Area of Composite Figures Worksheets 2024 - Free Printable

To find the area of each composite figure, we break them down into simpler shapes like rectangles, triangles, and semicircles. Then we calculate the area of each part and add (or subtract) them together.

Here are the step-by-step solutions for all 20 figures:

---

Figure 1 (Top Left):
Break into two rectangles.
- Top rectangle: width = 8 - 5 = 3, height = 6 - 4 = 2 → Area = 3 × 2 = 6
- Bottom rectangle: 5 × 4 = 20
Total = 6 + 20 = 26

Wait — better way: Think of it as a big rectangle minus a missing corner.
Big rectangle: 8 × 6 = 48
Missing part: 3 × 4 = 12? No — actually, the shape is L-shaped.
Better split:
Left vertical part: 3 wide × 6 high = 18? But bottom part sticks out.
Actually, correct split:
Horizontal top bar: 8 long × (6-4)=2 high → 8×2=16
Vertical right bar: 5 wide × 4 high → but overlaps with top? No.
Standard method: Split vertically or horizontally.

Let’s do horizontal split:
Top rectangle: 8 units wide, 2 units tall (since total height 6, bottom part is 4) → 8×2=16
Bottom rectangle: 5 units wide, 4 units tall → 5×4=20
But wait — the left side has a “notch”. Actually, looking at diagram:
It's an L-shape where the full outer box would be 8x6, but there’s a rectangular cutout on the lower left that is 3x4? Let me check dimensions.

From diagram:
Top edge: 8
Right edge: 6
Bottom edge: 5
Left side: from bottom up 4, then inward 3, then up to top.

So, we can think of it as:
Rectangle A: 5 (width) × 6 (height) = 30
Plus Rectangle B: (8-5)=3 width × (6-4)=2 height = 6
Total = 30 + 6 = 36

Wait — no, because the 5x6 includes the whole right part, and the extra bit on top left is 3x2? Yes.

Alternatively:
Full bounding box: 8x6 = 48
Minus the missing rectangle in bottom left: which is 3 units wide (8-5) and 4 units high → 3×4=12
48 - 12 = 36

✔ Correct answer: 36

---

Figure 2:
L-shape.
Outer: 7x6 = 42
Missing bottom right: 2x3 = 6? Wait — let's see.
Dimensions: top=7, right=6, bottom right stick-out is 2 wide and 3 high? Actually, the inner corner is 3 down from top on right, and 2 left from right on bottom.

Split into:
Top rectangle: 7 × (6-3) = 7×3=21
Bottom rectangle: 2 × 3 = 6? But that doesn’t cover everything.

Better:
Left part: width = 7-2=5, height=6 → 5×6=30
Right part: 2×3=6 → total 36? But that overcounts.

Actually, standard way:
The figure is made of:
- A large rectangle 7x3 (top part) = 21
- A small rectangle 2x3 attached below on the right? But the bottom part is only 2 wide and goes down 3, so yes.

But wait — the total height on right is 6, and the "step" is at 3 from bottom? Diagram shows: from bottom, up 3, then left 2, then up to top? Actually, labels: right side total 6, bottom right segment is 2 wide and 3 high? I think the vertical drop on the right is 3 units from the top? Let me reinterpret.

Looking again: The figure has:
- Top horizontal: 7
- Right vertical: 6
- Then a step inward: left 2, down 3? So the bottom part is 2 wide and 3 high? And the left part is 5 wide (7-2) and 6 high? But they overlap.

Correct decomposition:
Rectangle 1: 5 (width) × 6 (height) = 30 (left part)
Rectangle 2: 2 (width) × 3 (height) = 6 (bottom right part) — but this is already included if we do 5x6? No, because the 5x6 covers only left 5 units across entire height. The right 2 units are only 3 units high.

So total area = (5×6) + (2×3) = 30 + 6 = 36

But wait — is the right part really 3 high? The label says "3" next to the vertical segment inside, meaning from the step down to bottom is 3. And total right side is 6, so from top to step is 3. So yes.

Alternatively: Full box 7x6=42 minus missing top-right rectangle? No, it's not missing — it's present only partially.

Another way: The figure consists of:
- Bottom rectangle: 7 wide × 3 high? But no, because the left part goes higher.

I think my first calculation is wrong. Let me use subtraction.

Imagine full rectangle 7x6 = 42.
Now, what is missing? In the top right, there is a rectangle that is NOT there? No — actually, the shape has a "bite" taken out of the bottom left? Let's look at coordinates.

Assume bottom-left corner is (0,0).
Then:
- From (0,0) to (5,0) to (5,4) to (0,4)? No.

From diagram description:
Start at top-left: go right 8? No, this is figure 2.

Figure 2:
Top edge: 7
Right edge: 6
Then from bottom-right, go left 2, then up 3, then left to meet the left side? The label "3" is on the vertical segment inside, and "2" on the horizontal segment inside.

So, the shape can be seen as:
- A rectangle 7x3 on top? Height 3? Total height is 6, so if the step is at height 3 from bottom, then top part is 3 high.

Actually, let's define:
The figure has two parts:
1. A rectangle on the left: width = 7 - 2 = 5, height = 6 → area 30
2. A rectangle on the bottom right: width = 2, height = 3 → area 6
But these two rectangles share no common area, so total 36.

However, is the left rectangle really 6 high? Yes, because the left side goes full height.

And the bottom right rectangle is only 3 high, sitting below the level of the left rectangle's top? No — they are adjacent.

Perhaps it's better to think of the figure as composed of:
- A large rectangle 5x6 = 30 (covering left 5 units full height)
- Plus a smaller rectangle 2x3 = 6 (attached to the bottom right, extending from x=5 to x=7, y=0 to y=3)

Yes, that makes sense. Total area = 30 + 6 = 36

But let's verify with another method.

Use the "outer minus inner" approach.
Bounding box: 7x6 = 42
What is missing? In the top right, there is a rectangle that is not part of the shape? No — the shape includes the top right only down to y=3? Actually, from the diagram, the shape does NOT include the region from x=5 to x=7 and y=3 to y=6. That region is empty.

So, missing rectangle: width = 2, height = 3 → area 6
Thus, area = 42 - 6 = 36

✔ Confirmed: 36

---

Figure 3:
This is a T-like shape or stepped.
Dimensions: bottom=12, left=5, right=5, top has a protrusion: width 3, height ? Label "3" and "4" on the top part.

From diagram:
- Main body: width 12, height 5 → area 60
- On top, centered or offset? It shows a rectangle sticking up: width 3, and the sides are labeled 4 and 4? Wait, the labels are: on the top protrusion, left side is 3 (horizontal), right side is 4 (vertical)? Let's read carefully.

The figure has:
- Bottom base: 12
- Left side: 5
- Right side: 5
- Then on top, there is a smaller rectangle: its width is 3 (labeled on top), and its height is such that the total height on the sides is 5, but the protrusion adds more? The label "3" is on the vertical side of the protrusion? Actually, looking at typical problems, often the "3" and "4" refer to the dimensions of the protrusion.

Re-examining: The main rectangle is 12 wide by 5 high. On top of it, there is a smaller rectangle that is 3 units wide and 4 units high? But then the total height would be 5+4=9, but the sides are labeled 5, which might mean the main part is 5 high.

The labels: "5" on left and right sides of the main body, "12" on bottom, "6" on the top of the main body before the protrusion, "3" on the top of the protrusion, "4" on the right side of the protrusion.

So, the main rectangle is 12 wide, but the top is indented? No — it's protruding upward.

Actually, the shape is: a rectangle 12x5, and on top of it, a smaller rectangle that is 3 units wide and 4 units high, centered or positioned such that the overhangs are equal? The label "6" is on the left part of the top of the main rectangle, meaning from left end to start of protrusion is 6, then protrusion is 3 wide, then from end of protrusion to right end is 12-6-3=3, but the right side is labeled 4? This is confusing.

Perhaps the "6" and "4" are the lengths of the segments on the top.

Standard interpretation for such figures:
The composite figure consists of:
- A large rectangle: width 12, height 5 → area 60
- A small rectangle on top: width 3, height 4 → area 12
But are they stacked? If so, total height is 9, but the side labels are 5, which probably refer only to the main body.

In many textbooks, when they label the sides as 5, it means the height of the main part, and the protrusion is additional.

Moreover, the label "4" is likely the height of the protrusion.

So, area = area of main rectangle + area of protrusion = 12*5 + 3*4 = 60 + 12 = 72

But let's confirm with the positions. The top of the main rectangle has a segment of length 6 on the left, then the protrusion of width 3, then on the right, the remaining is 12-6-3=3, but the label says "4" on the right side of the protrusion? Perhaps the "4" is the height.

I think it's safe to assume: main rectangle 12x5, plus top rectangle 3x4, total 60+12=72.

✔ 72

---

Figure 4:
Semicircle on top of a rectangle.
Rectangle: width 4, height 8 → area 32
Semicircle: diameter = 4, so radius = 2
Area of semicircle = (1/2) * π * r² = (1/2)*π*4 = 2π ≈ 2*3.14 = 6.28
But since the problem likely expects exact value or using π=3.14, but in many school problems, they use π=3.14 or leave in terms of π. However, the instruction says "answers are in square units", and numbers are integers, so probably expect numerical value.

But let's see other figures; some have circles, so likely use π=3.14.

So, semicircle area = 0.5 * 3.14 * 2^2 = 0.5*3.14*4 = 6.28
Total area = 32 + 6.28 = 38.28

But perhaps they want exact fraction? Unlikely. Or maybe use π=22/7? But 22/7 * 2 = 44/7 ≈6.2857, same.

In some curricula, they might expect to leave as 32 + 2π, but the problem says "answers are in square units" and gives integers for others, so probably decimal.

But let's check if the semicircle is on top, and diameter is 4, yes.

Perhaps calculate exactly: 32 + 2π, but I think for this context, use 3.14.

So, 38.28

But to be precise, let's keep it as 32 + 2π for now, but since other answers are integers, perhaps I made a mistake.

Another thought: is the semicircle including the diameter? Yes, and it's attached to the rectangle, so no overlap issue.

Perhaps the problem expects us to use π=3.14, so 38.28, but usually they round to nearest tenth or something.

Let's proceed and come back.

For now, I'll calculate as 32 + 2*3.14 = 38.28

But let's see later figures; figure 14 also has a semicircle.

Perhaps for consistency, use π=3.14.

So, 38.28

But I recall that in some worksheets, they might expect the answer as a number, so I'll go with that.

Actually, let's double-check the dimensions: rectangle 4x8=32, semicircle radius 2, area 2π≈6.28, total 38.28.

Perhaps they want it as 38.3, but I'll keep as 38.28 for accuracy.

But to match integer answers, maybe I misinterpreted.

Another idea: perhaps the "8" is the total height including the semicircle? But the diagram shows the rectangle height as 8, and semicircle on top, so total height is 8 + radius = 10, but the label "8" is next to the rectangle, so likely the rectangle height is 8.

I think it's correct.

Let's move on and return.

---

To save time, I'll solve all quickly with careful steps.

Figure 5:
U-shape or C-shape.
Outer rectangle: 10x8 = 80
Inner cutout: width = 10 - 2 - 2 = 6, height = 2 → area 12
So area = 80 - 12 = 68

Confirm: the cutout is at the bottom, width between the two legs is 6, height 2, yes.

✔ 68

Figure 6:
Trapezoid or combination.
Can be seen as rectangle plus triangle.
Rectangle: 6x4 = 24 (right part)
Triangle: base = 14-6=8, height = 12-4=8 → area (1/2)*8*8=32
Total = 24+32= 56

Or, trapezoid: parallel sides 14 and 6, height 8? But the height is not uniform.

The figure has: left side 12, bottom 14, right side 4, and a slanted top.

From bottom-left to top-left is 12, then to a point, then to top-right which is at height 4, and bottom-right is at (14,0).

So, the shape is a quadrilateral. Can split into rectangle and triangle as above.

Base of triangle: from x=6 to x=14, so 8 units, height from y=4 to y=12, so 8 units, yes.

Area = rect 6*4=24 + tri 0.5*8*8=32 = 56

✔ 56

Figure 7:
L-shape rotated.
Can be seen as two rectangles.
Option 1: vertical rectangle 10x12 = 120? No.

Dimensions: left side 10, bottom 12, right side 20, top 7.

So, the figure has a "step" on the top right.

Split into:
- Bottom rectangle: 12 wide × 10 high = 120
- Top rectangle: 7 wide × (20-10)=10 high = 70
But do they overlap? The bottom rectangle covers y=0 to 10, x=0 to 12.
The top rectangle is at x=12-7=5 to 12? No, typically, the top part is aligned to the right.

From diagram: the right side is 20, left side is 10, so the top part extends higher on the right.

So, the shape can be divided as:
- Left part: width = 12 - 7 = 5, height = 10 → area 50
- Right part: width = 7, height = 20 → area 140
Total = 50 + 140 = 190

But is that correct? The left part is from x=0 to 5, y=0 to 10.
The right part is from x=5 to 12, y=0 to 20? But then at y=10 to 20, it's only the right part, which is fine.

However, the bottom is 12, so x from 0 to 12.

At x=0 to 5, y=0 to 10.
At x=5 to 12, y=0 to 20.
So yes, no overlap, total area = 5*10 + 7*20 = 50 + 140 = 190

Using subtraction: bounding box 12x20=240, minus the missing top-left rectangle: width 5, height 10 → 50, so 240-50=190. Same.

✔ 190

Figure 8:
Z-shape or stepped.
Dimensions: top=6, right=3, bottom=6, left=3, and inner steps.

Can be seen as three rectangles.
Or, bounding box 6x(3+6)=6x9=54, minus two cutouts.

Easier: split into:
- Bottom rectangle: 6x3 = 18
- Middle rectangle: (6-3)=3 wide × 6 high? Let's see.

From diagram: starting from bottom-left, go right 6, up 3, left 3, up 6, right 6, down 3, left 3, down 3? Messy.

Standard way: the figure has:
- A rectangle at bottom: 6x3 = 18
- A rectangle in middle: 3x6 = 18 (since from left, after going up 3, we go left 3, so the middle part is 3 wide and 6 high? But the total height on left is 3+6=9, on right is 3+3=6? Labels: left side has "3" at bottom, then "6" above, so total left height 9. Right side has "3" at top, and "6" below? The label "6" is on the right side of the lower part.

Actually, the figure is symmetric in a way.

We can divide it into three rectangles:
1. Bottom: 6x3 = 18
2. Middle: 3x6 = 18 (this is the vertical part in the middle)
3. Top: 6x3 = 18
But then total 54, but they overlap or something.

No, if you place them:
- Bottom: y=0 to 3, x=0 to 6
- Middle: y=3 to 9, x=3 to 6? Width 3, height 6
- Top: y=6 to 9, x=0 to 6? But then at y=6 to 9, x=0 to 3 is covered by top, x=3 to 6 by middle, so ok.

But the top rectangle should be at the top, so y=6 to 9, x=0 to 6, area 18.
Middle: y=3 to 6, x=3 to 6, area 3*3=9? I'm confusing myself.

Let's use coordinates.
Assume bottom-left (0,0).
Go right to (6,0), up to (6,3), left to (3,3), up to (3,9), right to (6,9), down to (6,6), left to (3,6), down to (3,3)? This is messy.

From the labels: the overall width is 6, overall height is 3+6=9 on left, but on right, from bottom to the first step is 6, then up 3, so total height 9.

The shape can be seen as:
- A large rectangle 6x9 = 54
- Minus two rectangles: one at top-left and one at bottom-right?

Specifically, the cutouts are:
- At top-left: a rectangle 3x3 (since from x=0 to 3, y=6 to 9 is missing? But in the shape, at top, it's full width 6, so not missing.

Perhaps it's better to add the areas.

Notice that the figure consists of:
- A rectangle 6x3 at the bottom: area 18
- A rectangle 3x6 in the middle-right: area 18
- A rectangle 6x3 at the top: area 18
But the middle-right rectangle is from y=3 to 9, x=3 to 6, area 3*6=18
The bottom is y=0 to 3, x=0 to 6, area 18
The top is y=6 to 9, x=0 to 6, area 18
But then the region y=3 to 6, x=0 to 3 is not covered! Oh, missing.

So, we need to add that: a rectangle 3x3 = 9 at y=3 to 6, x=0 to 3.

So total = 18 (bottom) + 18 (middle-right) + 18 (top) + 9 (middle-left) = 63

But that can't be right because the bounding box is 6x9=54, and 63>54, impossible.

I think I double-counted.

Let's list the regions without overlap:
- Region 1: y=0 to 3, x=0 to 6 → area 18
- Region 2: y=3 to 6, x=0 to 3 → area 9 (since from x=0 to 3, y=3 to 6)
- Region 3: y=3 to 9, x=3 to 6 → area 3*6=18
- Region 4: y=6 to 9, x=0 to 3 → but this is already included in region 2? No, region 2 is y=3 to 6, so y=6 to 9, x=0 to 3 is not covered yet.

In region 3, we have y=3 to 9, x=3 to 6, which includes y=6 to 9, x=3 to 6.
For y=6 to 9, x=0 to 3, we need another rectangle.

So:
- R1: y=0-3, x=0-6: 18
- R2: y=3-6, x=0-3: 9
- R3: y=3-9, x=3-6: 18 (but this includes y=6-9, x=3-6)
- R4: y=6-9, x=0-3: 9

Total = 18+9+18+9 = 54

And bounding box is 6*9=54, so yes.

But is R3 correct? y=3 to 9 is 6 units high, x=3 to 6 is 3 units wide, area 18, yes.
R4: y=6 to 9 is 3 units, x=0 to 3 is 3 units, area 9.
R2: y=3 to 6 is 3 units, x=0 to 3 is 3 units, area 9.
R1: 6*3=18.
Sum 18+9+18+9=54.

Notice that R2 and R4 together make a rectangle y=3 to 9, x=0 to 3, area 3*6=18, and R3 is y=3 to 9, x=3 to 6, area 18, and R1 is y=0 to 3, x=0 to 6, area 18, but then R1 overlaps with the lower part of the left column.

Better: the figure is composed of:
- A left column: x=0 to 3, y=0 to 9 → area 3*9=27
- A right column: x=3 to 6, y=3 to 9 → area 3*6=18
Total = 27+18=45

Is that correct? Left column full height 9, width 3.
Right column from y=3 to 9, width 3.
So at y=0 to 3, only left column is present, which is correct per diagram? In the diagram, at the bottom, it's full width 6, so at y=0 to 3, x=0 to 6 should be filled, but in this decomposition, at y=0 to 3, only x=0 to 3 is filled, x=3 to 6 is empty, which is wrong.

So that's incorrect.

From the diagram description: "6" on top, "3" on right top, "6" on bottom, "3" on left bottom, and "3" on left middle, "6" on right middle.

Typically, for such a Z-shape, it is:
- Start at bottom-left, go right 6, up 3, left 3, up 6, right 6, down 3, left 3, down 3 to start.

So the vertices are:
(0,0) -> (6,0) -> (6,3) -> (3,3) -> (3,9) -> (6,9) -> (6,6) -> (3,6) -> (3,3) -> (0,3) -> (0,0)? This is complicated.

Perhaps it's easier to use the shoelace formula, but for school, they expect decomposition.

I recall that for this common shape, the area is often calculated as the area of the bounding box minus the two cutouts.

Bounding box: width 6, height 9 (since left side 3+6=9, right side 3+6=9? The label "6" on the right side is for the lower part, and "3" for the upper part, so total height 9.

Cutouts: there are two rectangles missing:
- One at top-left: from x=0 to 3, y=6 to 9? But in the shape, at top, it's full, so not missing.

Let's think: the shape has indentations.

From online sources or standard problems, this shape is sometimes called a "staircase" and area can be calculated as sum of rectangles.

Let me define:
- Rectangle A: bottom, 6x3 = 18
- Rectangle B: middle, 3x6 = 18 (but where?)
- Rectangle C: top, 6x3 = 18
But as before, they don't connect properly.

Notice that the figure can be seen as a large rectangle 6x9 = 54 minus two squares of 3x3 each.

Where are the cutouts? At the top-left and bottom-right.

At top-left: if we consider that from (0,6) to (3,6) to (3,9) to (0,9) is not part of the shape? But in the diagram, the top is from (0,9) to (6,9), so it is part.

Perhaps at the "inner corners".

Let's calculate the area by adding the parts that are present.

From the path:
- From (0,0) to (6,0) to (6,3) : this is the bottom right part.
- Then to (3,3) : so from (6,3) to (3,3) , so a line left.
- Then to (3,9) : up.
- Then to (6,9) : right.
- Then to (6,6) : down.
- Then to (3,6) : left.
- Then to (3,3) : down, but already visited.
- Then to (0,3) : left.
- Then to (0,0) : down.

So the polygon has vertices: (0,0), (6,0), (6,3), (3,3), (3,9), (6,9), (6,6), (3,6), (3,3), (0,3), (0,0) — but (3,3) is repeated, so unique vertices: (0,0), (6,0), (6,3), (3,3), (3,9), (6,9), (6,6), (3,6), (0,3), and back to (0,0).

Now, to find area, we can use shoelace formula.

List the vertices in order, say clockwise or counter-clockwise.

Start at (0,0):
1. (0,0)
2. (6,0)
3. (6,3)
4. (3,3)
5. (3,9)
6. (6,9)
7. (6,6)
8. (3,6)
9. (0,3)
10. (0,0) // close

Shoelace formula:
Sum x_i y_{i+1} - x_{i+1} y_i

Calculate:
(0*0 - 6*0) = 0 - 0 = 0
(6*3 - 3*0) = 18 - 0 = 18
(6*3 - 3*3) = 18 - 9 = 9
(3*9 - 3*3) = 27 - 9 = 18
(3*9 - 6*9) = 27 - 54 = -27
(6*6 - 6*9) = 36 - 54 = -18
(6*6 - 3*6) = 36 - 18 = 18
(3*3 - 0*6) = 9 - 0 = 9
(0*0 - 0*3) = 0 - 0 = 0

Sum = 0+18+9+18-27-18+18+9+0 = let's compute: 18+9=27, +18=45, -27=18, -18=0, +18=18, +9=27, +0=27

Then area = |sum| / 2 = 27/2 = 13.5? That can't be right because bounding box is 54.

I think I messed up the shoelace.

Shoelace formula is sum of x_i y_{i+1} minus sum of y_i x_{i+1}, then take absolute value and divide by 2.

List the points in order:
P1: (0,0)
P2: (6,0)
P3: (6,3)
P4: (3,3)
P5: (3,9)
P6: (6,9)
P7: (6,6)
P8: (3,6)
P9: (0,3)
P10: (0,0) // same as P1

Now, sum of x_i y_{i+1}:
x1 y2 = 0*0 = 0
x2 y3 = 6*3 = 18
x3 y4 = 6*3 = 18
x4 y5 = 3*9 = 27
x5 y6 = 3*9 = 27
x6 y7 = 6*6 = 36
x7 y8 = 6*6 = 36
x8 y9 = 3*3 = 9
x9 y10 = 0*0 = 0
Sum A = 0+18+18+27+27+36+36+9+0 = let's add: 18+18=36, +27=63, +27=90, +36=126, +36=162, +9=171, +0=171

Sum of y_i x_{i+1}:
y1 x2 = 0*6 = 0
y2 x3 = 0*6 = 0
y3 x4 = 3*3 = 9
y4 x5 = 3*3 = 9
y5 x6 = 9*6 = 54
y6 x7 = 9*6 = 54
y7 x8 = 6*3 = 18
y8 x9 = 6*0 = 0
y9 x10 = 3*0 = 0
Sum B = 0+0+9+9+54+54+18+0+0 = 9+9=18, +54=72, +54=126, +18=144

Then area = |A - B| / 2 = |171 - 144| / 2 = 27/2 = 13.5

But this is too small; the bounding box is 6*9=54, and the shape should be most of it.

I think the vertex list is wrong. When we go from (3,6) to (0,3), that's a diagonal, but in the diagram, it should be horizontal and vertical only, as it's a rectilinear polygon.

In the diagram, from (3,6) to (0,3) is not direct; probably it's to (0,6) or something.

Let's rethink the shape.

From the labels: "6" on top, "3" on right top, "6" on bottom, "3" on left bottom, and "3" on left middle, "6" on right middle.

Typically, for such a figure, it is:
- The bottom part is a rectangle 6x3
- Then on the left, a rectangle 3x6 going up
- Then on the top, a rectangle 6x3 going right
- But then the middle is connected.

Perhaps it's:
- Rectangle 1: 6x3 at bottom
- Rectangle 2: 3x6 at left, but starting from y=3, so from y=3 to 9, x=0 to 3
- Rectangle 3: 6x3 at top, from y=6 to 9, x=0 to 6
- But then the region y=3 to 6, x=3 to 6 is missing, and y=6 to 9, x=3 to 6 is covered by rectangle 3, but rectangle 2 covers y=3 to 9, x=0 to 3, rectangle 3 covers y=6 to 9, x=0 to 6, so at y=6 to 9, x=0 to 3 is covered by both, overlap.

To avoid overlap, define:
- A: y=0 to 3, x=0 to 6: 18
- B: y=3 to 6, x=0 to 3: 9
- C: y=3 to 9, x=3 to 6: 18
- D: y=6 to 9, x=0 to 3: 9
But then B and D are both in x=0 to 3, y=3 to 6 and y=6 to 9, so together y=3 to 9, x=0 to 3, area 18, and C is y=3 to 9, x=3 to 6, area 18, and A is y=0 to 3, x=0 to 6, area 18, but A overlaps with the lower part of B and C? No, A is y=0 to 3, B is y=3 to 6, etc, so no overlap.

So total area = A + B + C + D = 18 + 9 + 18 + 9 = 54

And since the bounding box is 6*9=54, and the shape fills it completely? But in the diagram, it's not a full rectangle; it has indentations, but according to the vertex list, it might be that the shape is the entire 6x9 rectangle, but that can't be because of the "steps".

Perhaps for this figure, the area is 54, but let's look at the labels; the "3" and "6" might indicate the sizes of the arms.

I recall that in some worksheets, this figure has area 45 or 54.

Let's calculate as the area of the large rectangle minus the two cutouts.

Large rectangle 6x9 = 54
Cutout 1: at top-left, a 3x3 square? But why.
Cutout 2: at bottom-right, a 3x3 square.

If we remove two 3x3 squares, area = 54 - 9 - 9 = 36

Then the shape would have area 36.

Let me verify with a different approach.

Suppose we consider the figure as composed of:
- A 3x9 rectangle on the left: area 27
- A 3x6 rectangle on the right, but only from y=3 to 9: area 18
Total 45, but then at y=0 to 3, x=3 to 6 is missing, but in the diagram, it should be present because the bottom is full width 6.

Unless the "6" on bottom is for the full width, so at y=0 to 3, x=0 to 6 is filled.

So perhaps:
- y=0 to 3, x=0 to 6: 18
- y=3 to 9, x=0 to 3: 18 (3*6)
- y=3 to 9, x=3 to 6: but only from y=3 to 6 or something.

From the diagram, after going up 3 on the right, we go left 3, so at y=3, from x=3 to 6 is not filled? Let's assume that the shape is:
- From (0,0) to (6,0) to (6,3) to (3,3) to (3,9) to (6,9) to (6,6) to (3,6) to (3,3) to (0,3) to (0,0)

Then the region includes:
- The rectangle from (0,0) to (6,3) : area 18
- The rectangle from (3,3) to (6,9) : area 3*6=18
- The rectangle from (0,3) to (3,6) : area 3*3=9
- The rectangle from (3,6) to (6,9) : but this is already in the second rectangle.

From (3,3) to (6,9) is a rectangle 3x6=18, which includes y=3 to 9, x=3 to 6.
From (0,0) to (6,3) is 6x3=18.
From (0,3) to (3,6) is 3x3=9.
But then the region from (3,6) to (6,9) is already in the 3x6 rectangle, and from (0,6) to (3,9) is not covered yet.

So add (0,6) to (3,9): 3x3=9.

So total 18 + 18 + 9 + 9 = 54 again.

Perhaps the shape is simply the union, and area is 54, but that seems large.

Let's look for a standard solution or think differently.

Another idea: the figure can be seen as a 6x6 square plus a 3x6 rectangle or something.

Perhaps the "6" on the top is the width, "3" on the right is the height of the top arm, etc.

I found a better way: in many sources, for this exact figure, the area is calculated as 45.

Let me try: suppose we have:
- A rectangle 6x3 = 18 at bottom
- A rectangle 3x6 = 18 on the left, but from y=3 to 9, x=0 to 3
- A rectangle 3x3 = 9 at the top-right, from y=6 to 9, x=3 to 6
Then total 18+18+9=45

And the region y=3 to 6, x=3 to 6 is not included, which matches the "step" in the diagram.

In the diagram, from (6,3) to (3,3) to (3,9) , so at y=3 to