Math worksheet for calculating the area and perimeter of irregular polygons by decomposing them into smaller shapes.

Here are the step-by-step solutions for each problem on the worksheet.

Problem 1

Step 1: Find the Area (A)
To find the area of this L-shaped polygon, we can split it into two rectangles. Let's split it vertically down the middle.
* Left Rectangle: The width is given as $5\text{ cm}$. To find the height, look at the right side ($9\text{ cm}$) and subtract the top part ($4\text{ cm}$). So, the height is $9 - 4 = 5\text{ cm}$.
* $\text{Area} = 5\text{ cm} \times 5\text{ cm} = 25\text{ cm}^2$
* Right Rectangle: The width is $3\text{ cm}$ and the height is $4\text{ cm}$.
* $\text{Area} = 3\text{ cm} \times 4\text{ cm} = 12\text{ cm}^2$
* Total Area: Add the two areas together.
* $25 + 12 = 37\text{ cm}^2$

Step 2: Find the Perimeter (P)
The perimeter is the distance around the outside of the shape. We need to add up all the side lengths.
* Top side: $5 + 3 = 8\text{ cm}$
* Right side: $4\text{ cm}$
* Inner vertical side: $3\text{ cm}$ (This is the difference between total height 9 and left height 6? No, let's trace carefully).
* Let's just add the outer boundary segments provided or calculated:
* Left side: $9\text{ cm}$
* Bottom side: $5 + 3 = 8\text{ cm}$
* Right-most vertical side: $4\text{ cm}$
* Top horizontal segment: $3\text{ cm}$
* Inner vertical drop: $9 - 4 = 5\text{ cm}$? Wait, looking at the diagram:
* Left side = $9$
* Bottom side = $5 + 3 = 8$
* Right side bottom part = $4$
* Top right part = $3$
* The inner corner sides are: Top left part = $5$, Inner vertical = $9-4=5$? No, the label "4 cm" is on the top right vertical edge. The label "9 cm" is the total left height. The label "5 cm" is the bottom left width. The label "3 cm" is the bottom right width.
* Let's re-read the diagram labels carefully.
* Left vertical: $9\text{ cm}$
* Bottom horizontal: composed of $5\text{ cm}$ and $3\text{ cm}$ segments? No, the "5 cm" is the top-left horizontal width. The "3 cm" is the top-right horizontal width? No, usually these diagrams label specific segments.
* Let's assume standard labeling:
* Left vertical edge: $9\text{ cm}$
* Bottom horizontal edge: $5\text{ cm}$ (left part) + $3\text{ cm}$ (right part)? No, the "5 cm" is labeled on the top horizontal segment of the left block. The "3 cm" is labeled on the top horizontal segment of the right block? Or is "3 cm" the width of the right leg?
* Let's look at the numbers again.
* Left vertical: $9$.
* Top-left horizontal: $5$.
* Inner vertical drop: $4$.
* Top-right horizontal: $3$.
* This implies the shape is an inverted L or similar. Let's trace the perimeter based on typical problems of this type.
* Usually, opposite parallel sides sum up to the same total length.
* Total Height (Left) = $9\text{ cm}$.
* Total Width (Bottom) = $5\text{ cm} + 3\text{ cm} = 8\text{ cm}$.
* Perimeter = $2 \times (\text{Total Height} + \text{Total Width})$
* $P = 2 \times (9 + 8) = 2 \times 17 = 34\text{ cm}$.

Let's double check with individual sides:
1. Left: $9$
2. Bottom: $5 + 3 = 8$
3. Right: The vertical side on the far right. If the inner vertical drop is $4$, and total height is $9$, then the rightmost vertical side is $9 - 4 = 5$? Or is the "4" the rightmost side? The label "4 cm" is next to the inner vertical edge. The label "3 cm" is the top horizontal edge of the right protrusion.
* Let's assume:
* Left Side: $9$
* Top Left Horizontal: $5$
* Inner Vertical Down: $4$
* Top Right Horizontal: $3$
* Right Vertical Down: ?
* Bottom Horizontal: ?
* Actually, a simpler way for rectilinear shapes (all right angles) is that the perimeter is equal to the perimeter of the bounding rectangle.
* Bounding Box Height = $9\text{ cm}$ (from the left side).
* Bounding Box Width = $5\text{ cm} + 3\text{ cm} = 8\text{ cm}$ (sum of the top horizontal segments).
* Perimeter = $9 + 9 + 8 + 8 = 34\text{ cm}$.

Answer for #1:
* A = $37\text{ cm}^2$
* P = $34\text{ cm}$

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Problem 2

Step 1: Find the Area (A)
Split the shape into two rectangles. Let's split it horizontally.
* Top Rectangle: Width = $4\text{ in}$, Height = $3\text{ in}$.
* $\text{Area} = 4 \times 3 = 12\text{ in}^2$
* Bottom Rectangle:
* The total width at the bottom is $10\text{ in}$.
* The height of the bottom section is $7\text{ in}$.
* Wait, the label "7 in" is on the right vertical side of the bottom part. The label "10 in" is the total bottom width. The label "4 in" is the top width. The label "3 in" is the top height.
* Let's split it vertically instead, it might be clearer.
* Left Rectangle: Width = $4\text{ in}$. Total Height = $3\text{ in} + 7\text{ in} = 10\text{ in}$.
* $\text{Area} = 4 \times 10 = 40\text{ in}^2$
* Right Rectangle:
* Total Width = $10\text{ in}$. Left Width = $4\text{ in}$. So Right Width = $10 - 4 = 6\text{ in}$.
* Height = $7\text{ in}$.
* $\text{Area} = 6 \times 7 = 42\text{ in}^2$
* Total Area: $40 + 42 = 82\text{ in}^2$

*Alternative Split (Horizontal):*
* Top Rectangle: $4\text{ in} \times 3\text{ in} = 12\text{ in}^2$.
* Bottom Rectangle: Width = $10\text{ in}$. Height = $7\text{ in}$.
* $\text{Area} = 10 \times 7 = 70\text{ in}^2$.
* Total Area: $12 + 70 = 82\text{ in}^2$.

Step 2: Find the Perimeter (P)
Use the bounding box method.
* Total Height = $3\text{ in} + 7\text{ in} = 10\text{ in}$.
* Total Width = $10\text{ in}$.
* Perimeter = $2 \times (10 + 10) = 40\text{ in}$.

Let's verify by adding sides:
* Left: $10$ ($3+7$)
* Bottom: $10$
* Right: $7$
* Inner Horizontal: $6$ ($10-4$)
* Inner Vertical: $3$
* Top: $4$
* Sum: $10 + 10 + 7 + 6 + 3 + 4 = 40\text{ in}$.

Answer for #2:
* A = $82\text{ in}^2$
* P = $40\text{ in}$

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Problem 3

Step 1: Find the Area (A)
Split the shape into three vertical rectangles or one large one minus a cutout. Let's use addition.
* Left Rectangle: Width = $2\text{ m}$, Height = $7\text{ m}$.
* $\text{Area} = 2 \times 7 = 14\text{ m}^2$
* Middle Rectangle:
* Width = $6\text{ m}$.
* Height? The total height on the left is $7$. The top part sticks up. The label "$3\text{ m}$" is the height of the top protrusion above the side wings? Or is it the height of the middle section?
* Let's look at the labels:
* Left wing: Width $2$, Height $7$.
* Right wing: Width $3$, Height $7$ (implied symmetric or same base level? The label $7\text{ m}$ is on the far left. The label $7\text{ m}$ is NOT on the right. But there is a $7\text{ m}$ on the bottom right vertical? No, that's a $7\text{ m}$ on the left.
* Let's re-examine image 3.
* Left vertical side: $7\text{ m}$.
* Bottom left width: $2\text{ m}$.
* Middle bottom width: $6\text{ m}$.
* Right bottom width: $3\text{ m}$.
* Top middle height protrusion: $3\text{ m}$.
* Right vertical side: $7\text{ m}$? No, the label on the far right vertical is missing, but there is a label "$7\text{ m}$" on the left. There is a label "$7\text{ m}$" on the right vertical side of the *bottom* section? It looks like the side wings are height $7$? No, the $7$ is the total height of the left side.
* Let's assume the "base" height is consistent.
* Actually, usually these "U" or "T" shapes have a base.
* Let's try splitting it into a bottom rectangle and top rectangles.
* Or better, split into 3 vertical columns.
* Column 1 (Left): Width $2\text{ m}$. Height? The label $7\text{ m}$ is the full height of the left side. So Area = $2 \times 7 = 14\text{ m}^2$.
* Column 3 (Right): Width $3\text{ m}$. Height? The shape looks symmetric in height for the wings, but let's check. The label "$7\text{ m}$" is on the left. Is there a label on the right? Yes, there is a "$7\text{ m}$" on the right vertical edge too. So the right wing is also $7\text{ m}$ high. Area = $3 \times 7 = 21\text{ m}^2$.
* Column 2 (Middle): Width $6\text{ m}$. Height? The middle part sticks *up* by $3\text{ m}$? Or is the $3\text{ m}$ the height of the middle part *above* the wings?
* Looking at the diagram, the middle section is taller. The label "$3\text{ m}$" indicates the extra height of the middle section compared to the sides.
* So, Height of Middle = Height of Side ($7\text{ m}$) + Extra ($3\text{ m}$) = $10\text{ m}$.
* Area of Middle = $6\text{ m} \times 10\text{ m} = 60\text{ m}^2$.
* Total Area: $14 + 21 + 60 = 95\text{ m}^2$.

*Alternative Interpretation:*
Maybe the $7\text{ m}$ is just the height of the side wings, and the middle is separate?
Let's try splitting horizontally.
* Bottom Rectangle: Spans the whole width. Width = $2 + 6 + 3 = 11\text{ m}$. Height = $7\text{ m}$ (assuming the sides define the base height). Area = $11 \times 7 = 77\text{ m}^2$.
* Top Rectangle: Sits on top of the middle section. Width = $6\text{ m}$. Height = $3\text{ m}$. Area = $6 \times 3 = 18\text{ m}^2$.
* Total Area: $77 + 18 = 95\text{ m}^2$.
This confirms the previous calculation.

Step 2: Find the Perimeter (P)
Add all outer sides.
* Bottom: $2 + 6 + 3 = 11\text{ m}$
* Left Side: $7\text{ m}$
* Right Side: $7\text{ m}$
* Top Left Horizontal: $2\text{ m}$
* Top Right Horizontal: $3\text{ m}$
* Middle Vertical Up: $3\text{ m}$
* Middle Top Horizontal: $6\text{ m}$
* Middle Vertical Down: $3\text{ m}$
* Sum: $11 + 7 + 7 + 2 + 3 + 3 + 6 + 3 = 42\text{ m}$.

Let's use the bounding box method to check.
* Total Width = $11\text{ m}$.
* Total Height = $7 + 3 = 10\text{ m}$.
* Perimeter of bounding box = $2 \times (11 + 10) = 42\text{ m}$.
* Since the indents are "pushed in" but the vertical/horizontal segments still sum to the bounding box dimensions (for this specific convex-like orthogonal shape where no holes exist), the perimeter is indeed $42\text{ m}$.

Answer for #3:
* A = $95\text{ m}^2$
* P = $42\text{ m}$

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Final Answer:

1.
A = 37 cm²
P = 34 cm

2.
A = 82 in²
P = 40 in

3.
A = 95 m²
P = 42 m