal: 6
- Left vertical: 4

So, similar to Shape 5.

Full rectangle 7x7=49

Missing part on bottom-left: width = 7 - 6 = 1, height = ? The left side has a segment of 4, but that's from bottom to the inner corner?

In the diagram, the left side is labeled 4, which is the height from bottom to the step.

Then from there to top is 7 - 4 = 3, but not needed.

The missing part is a rectangle on the bottom-left: width = 7 - 6 = 1 (since bottom is 6, top is 7), height = 4 (the labeled left segment).

So area missing = 1 * 4 = 4

Area of shape = 49 - 4 = 45

Split:
- Top part: width 7, height 7 - 4 = 3 → area 21
- Bottom part: width 6, height 4 → area 24
Total = 45

Yes.

So Shape 7: 45 mm²

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Shape 8:
Irregular shape.

Labels:
- Top-left horizontal: 2
- Then down 3
- Then right to end, but labeled 5 on left? Let's see.

From diagram:
- Start at top-left: go right 2, then down 3, then right to the end of the bottom, which is labeled 6 at bottom.
- Left side: from bottom, up 5, then right 2? The label "5" is on the left vertical, but it's from bottom to the inner corner.

Actually, labels:
- Top horizontal segment: 2
- Vertical segment down from there: 3
- Left vertical segment from bottom: 5
- Bottom horizontal: 6

So, the shape has a step on the top-left.

Split into rectangles.

One way:
- Bottom rectangle: width 6, height ? The left side has 5 from bottom to the step, but the step is at height 5, and above it, there is a part.

From the top: the top-left has a rectangle 2 wide and 3 high? But then it connects.

Define points.

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

Go right 6 to (6,0) — bottom.

Go up to (6,h) — but right side not labeled.

From left: from (0,0) up to (0,5) — labeled 5.

Then from (0,5) right to (2,5) — but not labeled, but then down to (2,2)? The vertical segment is labeled 3, which is from (2,5) down to (2,2)? Then from (2,2) right to (6,2)? But bottom is at y=0.

This is messy.

From the labels:
- The vertical segment on the left is 5, from bottom to the inner corner.
- From that inner corner, it goes right 2? Not labeled, but then down 3 to the bottom level? But the bottom is already there.

Perhaps the 3 is the height of the top part.

Let's think of the shape as having two parts:

Part A: a rectangle on the bottom: width 6, height h1

Part B: a rectangle on the top-left: width 2, height 3

But how are they connected?

From the diagram, it seems that from the bottom-left, you go up 5, then right 2, then down 3, then right to the end.

So, the point after going up 5 and right 2 is at (2,5), then down 3 to (2,2), then right to (6,2), then down to (6,0), then left to (0,0).

But then from (0,0) to (0,5) is the left side.

So the shape has vertices at: (0,0), (6,0), (6,2), (2,2), (2,5), (0,5), back to (0,0).

Yes.

So, we can split into:
- Rectangle 1: from x=0 to 6, y=0 to 2 → area 6*2 = 12
- Rectangle 2: from x=0 to 2, y=2 to 5 → area 2*3 = 6
Total = 18

Is that correct? From y=2 to 5, only from x=0 to 2, yes.

And from y=0 to 2, full width 6.

No overlap.

So area = 12 + 6 = 18

Another way: the whole thing can be seen as a large rectangle 6x5=30, minus the missing part on the top-right.

Missing part: from x=2 to 6, y=2 to 5, which is 4 wide and 3 high, area 12, so 30-12=18.

Same.

So Shape 8: 18 mm²

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Shape 9:
T-shape or something.

Labels:
- Top horizontal: 2
- Vertical down: 5
- Then right to end, bottom horizontal: 6
- Left vertical: 3

From diagram: it's like a capital T but rotated or something.

Vertices: start at top-left: go right 2, down 5, then right to end of bottom, which is 6, but the bottom is labeled 6, and left side has 3.

Probably: from bottom-left, go up 3, then right to the stem, then up 5, etc.

Assume:
- Bottom rectangle: width 6, height 3 → area 18
- Top rectangle: width 2, height 5, centered or attached.

But where is it attached? The top is labeled 2, and it's above the bottom.

The vertical segment is 5, from the top of the bottom part to the top.

So, if the bottom is 3 high, and the stem is 5 high, total height 8.

The top rectangle is 2 wide, and it's attached to the bottom rectangle.

Since the bottom is 6 wide, and top is 2 wide, likely centered or at some position, but for area, as long as no overlap, we add.

Area of bottom: 6*3=18

Area of top: 2*5=10

Total = 28

Is there overlap? Only if the top is within the bottom, but typically, the stem is on top, so no overlap.

In the diagram, the top part is above the bottom part, so yes.

So Shape 9: 28 mm²

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Shape 10:
Complex shape.

Labels:
- Left vertical: 2
- Then right 4
- Then up? Label "4" on top horizontal
- Then down 9 on right

From diagram: it's like a staircase or zigzag.

Vertices: start at bottom-left: go up 2, then right 4, then up to top, but top has a horizontal labeled 4, then down 9 on right.

Also, the bottom is not labeled, but we can infer.

From the path:
- From (0,0) up to (0,2) — labeled 2
- Then right to (4,2) — labeled 4
- Then up to (4,h) — but then right to (8,h)? The top horizontal is labeled 4, so from (4,h) to (8,h)
- Then down to (8,0) — labeled 9, so height is 9.

But from (0,0) to (0,2) is 2, then to (4,2), then to (4,9)? Because down 9 from top to bottom, so if bottom is y=0, top is y=9.

Then from (4,9) to (8,9), then down to (8,0).

But then from (8,0) to (0,0), but there's a gap.

The shape is only the outlined part, so from (0,0) to (0,2) to (4,2) to (4,9) to (8,9) to (8,0) to (0,0)? But that would include the area under, but in the diagram, it's probably only the polygon described.

In rectilinear shapes, it's the boundary.

So the shape has vertices at: (0,0), (0,2), (4,2), (4,9), (8,9), (8,0), and back to (0,0).

But from (8,0) to (0,0) is the bottom, so it's a single polygon.

To find area, we can split into rectangles.

One way:
- Rectangle A: from x=0 to 4, y=0 to 2 → area 4*2=8
- Rectangle B: from x=0 to 4, y=2 to 9 → but wait, from y=2 to 9, x=0 to 4, but in the shape, from (0,2) to (4,2) to (4,9), so yes, a rectangle 4 wide (x=0 to 4) and 7 high (y=2 to 9), area 4*7=28
- Rectangle C: from x=4 to 8, y=0 to 9 → area 4*9=36

But this counts the region from x=0 to 4, y=0 to 9 twice? No.

Rectangle A is y=0 to 2, x=0 to 4

Rectangle B is y=2 to 9, x=0 to 4

Rectangle C is y=0 to 9, x=4 to 8

So together, they cover:
- x=0 to 4, y=0 to 9: covered by A and B
- x=4 to 8, y=0 to 9: covered by C

No overlap, and it matches the shape.

Total area = 8 + 28 + 36 = 72

But is that correct? The shape is only the area enclosed, which is exactly this.

We can calculate as a large rectangle minus nothing, but it's irregular.

Another way: the shape can be seen as a rectangle 8 wide and 9 high, but with a notch on the top-left? No.

From x=0 to 4, the height is from y=0 to 9, but in our case, it is, except that from y=0 to 2, it's included, and y=2 to 9 also.

In fact, for x=0 to 4, the shape goes from y=0 to 9, and for x=4 to 8, also y=0 to 9, so actually it's a full rectangle 8x9=72.

Is that right? In the description, from (0,0) to (0,2) to (4,2) to (4,9) to (8,9) to (8,0) to (0,0).

From (0,0) to (0,2) is up, then to (4,2) is right, then to (4,9) is up, then to (8,9) is right, then to (8,0) is down, then to (0,0) is left.

So the path encloses the area where for x from 0 to 4, y from 0 to 9? No.

Let's plot the points:

- (0,0)
- (0,2) — so at x=0, y from 0 to 2
- (4,2) — so at y=2, x from 0 to 4
- (4,9) — at x=4, y from 2 to 9
- (8,9) — at y=9, x from 4 to 8
- (8,0) — at x=8, y from 9 to 0
- (0,0) — at y=0, x from 8 to 0

So the enclosed region is:
- For x from 0 to 4: y from 0 to 9? But at x=2, for example, what is the range of y?

Actually, the boundary is: left side x=0 from y=0 to 2, then bottom of the "step" at y=2 from x=0 to 4, then right side of the step at x=4 from y=2 to 9, then top at y=9 from x=4 to 8, then right side x=8 from y=9 to 0, then bottom y=0 from x=8 to 0.

So, the region inside is:
- Below y=2, for all x from 0 to 8 (since bottom is y=0 to 8)
- Above y=2, only for x from 4 to 8? No.

Let's think: when you go from (0,0) up to (0,2), then right to (4,2), then up to (4,9), then right to (8,9), then down to (8,0), then left to (0,0).

So, the area to the left of this path.

For a given x, the y-range.

For x from 0 to 4:
- The lower boundary is y=0 (from the bottom)
- The upper boundary: from x=0 to 4, the top is not direct; actually, for x in [0,4], the upper boundary is y=2 for the first part, but then at x=4, it goes up.

Actually, the shape has a "bay" on the left.

Specifically, for x from 0 to 4, the shape goes from y=0 up to y=2, and then stops, but no—when you go from (0,2) to (4,2) to (4,9), that means that for x<4, above y=2, is it included? In the path, after (4,2) it goes up to (4,9), so for x<4, the region above y=2 is not bounded on the right until x=4.

I think I made a mistake.

In a simple polygon, the area enclosed by the path.

The path is: start at (0,0), move to (0,2), then to (4,2), then to (4,9), then to (8,9), then to (8,0), then back to (0,0).

This path encloses a region that includes:
- The rectangle from x=0 to 8, y=0 to 2 (because from (0,0) to (0,2) to (4,2) to (8,2)? No, it goes to (4,2) then up, not to (8,2).

Actually, from (4,2) it goes up to (4,9), not right to (8,2).

So, the region is:
- From x=0 to 4, y=0 to 2 (a rectangle)
- Plus from x=4 to 8, y=0 to 9 (another rectangle)
- Plus from x=0 to 4, y=2 to 9? But that would be connected, but in the path, when it goes from (0,2) to (4,2) to (4,9), it means that the area between x=0 to 4 and y=2 to 9 is not directly connected; actually, it is included because the path goes around it.

Let's use the shoelace formula to be sure.

List the vertices in order:

1. (0,0)
2. (0,2)
3. (4,2)
4. (4,9)
5. (8,9)
6. (8,0)
7. back to (0,0)

Shoelace formula:

Sum1 = sum of x_i * y_{i+1}
= 0*2 + 0*2 + 4*9 + 4*9 + 8*0 + 8*0 = 0 + 0 + 36 + 36 + 0 + 0 = 72? Let's calculate properly.

Vertices in order:
A: (0,0)
B: (0,2)
C: (4,2)
D: (4,9)
E: (8,9)
F: (8,0)
G: (0,0) // back

Shoelace:
List coordinates in order, repeat first at end:

x y
0 0
0 2
4 2
4 9
8 9
8 0
0 0 // back

Now, sum of x_i * y_{i+1}:
0*2 = 0 (x1*y2)
0*2 = 0 (x2*y3)
4*9 = 36 (x3*y4)
4*9 = 36 (x4*y5)
8*0 = 0 (x5*y6)
8*0 = 0 (x6*y7)
Sum1 = 0+0+36+36+0+0 = 72

Sum of y_i * x_{i+1}:
0*0 = 0 (y1*x2)
2*4 = 8 (y2*x3)
2*4 = 8 (y3*x4)
9*8 = 72 (y4*x5)
9*8 = 72 (y5*x6)
0*0 = 0 (y6*x7)
Sum2 = 0+8+8+72+72+0 = 160

Area = |Sum1 - Sum2| / 2 = |72 - 160| / 2 = 88/2 = 44

Oh! So 44.

I see, I misunderstood the shape.

With shoelace, it's 44.

How to split:

The shape can be divided into:
- Rectangle 1: x=0 to 4, y=0 to 2 → area 8
- Rectangle 2: x=4 to 8, y=0 to 9 → area 4*9=36
- But then the part from x=0 to 4, y=2 to 9 is not included? But in the path, it is included because from (0,2) to (4,2) to (4,9), so the area between x=0 to 4 and y=2 to 9 is inside the polygon.

In the shoelace, we got 44, and 8 + 36 = 44, but that misses the top-left part.

8 + 36 = 44, but what about the rectangle from x=0 to 4, y=2 to 9? That would be additional 4*7=28, too much.

I think I have the wrong interpretation.

From the shoelace calculation, area is 44.

Let me verify the shoelace.

Points:
1. (0,0)
2. (0,2)
3. (4,2)
4. (4,9)
5. (8,9)
6. (8,0)
7. (0,0)

Sum of x_i y_{i+1}:
x1y2 = 0*2 = 0
x2y3 = 0*2 = 0
x3y4 = 4*9 = 36
x4y5 = 4*9 = 36
x5y6 = 8*0 = 0
x6y7 = 8*0 = 0
Sum1 = 0+0+36+36+0+0 = 72

Sum of y_i x_{i+1}:
y1x2 = 0*0 = 0
y2x3 = 2*4 = 8
y3x4 = 2*4 = 8
y4x5 = 9*8 = 72
y5x6 = 9*8 = 72
y6x7 = 0*0 = 0
Sum2 = 0+8+8+72+72+0 = 160

|72 - 160| = 88, divided by 2 is 44. Yes.

Now, geometrically, the shape is:
- A rectangle from x=0 to 8, y=0 to 2: area 16
- Plus a rectangle from x=4 to 8, y=2 to 9: area 4*7=28
Total 16+28=44

Yes! Because from y=0 to 2, full width 8.
From y=2 to 9, only from x=4 to 8, since the left part is cut off.

In the path, after (0,2) it goes to (4,2), then up to (4,9), so for y>2, only x>=4 is included.

And from (4,9) to (8,9) to (8,0), so yes.

So area = (8 * 2) + (4 * 7) = 16 + 28 = 44

Perfect.

So Shape 10: 44 mm²

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Now, summarize all answers:

1) 15
2) 24
3) 10
4) 41
5) 42
6) 28
7) 45
8) 18
9) 28
10) 44

Final Answer:
1. 15
2. 24
3. 10
4. 41
5. 42
6. 28
7. 45
8. 18
9. 28
10. 44