Areas of Irregular Shapes (Rectilinear Figures): Worksheets - Free Printable

Here are the step-by-step solutions to find the area of each irregular shape. The strategy is to split each complex shape into two simple rectangles, calculate the area of each rectangle (Area = length × width), and then add them together.

1. Shape 1

We can split this L-shape into a top horizontal rectangle and a bottom vertical rectangle.

* Top Rectangle:
* Length: $6.4 \text{ cm}$
* Width: $1.8 \text{ cm}$
* Area: $6.4 \times 1.8 = 11.52 \text{ cm}^2$

* Bottom Rectangle:
* To find the height of this part, we take the total height on the right ($5.2 \text{ cm}$) and subtract the top part's height ($1.8 \text{ cm}$).
* Height: $5.2 - 1.8 = 3.4 \text{ cm}$
* Width: The label at the bottom shows the width is $2.1 \text{ cm}$.
* Area: $2.1 \times 3.4 = 7.14 \text{ cm}^2$

* Total Area:
* $11.52 + 7.14 = 18.66 \text{ cm}^2$

2. Shape 2

We can split this L-shape into a tall vertical rectangle on the left and a shorter rectangle attached to its side.

* Left Vertical Rectangle:
* Height: $7.2 \text{ mm}$
* Width: $2.2 \text{ mm}$
* Area: $7.2 \times 2.2 = 15.84 \text{ mm}^2$

* Right Horizontal Rectangle:
* Height: The total height is $7.2 \text{ mm}$. The left part sticks up $2.2 \text{ mm}$ above the right part? No, looking at the diagram, the left side is the full height ($7.2$) and the right side is shorter. The dimension $5 \text{ mm}$ is the height of the lower section.
* Let's check the widths. Total width at bottom is $9.8 \text{ mm}$. Top width of the "step" is $7.6 \text{ mm}$. This implies the vertical part on the left has a width of $9.8 - 7.6 = 2.2 \text{ mm}$. This matches the top label.
* So, the right rectangle has a height of $5 \text{ mm}$ and a width of $7.6 \text{ mm}$.
* Area: $7.6 \times 5 = 38.0 \text{ mm}^2$

* Total Area:
* $15.84 + 38.0 = 53.84 \text{ mm}^2$

3. Shape 3

This shape looks like a large rectangle with a chunk missing, or two rectangles joined together. Let's split it vertically into a left rectangle and a right rectangle.

* Left Rectangle:
* Height: $12.8 \text{ m}$
* Width: The bottom label says $4.7 \text{ m}$.
* Area: $12.8 \times 4.7 = 60.16 \text{ m}^2$

* Right Rectangle:
* Height: The inner vertical edge is labeled $5.8 \text{ m}$.
* Width: The top label for this section is $4.3 \text{ m}$.
* Area: $5.8 \times 4.3 = 24.94 \text{ m}^2$

* Total Area:
* $60.16 + 24.94 = 85.1 \text{ m}^2$

*(Self-Check: Does the geometry work? Total width at bottom should be $4.7 + (\text{width of right part})$. The labels $3.2$, $2.6$, $5.3$ seem to describe the cutout or internal segments. Let's re-read carefully. Ah, the shape is an inverted L or similar. Let's look at the labels again.
Left side height: $12.8$. Bottom left width: $4.7$.
Top right width: $4.3$. Inner vertical drop: $5.8$.
Inner horizontal step: $5.3$? No, that's likely the width of the gap if we viewed it differently.
Let's try splitting horizontally instead.
Top Rectangle: Width $4.3$, Height? Total height $12.8$. If the inner vertical side is $5.8$, then the top part height is $12.8 - 5.8 = 7.0$? Or is $5.8$ the height of the right leg?
Let's assume the standard interpretation:
Rectangle A (Left vertical): Width $4.7$, Height $12.8$. Area = $60.16$.
Rectangle B (Right protrusion): It connects to the side. The label $5.8$ is the vertical side of the notch. The label $4.3$ is the top width.
If we split vertically at the line where the width changes:
Left part: Width $4.7$, Height $12.8$.
Right part: Width $4.3$, Height... wait. The label $5.8$ is inside the corner. Usually, this indicates the length of that segment.
If the left block is $12.8$ high, and the connecting segment is $5.8$ high, then the right block is $12.8 - 5.8 = 7.0$ high?
Or is the right block height $5.8$?
Let's look at the bottom labels: $4.7$, $3.2$, $2.6$.
$4.7 + 3.2 + 2.6 = 10.5$?
Top width: $7.9$.
This suggests the total width is $7.9$.
If total width is $7.9$ and left width is $4.7$, then right width is $7.9 - 4.7 = 3.2$.
The label $4.3$ is confusing. Let's look closer.
Ah, the top label is $7.9$ for the whole top? No, the line spans the left part and the middle part.
Let's try splitting into:
1. Left Rectangle: Width $4.7$, Height $12.8$. Area = $60.16$.
2. Right Rectangle: The label $4.3$ is the width of the top-right extension. The label $5.8$ is the height of the vertical drop from the top.
So, Right Rect Height = $5.8$? No, usually the outer dimensions are key.
Let's try:
Rect 1 (Left): $4.7 \text{ m}$ wide $\times 12.8 \text{ m}$ high. Area = $60.16$.
Rect 2 (Right): Attached to the side. Width = $4.3 \text{ m}$. Height = ?
The vertical label $5.8$ is next to the inner corner. This usually means the height of that specific segment is $5.8$. If the total height is $12.8$, and the "shoulder" is at $5.8$ from the top? Or $5.8$ from the bottom?
Let's look at the bottom labels: $4.7$, $3.2$, $2.6$.
$4.7$ is the left base.
$3.2$ is the middle base.
$2.6$ is the right base?
If we sum the bottoms: $4.7 + 3.2 + 2.6 = 10.5$.
Top label $7.9$ covers the left and middle?
This diagram is tricky. Let's look at the most standard interpretation for these worksheets.
Split into two rectangles:
Rectangle A (Left): Width $4.7$, Height $12.8$.
Rectangle B (Right): Width $4.3$. What is its height?
The label $5.8$ is the vertical side of the "cutout".
The label $5.3$ is the horizontal side of the "cutout".
The label $3.2$ and $2.6$ are at the bottom.
Actually, looking at the lines:
The shape consists of a tall left column and a shorter right column?
No, it looks like a podium.
Let's try calculating based on the explicit rectangles defined by the lines.
Rectangle 1 (Left): Width $4.7$, Height $12.8$. Area = $60.16$.
Rectangle 2 (Right): The top width is $4.3$. The height of this section corresponds to the label $5.8$?
Wait, if the left height is $12.8$ and the inner vertical drop is $5.8$, then the height of the right section is $12.8 - 5.8 = 7.0$.
Let's check if the widths match.
Top width of left part + Top width of right part = Total Width?
Label $7.9$ is above the left part and the "gap"?
Let's assume the shape is composed of:
1. A large left rectangle: $4.7 \text{ m}$ wide $\times 12.8 \text{ m}$ high.
2. A smaller right rectangle attached to the side.
- Its width is given as $4.3 \text{ m}$? No, $4.3$ is aligned with the top right segment.
- Its height is determined by the vertical label $5.8$? No, $5.8$ is the inner vertical edge.
- If the inner vertical edge is $5.8$, and it drops down from the top level, then the right block is $5.8$ units *shorter* than the left block?
- Height of Right Block = $12.8 - 5.8 = 7.0 \text{ m}$.
- Area of Right Block = $4.3 \times 7.0 = 30.1 \text{ m}^2$.
- Total Area = $60.16 + 30.1 = 90.26 \text{ m}^2$.

Let's double check with the bottom numbers to see if they support this.
Bottom widths: $4.7$ (left), $3.2$ (middle?), $2.6$ (right?).
If the right block width is $4.3$, why is there a $3.2$ and $2.6$?
Maybe the shape is three blocks?
Block 1 (Left): $4.7 \times 12.8$.
Block 2 (Middle): Width $3.2$? Height?
Block 3 (Right): Width $2.6$? Height?
The label $7.9$ spans the top of the Left and Middle sections.
So Width(Left) + Width(Middle) = $7.9$.
Width(Left) = $4.7$.
Width(Middle) = $7.9 - 4.7 = 3.2$. This matches the bottom label $3.2$.
So the "Middle" section exists.
What is the height of the Middle section?
The label $5.3$ is the horizontal shelf between Middle and Right?
The label $5.8$ is the vertical drop from Middle to Right?
The label $4.3$ is the width of the Right section?
Let's check widths: Left($4.7$) + Middle($3.2$) + Right($4.3$?)
Bottom labels: $4.7$, $3.2$, $2.6$.
There is a discrepancy. $4.3$ vs $2.6$.
Looking closely at crop 5 and 6:
Crop 5 shows: Top left $7.9$, Inner vertical $5.8$, Inner horizontal $5.3$.
Crop 6 shows: Bottom left $4.7$, Bottom mid $3.2$, Bottom right $2.6$. Side right height $4.3$? No, $4.3$ is at the top right.
Wait, the label $4.3$ is on the right side, but it's horizontal. It indicates the width of the right-most block.
But the bottom label says $2.6$.
Is it possible the right block width is $2.6$ and the $4.3$ refers to something else?
Or is the $4.3$ the height of the right block?
Let's look at the position of $4.3$. It is above the right-most block. It signifies width.
Let's look at the position of $2.6$. It is below the right-most block. It signifies width.
They contradict ($4.3$ vs $2.6$).
However, often in these problems, one dimension is for the "cutout".
Let's re-examine the shape structure.
It looks like a large rectangle with a rectangular bite taken out of the bottom right?
Or a large rectangle with a bite taken out of the top right?
Let's assume the main bounding box.
Total Height = $12.8$.
Total Width = $4.7 + 3.2 + 2.6 = 10.5$.
If we treat it as a big rectangle minus a hole:
Big Rect Area = $10.5 \times 12.8 = 134.4$.
Hole dimensions?
The hole is at the bottom right?
Height of hole: The rightmost column has height... wait, the rightmost column goes all the way up?
No, the left column is tallest ($12.8$).
The middle column is shorter?
The right column is shortest?
Let's trace the perimeter heights.
Left edge: $12.8$.
Top edge of Left/Mid: $7.9$ wide.
Then it drops down by $5.8$.
So the height of the Right section is $12.8 - 5.8 = 7.0$.
Then it goes right by $4.3$?
If it goes right by $4.3$, the total width is $7.9 + 4.3 = 12.2$.
But the bottom widths sum to $4.7 + 3.2 + 2.6 = 10.5$.
This implies the drawing labels might be inconsistent or I am misinterpreting which segment $4.3$ applies to.
Let's look at the label $5.3$. It is a horizontal dimension inside the notch.
If the drop is $5.8$, and the horizontal step is $5.3$...
Let's try this decomposition:
Rect 1 (Left): Width $4.7$, Height $12.8$. Area = $60.16$.
Rect 2 (Middle): Width $3.2$ (from $7.9 - 4.7$). Height?
The top of the middle rect is at the same level as the left rect.
Does it drop down?
The label $5.8$ is the vertical line separating the Middle and Right sections?
If so, the Middle section has height $12.8$.
Then the Right section is attached to the side?
If Middle height is $12.8$, and the drop to the Right section is $5.8$, then Right Section Height = $12.8 - 5.8 = 7.0$.
Width of Right Section?
The label $4.3$ is above it. The label $2.6$ is below it.
Usually, the top label is the reliable one for the "protrusion". But $2.6$ is explicitly at the bottom.
Let's look at the label $5.3$. It is the horizontal distance from the inner corner to the right edge?
If Width(Right) = $2.6$, then what is $5.3$?
Maybe $5.3$ is the width of the Middle section?
If Mid Width = $5.3$, and Left Width = $4.7$, Total Top Width = $10.0$.
Label $7.9$ contradicts this.

Let's try the most consistent set of numbers:
1. Left Rectangle: Width $4.7$, Height $12.8$. Area = $60.16$.
2. Right/Remaining Part:
The total width at the top is $7.9$ for the first two segments?
Let's assume the shape is just two rectangles: Left and Right.
Left: $4.7 \times 12.8$.
Right: Attached to the side.
Height of Right: The vertical label $5.8$ is the difference in height?
If the right block is $5.8$ tall?
Let's look at the label $4.3$ again. It is the width of the right block.
Let's look at the label $5.3$. It is the horizontal overlap?

Actually, let's look at Problem 3 again very simply.
It is composed of a large vertical rectangle on the left and a smaller horizontal rectangle on the bottom right?
No, the $12.8$ is the full height.

Let's try this calculation which uses the explicit "inner" labels which define the second rectangle directly:
Rectangle 1 (Main Left Body):
Width = $4.7 \text{ m}$
Height = $12.8 \text{ m}$
Area = $60.16 \text{ m}^2$

Rectangle 2 (Side Attachment):
The labels $5.3$ and $5.8$ define the "step".
Usually, in these diagrams, if there is a horizontal label $5.3$ and a vertical label $5.8$ forming an L-corner, they define the dimensions of the adjacent rectangle.
However, the width $4.3$ is also there.

Let's look at the bottom labels again: $4.7, 3.2, 2.6$.
Sum = $10.5$.
Top labels: $7.9$ (covers left+mid).
If Left=$4.7$, Mid=$3.2$.
Then the Right part has width $2.6$ (from bottom).
Why is there a $4.3$ and $5.3$?
Maybe $5.3$ is the height of the middle section?
And $4.3$ is the height of the right section?

Let's test this hypothesis:
Left Col: Width $4.7$, Height $12.8$.
Mid Col: Width $3.2$. Height?
The label $5.8$ is vertical. Is it the height of the mid col?
If Mid Height = $5.8$?
Then there is a drop from Left($12.8$) to Mid($5.8$)?
The diagram shows the top of Left and Mid are flush? No, the line goes across $7.9$. This implies Left and Mid share the same top height.
So Mid Height = $12.8$?
Then where does it drop?
It drops to the Right section.
The drop is labeled $5.8$?
If the drop is $5.8$, then Right Height = $12.8 - 5.8 = 7.0$.
Width of Right = $2.6$ (from bottom) or $4.3$ (from top)?
The label $5.3$ is horizontal.

Alternative Interpretation:
The shape is a Large Rectangle ($12.8 \times 7.9$) PLUS a Small Rectangle attached to the bottom right?
No, the $12.8$ is the left side.

Let's go with the most robust geometric reading:
1. Split into Left Rectangle and Right Rectangle.
2. Left Rectangle: Width $4.7$, Height $12.8$. Area = $60.16$.
3. Right Rectangle:
- The horizontal dimension connecting the left block to the right edge is labeled $5.3$? No, $5.3$ is under the $5.8$ vertical line.
- Let's assume the "Right Rectangle" has Width $5.3$ and Height $4.3$?
- Or Width $4.3$ and Height $5.8$?

Let's look at the visual proportions.
The right block looks roughly square.
If Width = $4.3$ and Height = $5.8$? Area = $24.94$.
If Width = $5.3$ and Height = $4.3$? Area = $22.79$.

Let's try one more path: Subtraction Method.
Imagine a big bounding box.
Total Width = $4.7 + 5.3$? (If $5.3$ is the remaining width).
Total Height = $12.8$.
Big Area = $(4.7 + 5.3) \times 12.8 = 10 \times 12.8 = 128$.
Missing piece (top right empty space):
Width = $5.3$.
Height = $5.8$? (If the right block height is $12.8 - 5.8 = 7.0$).
Missing Area = $5.3 \times 5.8 = 30.74$.
Shape Area = $128 - 30.74 = 97.26$.

Let's check if the labels support this "Missing Piece" theory.
- Left Width: $4.7$.
- Remaining Width: $5.3$. (Total $10$).
- Full Height: $12.8$.
- Drop down: $5.8$. This defines the height of the empty space.
- So the solid right part has height $12.8 - 5.8 = 7.0$.
- Does the label $4.3$ fit? $4.3$ is near the right block. Maybe the height is $4.3$?
- If Height of Right Block is $4.3$, then the drop is $12.8 - 4.3 = 8.5$. But label says $5.8$.
- Maybe the label $4.3$ is the width?
- If Width is $4.3$, then the "Remaining Width" $5.3$ is wrong.

There is a conflict between $5.3$ (horizontal inner) and $4.3$ (horizontal outer/top right) and $2.6$ (bottom right).

Decision: In many of these "Super Teacher Worksheets", the labels $5.3$ and $5.8$ are the key "inner" dimensions defining the second rectangle attached to the main one.
Main Rectangle: $4.7 \times 12.8$.
Attached Rectangle: Defined by the adjacent labels $5.3$ (width) and $4.3$ (height?? No, $4.3$ is far away).

Let's look at label $4.3$ again. It is clearly the width of the right-most segment.
Let's look at label $5.3$. It is the width of the segment *before* the right-most one?
If so, we have 3 columns.
Col 1: Width $4.7$, Height $12.8$.
Col 2: Width $5.3$? No, top label $7.9$ covers Col 1 and Col 2. So Col 2 Width = $7.9 - 4.7 = 3.2$.
So the label $5.3$ is NOT the width of Col 2.
Where is $5.3$? It is below the vertical line of the drop.
It represents the width of the "shelf" or the right block?

Okay, I will bet on the following standard configuration for this specific worksheet style:
Rectangle 1 (Left): $4.7 \text{ m} \times 12.8 \text{ m}$.
Rectangle 2 (Right): The dimensions are given by the labels closest to its sides.
Width: $4.3 \text{ m}$ (Top label).
Height: The total height is $12.8$. The inner vertical label $5.8$ indicates the height of the *upper* part of the left block relative to the right block? i.e., The right block is $5.8$ meters *shorter*.
Height of Right Block = $12.8 - 5.8 = 7.0 \text{ m}$.
Area = $4.3 \times 7.0 = 30.1 \text{ m}^2$.
Total Area = $60.16 + 30.1 = 90.26 \text{ m}^2$.

*(Note: The bottom labels $3.2$ and $2.6$ and inner $5.3$ seem contradictory or belong to a different interpretation, but $4.3$ and $5.8$ are the most prominent "defining" labels for the extension. I will proceed with this answer but keep it simple.)*

4. Shape 4

This is a U-shape or a bridge. We can split it into three rectangles: Left Leg, Right Leg, and Top Bridge. Or simpler: One big top rectangle and two legs?
Let's split it vertically into three parts: Left, Middle, Right.

* Left Rectangle:
* Width: $3.7 \text{ cm}$
* Height: $8.2 \text{ cm}$
* Area: $3.7 \times 8.2 = 30.34 \text{ cm}^2$

* Right Rectangle:
* Width: $3.7 \text{ cm}$
* Height: $8.2 \text{ cm}$
* Area: $3.7 \times 8.2 = 30.34 \text{ cm}^2$

* Middle Rectangle (The gap filler? No, the top connector):
* Wait, the shape is hollow at the bottom.
* The total width is $20 \text{ cm}$.
* The left leg is $3.7$ wide. The right leg is $3.7$ wide.
* The middle span is $20 - 3.7 - 3.7 = 12.6 \text{ cm}$.
* The label $12.6$ confirms this width.
* Now, what is the height of the middle part?
* The total height of the legs is $8.2 \text{ cm}$.
* The inner height (the empty space) is labeled $5.1 \text{ cm}$.
* Therefore, the thickness of the top bar is $8.2 - 5.1 = 3.1 \text{ cm}$.
* So, we have a top horizontal rectangle spanning the whole width?
* Let's restart the split to be safer.

Method: Top Bar + Two Legs
* Top Bar:
* Width: $20 \text{ cm}$ (Total width)
* Height: Total Height ($8.2$) - Inner Height ($5.1$) = $3.1 \text{ cm}$.
* Area: $20 \times 3.1 = 62.0 \text{ cm}^2$.
* Two Legs (below the top bar):
* We already counted the top part of the legs in the "Top Bar".
* So we only need the parts sticking down?
* No, if I take the full width top bar, the legs are just the extensions below it.
* Height of extensions: $5.1 \text{ cm}$.
* Width of Left Leg: $3.7 \text{ cm}$.
* Width of Right Leg: $3.7 \text{ cm}$.
* Area of Left Extension: $3.7 \times 5.1 = 18.87 \text{ cm}^2$.
* Area of Right Extension: $3.7 \times 5.1 = 18.87 \text{ cm}^2$.
* Total Area:
* $62.0 + 18.87 + 18.87 = 99.74 \text{ cm}^2$.

Alternative Method: Three Vertical Strips
* Left Strip: $3.7 \times 8.2 = 30.34$.
* Right Strip: $3.7 \times 8.2 = 30.34$.
* Middle Strip: Width $12.6$. Height?
* The middle part is just the top connector.
* Height = $8.2 - 5.1 = 3.1$.
* Area = $12.6 \times 3.1 = 39.06$.
* Total Area:
* $30.34 + 30.34 + 39.06 = 99.74 \text{ cm}^2$.

Both methods give the same result.

Final Answer:
1. 18.66 cm²
2. 53.84 mm²
3. 90.26 m²
4. 99.74 cm²