Math worksheet for calculating the area and perimeter of irregular polygons.

Let’s solve each problem step by step. We’ll find the area and perimeter for each irregular polygon.

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Problem 1:

Shape is like an “L” made of two rectangles.

We can split it into:
- Left rectangle: 8 cm wide × 10 cm tall → area = 8 × 10 = 80 cm²
- Top right rectangle: (18 - 8) = 10 cm wide × 4 cm tall → area = 10 × 4 = 40 cm²
Total Area = 80 + 40 = 120 cm²

Perimeter: Add all outer sides.
Start from bottom left, go clockwise:
- Bottom: 8 cm
- Right side up: 10 cm
- Top right horizontal: 10 cm (since 18 - 8)
- Down vertical: 4 cm
- Left horizontal: 4 cm? Wait — let's trace carefully.

Actually, better to list all outer edges:

From bottom-left corner:
→ Right along bottom: 8 cm
↑ Up left side: 10 cm
→ Right top of left part: but wait — shape goes right 18 cm total at top.

Better method: Use "bounding box" trick or add all visible segments.

List all outer sides in order (clockwise):

Start at bottom-left:
1. Right → 8 cm
2. Up → 10 cm
3. Right → (18 - 8) = 10 cm
4. Down → 4 cm
5. Left → ? Wait — no, after going down 4 cm, we’re at the inner corner. Then we go left? No — actually, from that point, we go left along the bottom of the top-right rectangle? That’s not outer.

Wait — let me draw mentally:

The full height on left is 10 cm. The right part sticks out only 4 cm high. So from top-right of the whole shape, we go down 4 cm, then left 10 cm (to meet the left block), then down another 6 cm (because 10 - 4 = 6) to bottom? But bottom is already drawn as 8 cm.

Actually, perimeter should be same as if it were a full rectangle minus nothing? No — because it’s indented? Actually, this shape has no indentation — it’s convex? Wait, no — it’s L-shaped, so perimeter includes the inner corner.

Correct way: Trace the entire boundary.

Start at bottom-left corner:

→ Move right 8 cm (bottom of left rectangle)
↑ Move up 10 cm (left side)
→ Move right 10 cm (top of left rectangle to start of right extension) — but wait, total top width is 18 cm, so yes, 18 - 8 = 10 cm
↓ Move down 4 cm (right side of top extension)
← Move left 10 cm? No — that would be inside. Actually, from there, we move left along the bottom of the top-right piece? But that’s not outer — the outer path continues down the right side of the lower part? I think I’m confusing myself.

Alternative: For any rectilinear shape (all angles 90°), perimeter = sum of all horizontal segments × 2? Not exactly.

Best: List every outer edge segment.

Looking at figure 1:

Vertical sides:
- Leftmost: 10 cm
- Rightmost: 4 cm (only the top part)
- Inner vertical drop: from y=4 to y=0 on the right side of the left block? Actually, when you go from top-right of the big shape down 4 cm, then you have to go left 10 cm (but that’s horizontal), then down 6 cm (vertical) to reach bottom.

So vertical segments:
- Left: 10 cm
- Right-top: 4 cm
- Inner-down: 6 cm (from y=4 to y=0 on the right side of the left rectangle)

Horizontal segments:
- Bottom: 8 cm
- Top: 18 cm
- Middle-horizontal (connecting the two parts): 10 cm? But that’s internal? No — in perimeter, we don’t count internal lines.

I think I need to visualize the outline.

Actually, standard way for L-shape:

Imagine walking around the outside:

Start at bottom-left:
1. Go right 8 cm → end at (8,0)
2. Go up 6 cm? No — the left part is 10 cm tall, but the right part starts at x=8 and goes to x=18, and is only 4 cm tall.

So from (8,0), we go up to (8,10)? But then how do we connect to the right part?

Actually, the shape is:

- From (0,0) to (8,0) to (8,10) to (18,10) to (18,4) to (8,4) to (8,0)? No, that would make a hole.

I think the correct vertices are:

Assume bottom-left is (0,0)

Then:
- (0,0) → (8,0) → (8,4) → (18,4) → (18,10) → (0,10) → back to (0,0)? But that doesn't match the diagram.

Looking back at the diagram description:

It says:
Left side: 10 cm
Bottom: 8 cm
Top: 18 cm
Right-top: 4 cm

And there's a step-in on the right side.

Actually, typical L-shape for such problems:

The shape has:
- A large rectangle 18 cm wide and 10 cm tall, but with a rectangle cut out from the bottom-right? Or added?

No — from the labels:

The left part is 8 cm wide and 10 cm tall.
The top part extends to 18 cm wide and is 4 cm tall.

So the shape consists of:
- Rectangle A: 8 cm × 10 cm (left part)
- Rectangle B: 10 cm × 4 cm (top-right part, attached to the top of rectangle A)

But they overlap? No — rectangle B is sitting on top of the right part of rectangle A? But rectangle A is only 8 cm wide, so rectangle B starts at x=8, y=6? This is messy.

Perhaps it's better to calculate area first, which is easier.

Area:
As I did earlier:
- Left rectangle: 8 * 10 = 80
- Top-right rectangle: (18-8) * 4 = 10 * 4 = 40
Total area = 120 cm² — that seems correct.

For perimeter, let's list all outer edges without double-counting.

The outer boundary consists of:

- Bottom: 8 cm (from left to where the right part starts)
- Then up the right side of the left part: but only to y=4? No.

Let's define coordinates properly.

Set bottom-left corner as (0,0).

Then:
- The left rectangle goes from x=0 to x=8, y=0 to y=10.
- The top-right rectangle goes from x=8 to x=18, y=6 to y=10? But the label says the right-top height is 4 cm, and the total height on left is 10 cm, so probably the top-right rectangle is from y=6 to y=10? But 10-6=4, yes.

But in the diagram, it shows the right-top side as 4 cm, and the left side as 10 cm, and bottom as 8 cm, top as 18 cm.

Also, there is a vertical segment on the right of the left part from y=4 to y=10? I'm confused.

Perhaps the shape is:

From (0,0) to (8,0) to (8,4) to (18,4) to (18,10) to (0,10) to (0,0). But then the left side is 10 cm, bottom is 8 cm, top is 18 cm, and the right side from (18,4) to (18,10) is 6 cm, but the diagram says 4 cm for the right-top. Contradiction.

Unless the 4 cm is the height of the protrusion, but in that case, from (8,4) to (18,4) is horizontal, then up to (18,10), so the right side is 6 cm, not 4 cm.

I think there's a mistake in my interpretation.

Let me read the diagram again as described:

In problem 1:
- Left side: 10 cm
- Bottom: 8 cm
- Top: 18 cm
- Right-top: 4 cm

And it's an L-shape, so likely, the 4 cm is the height of the right arm, and the left arm is 10 cm tall, so the difference is 6 cm, which is the height of the "step".

So the shape has:

- From bottom-left (0,0) to (8,0) [bottom]
- Up to (8,6) ? No.

Standard way: the perimeter can be calculated as the perimeter of the bounding box plus twice the depth of the indent, but here it's not indented; it's extended.

For an L-shape made of two rectangles sharing a corner, the perimeter is the sum of all outer sides.

Let's list the sides in order, starting from bottom-left, going clockwise:

1. Bottom: 8 cm (from x=0 to x=8 at y=0)
2. Up: 6 cm? Why 6? Because the total height is 10 cm, but the right part starts at y=4? Let's assume that the right-top rectangle sits on top of the left rectangle, but only covering the right part.

Perhaps the left rectangle is 8x10, and the right rectangle is 10x4, attached to the top-right of the left rectangle, so they share the line from (8,6) to (8,10)? No.

I recall that for such shapes, the perimeter is equal to the perimeter of the smallest rectangle that contains it, because the "indent" adds extra length.

For example, if you have a rectangle with a rectangular bite taken out, the perimeter increases by twice the depth of the bite.

But in this case, it's not a bite; it's an extension.

Let's calculate the actual path.

Assume the shape has the following vertices in order (clockwise):

Start at (0,0)
- To (8,0) // bottom
- To (8,4) // up the right side of the left part? But why 4?
- To (18,4) // right along the bottom of the top-right part
- To (18,10) // up the right side
- To (0,10) // left along the top
- To (0,0) // down the left side

But then the left side is from (0,10) to (0,0) = 10 cm, good.
Bottom from (0,0) to (8,0) = 8 cm, good.
Top from (0,10) to (18,10) = 18 cm, good.
Right-top from (18,4) to (18,10) = 6 cm, but the diagram says 4 cm for the right-top side. Inconsistency.

Unless the 4 cm is not the full right side, but only the part above the step.

In the diagram, it labels "4 cm" on the right-top vertical side, which suggests that from the top down to the step is 4 cm, so the step is at y=6 if top is y=10.

So let's set:
- Top at y=10
- Step at y=6 (since 10-4=6)
- Bottom at y=0

Then:
- Left side: from (0,0) to (0,10) = 10 cm
- Bottom: from (0,0) to (8,0) = 8 cm
- Then from (8,0) to (8,6) = 6 cm (up the right side of the left part)
- Then from (8,6) to (18,6) = 10 cm (right along the bottom of the top-right part)
- Then from (18,6) to (18,10) = 4 cm (up the right side)
- Then from (18,10) to (0,10) = 18 cm (left along the top)

But then we have a gap from (0,10) to (0,0), which is already included.

Now, is this closed? From (18,10) to (0,10) to (0,0) to (8,0) to (8,6) to (18,6) to (18,10) — yes, but we have a line from (8,6) to (18,6) which is internal? No, in this path, it's part of the boundary.

Let's list the segments in order for the boundary:

Start at (0,0):
1. Right to (8,0) : 8 cm
2. Up to (8,6) : 6 cm
3. Right to (18,6) : 10 cm
4. Up to (18,10) : 4 cm
5. Left to (0,10) : 18 cm
6. Down to (0,0) : 10 cm

Sum: 8+6+10+4+18+10 = let's calculate: 8+6=14, +10=24, +4=28, +18=46, +10=56 cm.

But is this correct? The segment from (8,6) to (18,6) is at y=6, and from (0,10) to (0,0) is left side.

However, in this path, when we go from (18,10) to (0,10), that's the top, and from (0,10) to (0,0) is left side, but then from (0,0) to (8,0) is bottom, etc.

But notice that between (8,6) and (18,6), and between (0,10) and (0,0), etc., all are outer.

But is there a missing segment? From (8,6) to (8,0) is already included as step 2.

In this configuration, the shape has a "notch" at the bottom-right, but in the diagram, it's usually the other way.

Perhaps for problem 1, the 4 cm is the height of the right arm, and the left arm is 10 cm, so the vertical drop on the right side of the left arm is 10 - 4 = 6 cm.

And the horizontal part connecting them is 18 - 8 = 10 cm.

So perimeter should be:

- Left: 10 cm
- Bottom: 8 cm
- Right-bottom: 6 cm (down from y=10 to y=4? Let's use y=0 at bottom.

Set y=0 at bottom.

Then:
- Left side: from y=0 to y=10 at x=0: 10 cm
- Bottom: from x=0 to x=8 at y=0: 8 cm
- Then up at x=8 from y=0 to y=4: 4 cm? But then how to connect to the top.

I think I found the issue. In many such worksheets, the L-shape is oriented with the long part on the left and the short part on the top-right, and the dimensions given are:

- Overall width: 18 cm
- Overall height: 10 cm
- Width of left part: 8 cm
- Height of top part: 4 cm

So the shape is composed of:
- A rectangle 8 cm wide and 10 cm tall on the left.
- A rectangle 10 cm wide (18-8) and 4 cm tall on the top-right, attached to the top of the left rectangle.

But since the left rectangle is 10 cm tall, and the top-right is only 4 cm tall, they are attached at the top, so the top-right rectangle sits on top of the left rectangle from x=8 to x=18, y=6 to y=10? No, if attached at the top, then the top-right rectangle would be from y=6 to y=10 if the left is from y=0 to y=10, but then the attachment is at y=6 to y=10 for the right part, but the left part has y=0 to y=10, so the common region is from x=8 to x=8, y=6 to y=10, which is a line, not area.

To avoid overlap, the top-right rectangle should be from y=6 to y=10, and the left rectangle from y=0 to y=10, so they share the line x=8, y=6 to y=10.

Then the area is still 8*10 + 10*4 = 80 + 40 = 120 cm², since no overlap in area.

For perimeter, the outer boundary is:

- From (0,0) to (8,0) : 8 cm
- (8,0) to (8,6) : 6 cm (up the right side of the left part, but only to y=6, because above that is covered by the top-right part? No, the top-right part is from y=6 to y=10, so from (8,6) to (8,10) is shared, not outer.

So the outer path:

Start at (0,0):
- Right to (8,0) : 8 cm
- Up to (8,6) : 6 cm (this is the right side of the left part below the top-right part)
- Right to (18,6) : 10 cm (bottom of the top-right part)
- Up to (18,10) : 4 cm (right side of top-right part)
- Left to (0,10) : 18 cm (top of both parts)
- Down to (0,0) : 10 cm (left side)

Sum: 8 + 6 + 10 + 4 + 18 + 10 = 56 cm.

And the segment from (8,6) to (8,10) is internal, not part of perimeter, similarly from (8,10) to (18,10) is part of the top, already included.

So perimeter = 56 cm.

Some people might think that the top is only from (0,10) to (18,10), which is 18 cm, and left is 10 cm, etc.

Yes, so for problem 1:
Area = 120 cm²
Perimeter = 56 cm

But let's verify with another method.

The perimeter can also be calculated as 2*(width + height) for the bounding box, but adjusted for the steps.

Bounding box is 18 cm wide, 10 cm tall, perimeter 2*(18+10) = 56 cm. Oh! Exactly the same.

Why? Because for a rectilinear shape that is "orthogonally convex" or has no holes, the perimeter is the same as the bounding box if it's monotonic, but in this case, since it's L-shaped, but in our calculation, it came out to 56 cm, same as bounding box.

Is that always true? For an L-shape made by adding a rectangle to the corner, the perimeter may be the same as the bounding box if the addition doesn't create new indentations.

In this case, from (0,0) to (18,0) to (18,10) to (0,10) back, but our shape doesn't have the bottom-right part; it has a cut-out or something.

In our vertex list, we have points at (8,0), (8,6), (18,6), etc., but the bounding box would be from (0,0) to (18,10), and our shape touches all four sides: left at x=0, right at x=18, bottom at y=0, top at y=10, and since it's connected, the perimeter should be the same as the bounding box only if it's a rectangle, but here it's not.

In our calculation, we got 56 cm, and 2*(18+10) = 56 cm, so coincidentally the same.

Let me check the segments again:

Segments:
1. (0,0)-(8,0): 8
2. (8,0)-(8,6): 6
3. (8,6)-(18,6): 10
4. (18,6)-(18,10): 4
5. (18,10)-(0,10): 18
6. (0,10)-(0,0): 10

Sum: 8+6=14, 14+10=24, 24+4=28, 28+18=46, 46+10=56 cm. Yes.

And the bounding box perimeter is 2*18 + 2*10 = 36 + 20 = 56 cm. Same number, but different shape.

So for problem 1:
Area = 120 cm²
Perimeter = 56 cm

Okay, moving on.

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Problem 2:

Shape is like a U or a rectangle with a bite taken out from the top.

Dimensions:
- Total width: 4 + 6 + 4 = 14 in? Let's see.
Labels: left top: 4 in, middle bottom of bite: 6 in, right top: 4 in, so total width = 4 + 6 + 4 = 14 in.
Height on sides: 8.5 in
Depth of bite: 2.5 in

So area: can be seen as large rectangle minus the bite.

Large rectangle: width 14 in, height 8.5 in, area = 14 * 8.5

Calculate: 14*8 = 112, 14*0.5=7, so 119 in²

Bite: width 6 in, height 2.5 in, area = 6 * 2.5 = 15 in²

So area of shape = 119 - 15 = 104 in²

Perimeter: outer boundary.

When you take a bite out of the top, the perimeter increases by twice the depth of the bite, because you remove a segment but add two sides.

Original rectangle perimeter: 2*(14 + 8.5) = 2*22.5 = 45 in

But when you cut out a rectangle from the top, you remove the top segment of length 6 in, but add two vertical sides of 2.5 in each and the bottom of the bite, which is 6 in, but since it's now part of the boundary, net change: you remove 6 in (the top part that was removed), but add 2*2.5 + 6 = 5 + 6 = 11 in, so net increase of 5 in.

Let's think:

Original top side: 14 in continuous.

After cutting out a 6 in wide bite of depth 2.5 in, the top is now: left 4 in, then down 2.5 in, then right 6 in (bottom of bite), then up 2.5 in, then right 4 in.

So compared to original top of 14 in, now we have: 4 + 2.5 + 6 + 2.5 + 4 = 19 in for the top part, whereas before it was 14 in, so increase of 5 in.

The other sides are unchanged: bottom 14 in, left 8.5 in, right 8.5 in.

So total perimeter = original perimeter + 5 in = 45 + 5 = 50 in.

Calculate directly:

List all outer segments:

Start at bottom-left:
- Right along bottom: 14 in
- Up right side: 8.5 in
- Left along top-right: 4 in (to the start of the bite)
- Down into bite: 2.5 in
- Left along bottom of bite: 6 in
- Up out of bite: 2.5 in
- Left along top-left: 4 in
- Down left side: 8.5 in

Sum: 14 + 8.5 + 4 + 2.5 + 6 + 2.5 + 4 + 8.5

Calculate step by step:
14 + 8.5 = 22.5
22.5 + 4 = 26.5
26.5 + 2.5 = 29
29 + 6 = 35
35 + 2.5 = 37.5
37.5 + 4 = 41.5
41.5 + 8.5 = 50 in

Yes.

Area: as above, 14*8.5 - 6*2.5 = 119 - 15 = 104 in²

So problem 2:
Area = 104 in²
Perimeter = 50 in

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Problem 3:

Another L-shape.

Dimensions:
- Top: 12 in
- Left: 7 in
- Right: 19 in
- Bottom-right: 6 in

Probably, the shape has a tall right part and a shorter left part.

Assume:
- The right part is 6 in wide and 19 in tall.
- The left part is 12 in wide? But 12 in is labeled on top, and 7 in on left.

Likely, the total width is 12 in for the top, but the right part extends down.

Set coordinates.

Suppose bottom-left is (0,0).

Then, the left part might be from x=0 to x=6? Let's see.

Commonly, for such L-shapes, the dimensions indicate:

- The horizontal part on top: 12 in
- The vertical part on left: 7 in
- The vertical part on right: 19 in
- The horizontal part on bottom-right: 6 in

So probably, the shape is:

- From (0,0) to (6,0) to (6,19) to (12,19) to (12,7) to (0,7) to (0,0)? But then left side is from (0,0) to (0,7) = 7 in, good.
Top from (0,7) to (12,7)? But labeled 12 in on top, which might be from (0,19) to (12,19)? Confusing.

Perhaps the 12 in is the width of the top arm, and 7 in is the height of the left arm, 19 in is the height of the right arm, 6 in is the width of the bottom of the right arm.

So likely, the right arm is 6 in wide and 19 in tall, and the left arm is attached to its left, but only 7 in tall, so the left arm is from x=0 to x=6? No.

Assume the right arm is from x=a to x=a+6, y=0 to y=19.

The left arm is from x=0 to x=b, y=c to y=d.

From the labels, the top is 12 in, so probably the total width at top is 12 in, so if the right arm is 6 in wide, the left arm must be 6 in wide at top, but the left side is labeled 7 in, which is height.

Perhaps the left part is 6 in wide and 7 in tall, and the right part is 6 in wide and 19 in tall, and they are adjacent, so total width 12 in, and the left part is at the bottom, right part is full height.

So shape:
- Left rectangle: 6 in wide × 7 in tall
- Right rectangle: 6 in wide × 19 in tall
Attached at the bottom, so they share the line x=6, y=0 to y=7.

Then area = 6*7 + 6*19 = 42 + 114 = 156 in²

Perimeter: outer boundary.

Vertices:
Start at (0,0):
- Right to (6,0) : 6 in (bottom of left part)
- Up to (6,7) : 7 in (right side of left part, but shared with right part? No, since right part starts at x=6, so from (6,0) to (6,7) is shared, not outer.

Outer path:
- (0,0) to (6,0) : 6 in
- (6,0) to (12,0) : 6 in (bottom of right part) — but is there a bottom for the right part? Yes, if it's full height.
- (12,0) to (12,19) : 19 in (right side)
- (12,19) to (0,19) : 12 in (top) — but the left part is only 7 in tall, so from (0,19) to (0,7) : 12 in? No.

If the left part is only up to y=7, then from (0,19) to (0,7) is not part of the shape; the shape has no material there.

So the top is only from (6,19) to (12,19) for the right part, and for the left part, the top is from (0,7) to (6,7).

So vertices in order:

Start at (0,0):
- Right to (6,0) : 6 in
- Up to (6,7) : 7 in (right side of left part)
- Right to (12,7) : 6 in (top of left part to start of right part? But the right part is taller, so from (6,7) to (12,7) is the top of the left part, but the right part starts at x=6, y=0, so at y=7, it's still there.

Actually, since the right part is from y=0 to y=19, at x=6 to x=12, so at y=7, it's present.

So from (6,7) , we can go right to (12,7) : 6 in, but that's along the top of the left part, which is also the bottom of the upper part of the right part? No, the right part is solid from y=0 to y=19, so from (6,7) to (12,7) is inside the right part if we consider it, but in terms of boundary, it's not outer.

I think the correct outer boundary is:

- (0,0) to (6,0) : 6 in (bottom left)
- (6,0) to (12,0) : 6 in (bottom right) — assuming the right part has bottom at y=0
- (12,0) to (12,19) : 19 in (right side)
- (12,19) to (6,19) : 6 in (top of right part)
- (6,19) to (6,7) : 12 in down? But that would be inside.

No, from (6,19) , since the left part only goes up to y=7, we need to go left to (0,7) or something.

Perhaps the left part is attached to the left of the right part, but only up to y=7, so the shape has:

- From (0,0) to (6,0) to (6,7) to (0,7) to (0,0) for the left part, but then the right part is from (6,0) to (12,0) to (12,19) to (6,19) to (6,0), but then they overlap on (6,0) to (6,7).

To avoid overlap, the combined shape has vertices:

Start at (0,0):
- Right to (6,0) : 6 in
- Right to (12,0) : 6 in (continuing along bottom)
- Up to (12,19) : 19 in
- Left to (6,19) : 6 in
- Down to (6,7) : 12 in (since 19-7=12)
- Left to (0,7) : 6 in
- Down to (0,0) : 7 in

Sum for perimeter: 6+6+19+6+12+6+7 = let's calculate: 6+6=12, +19=31, +6=37, +12=49, +6=55, +7=62 in.

Area: left part 6*7 = 42, right part 6*19 = 114, total 156 in², and no overlap since they share only the line x=6, y=0 to y=7, which has no area.

So area = 156 in²

Perimeter = 62 in

Check with labels: left side is from (0,0) to (0,7) = 7 in, good.
Top is from (0,7) to (6,7) to (6,19) to (12,19)? But the top is labeled 12 in, which might be the distance from left to right at the top, which is from x=0 to x=12 at y=19, but in our shape, at y=19, it's only from x=6 to x=12, so 6 in, not 12 in. Contradiction.

Perhaps the 12 in is the width of the top arm, which is the left part's top, but the left part is only 6 in wide.

I think I have it backward.

Let me look at the diagram description: "12 in" on top, "7 in" on left, "19 in" on right, "6 in" on bottom-right.

Probably, the shape is oriented with the long part on the right, and the short part on the top-left.

So:
- The top arm is 12 in wide and 7 in tall? But 7 in is labeled on left, which might be height.

Assume:
- The horizontal part on top: 12 in wide, 7 in tall? But then the right side is 19 in, which is taller.

Perhaps the 7 in is the height of the left vertical part, and 19 in is the height of the right vertical part, and 12 in is the width of the top horizontal part, and 6 in is the width of the bottom of the right part.

So likely, the shape has:
- A vertical rectangle on the right: 6 in wide, 19 in tall.
- A horizontal rectangle on the top: 12 in wide, 7 in tall, attached to the top of the right rectangle, but extending to the left.

So the top rectangle is from x=0 to x=12, y=12 to y=19? Let's set y=0 at bottom.

Suppose the right rectangle is from x=6 to x=12, y=0 to y=19. (width 6 in, height 19 in)

The top rectangle is from x=0 to x=12, y=12 to y=19? But then height is 7 in, so from y=12 to y=19 is 7 in, good.

But then they overlap on x=6 to x=12, y=12 to y=19.

Area would be area of right rectangle + area of top rectangle - overlap.

Right rectangle: 6*19 = 114
Top rectangle: 12*7 = 84
Overlap: 6*7 = 42 (since from x=6 to 12, y=12 to 19)
So area = 114 + 84 - 42 = 156 in² same as before.

For perimeter, outer boundary.

Vertices:
Start at (0,12) : top-left of top rectangle
- Right to (12,12) : 12 in (bottom of top rectangle, but may be internal)
Better to start at (0,0) or (0,12).

Start at (0,12):
- Right to (12,12) : 12 in ( but this is the bottom of the top rectangle, and if the right rectangle is below, this might be internal if they are attached, but in this case, the right rectangle is from y=0 to y=19, so at y=12, it's present, so from (0,12) to (12,12) is along the top of the right rectangle for x>6, but for x<6, it's the bottom of the top rectangle.

This is complicated.

Perhaps the intended configuration is that the top rectangle is only from x=0 to x=6, y=12 to y=19, and the right rectangle from x=6 to x=12, y=0 to y=19, so they share the line x=6, y=12 to y=19.

Then area = top: 6*7 = 42, right: 6*19 = 114, total 156 in².

Perimeter:
- (0,12) to (6,12) : 6 in (bottom of top rectangle)
- (6,12) to (6,19) : 7 in (right side of top rectangle, but shared)
Not good.

Start at (0,0):
- But the left part may not extend to y=0.

Assume the shape has:
- From (0,12) to (6,12) to (6,19) to (0,19) to (0,12) for the top-left part? But then the right part is separate.

I think for simplicity, in such problems, the perimeter can be calculated as the sum of all given outer sides, and for L-shapes, it's often 2*(sum of all "arms") or something.

Perhaps use the fact that for rectilinear polygons, perimeter is sum of all horizontal projections times 2, but let's calculate based on the labels.

From the diagram, the outer sides are:
- Top: 12 in
- Left: 7 in
- Right: 19 in
- Bottom-right: 6 in
- And there is a horizontal segment at the bottom-left, and a vertical segment at the top-right of the left part.

Typically, for this orientation, the missing sides can be inferred.

Notice that the total width at the bottom is the same as at the top for the right part, but the left part is shorter.

The difference in height between left and right is 19 - 7 = 12 in, which must be accounted for in the perimeter.

In many textbooks, for such an L-shape, the perimeter is 2*(length + width) where length and width are the overall dimensions, but here overall width is 12 in (top), overall height is 19 in (right), so 2*(12+19) = 62 in, same as my earlier calculation.

And area 156 in².

And in my first calculation for problem 3, I had perimeter 62 in, area 156 in², and it matched the labels if we interpret the 6 in as the width of the right part, 12 in as the width of the top part, etc.

So I'll go with that.

Problem 3:
Area = 156 in²
Perimeter = 62 in

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Problem 4:

L-shape with decimals.

Dimensions:
- Top: 16.8 ft
- Left: 8.1 ft
- Right: 13.2 ft
- Bottom: 12.4 ft
- Also, 4.4 ft and 5.1 ft on the left side.

Probably, the left side has a step.

From the labels, on the left, from bottom up: 5.1 ft, then 4.4 ft, then to top 8.1 ft, so 5.1 + 4.4 = 9.5, but 8.1 is given, inconsistency.

8.1 ft is labeled on the left, and 4.4 ft and 5.1 ft are also on the left, so perhaps 5.1 ft is the bottom part, 4.4 ft is the middle, but 5.1 + 4.4 = 9.5 > 8.1, impossible.

Perhaps 8.1 ft is the total left height, and 4.4 ft and 5.1 ft are segments.

Look: "8.1 ft" on left, "4.4 ft" and "5.1 ft" also on left, so likely, the left side is divided into 5.1 ft at bottom, then 4.4 ft, but 5.1 + 4.4 = 9.5, but 8.1 is given, so perhaps 8.1 is the height of the left part, and 4.4 and 5.1 are for other things.

Perhaps the 4.4 ft and 5.1 ft are the heights of the steps.

Another possibility: the shape has a notch on the left.

From the values, 5.1 + 4.4 = 9.5, and 13.2 - 8.1 = 5.1, oh! 13.2 - 8.1 = 5.1, and 5.1 is given, so perhaps the right side is 13.2 ft, left side is 8.1 ft, so the difference is 5.1 ft, which is the height of the step on the left.

Also, bottom is 12.4 ft, top is 16.8 ft, so difference 4.4 ft, which is given.

So likely