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

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

Problem 1

Shape: An L-shape made of two rectangles.
* Dimensions: Total height is $10\text{ cm}$, total width is $18\text{ cm}$. The bottom part is $8\text{ cm}$ wide. The top right part sticks out $4\text{ cm}$ high.

Area Calculation:
Let's split the shape into two vertical rectangles: a left one and a right one.
1. Left Rectangle:
* Width = $8\text{ cm}$
* Height = Total height ($10\text{ cm}$) minus the top right height ($4\text{ cm}$) = $6\text{ cm}$.
* Area = $8 \times 6 = 48\text{ cm}^2$.
2. Right Rectangle:
* Width = Total width ($18\text{ cm}$) minus the left width ($8\text{ cm}$) = $10\text{ cm}$.
* Height = $4\text{ cm}$.
* Area = $10 \times 4 = 40\text{ cm}^2$.
3. Total Area: $48 + 40 = \mathbf{88\text{ cm}^2}$.

Perimeter Calculation:
Add up all the outside edges. We need to find the missing inner edges first.
* Top edge: $18\text{ cm}$
* Right top edge: $4\text{ cm}$
* Inner horizontal edge: $18 - 8 = 10\text{ cm}$
* Inner vertical edge: $10 - 4 = 6\text{ cm}$
* Bottom edge: $8\text{ cm}$
* Left edge: $10\text{ cm}$
* Total Perimeter: $18 + 4 + 10 + 6 + 8 + 10 = \mathbf{56\text{ cm}}$.

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Problem 2

Shape: A rectangle with a rectangular notch cut out of the top.
* Dimensions: Total width is $4 + 6 + 4 = 14\text{ in}$. Total height is $8.5\text{ in}$. The notch is $6\text{ in}$ wide and $2.5\text{ in}$ deep.

Area Calculation:
Method: Calculate the area of the big full rectangle and subtract the empty notch.
1. Big Rectangle:
* Width = $14\text{ in}$
* Height = $8.5\text{ in}$
* Area = $14 \times 8.5 = 119\text{ in}^2$.
2. Notch (Empty Space):
* Width = $6\text{ in}$
* Height = $2.5\text{ in}$
* Area = $6 \times 2.5 = 15\text{ in}^2$.
3. Total Area: $119 - 15 = \mathbf{104\text{ in}^2}$.

Perimeter Calculation:
Add all outer edges. Note that the "indented" lines count towards the perimeter just like straight lines.
* Bottom: $14\text{ in}$
* Left Side: $8.5\text{ in}$
* Right Side: $8.5\text{ in}$
* Top segments: $4\text{ in}$ (left) + $4\text{ in}$ (right) = $8\text{ in}$.
* Notch sides: $2.5\text{ in}$ (down) + $2.5\text{ in}$ (up) = $5\text{ in}$.
* Notch bottom: $6\text{ in}$.
* Total Perimeter: $14 + 8.5 + 8.5 + 4 + 4 + 2.5 + 2.5 + 6 = \mathbf{50\text{ in}}$.

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Problem 3

Shape: An inverted L-shape.
* Dimensions: Top width $12\text{ in}$, Right height $19\text{ in}$, Left top height $7\text{ in}$, Bottom indent width $6\text{ in}$.

Area Calculation:
Split into a top horizontal rectangle and a bottom vertical rectangle.
1. Top Rectangle:
* Width = $12\text{ in}$
* Height = $7\text{ in}$
* Area = $12 \times 7 = 84\text{ in}^2$.
2. Bottom Rectangle:
* Height = Total height ($19$) - Top height ($7$) = $12\text{ in}$.
* Width = Total width ($12$) - Indent width ($6$) = $6\text{ in}$.
* Area = $6 \times 12 = 72\text{ in}^2$.
3. Total Area: $84 + 72 = \mathbf{156\text{ in}^2}$.

Perimeter Calculation:
* Top: $12\text{ in}$
* Right: $19\text{ in}$
* Bottom Right: $6\text{ in}$
* Inner Vertical: $12\text{ in}$ (calculated above)
* Inner Horizontal: $6\text{ in}$ (given as indent width? No, wait. The label '6 in' is under the sticking out part. Let's re-read carefully. The label '6 in' is under the protruding part on the bottom left? No, looking at diagram 3, the '6 in' is the width of the bottom leg. The top is 12. So the vertical leg width is $12-6=6$? Or is the 6in the gap? Usually, these labels indicate the segment length. Let's assume the bottom horizontal segment is 6in.
* Let's check side lengths:
* Top: 12
* Right: 19
* Bottom segment: 6
* Inner vertical rise: $19 - 7 = 12$
* Inner horizontal run: $12 - 6 = 6$
* Left top segment: 7
* Total Perimeter: $12 + 19 + 6 + 12 + 6 + 7 = \mathbf{62\text{ in}}$.

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Problem 4

Shape: A large rectangle with a smaller rectangle attached to the bottom left.
* Dimensions: Main block width $16.8\text{ ft}$, height $13.2\text{ ft}$. Attached block height $5.1\text{ ft}$, width unknown directly but we have a vertical segment of $4.4\text{ ft}$ and total left height $8.1+4.4$? No, let's look closer.
* Top width: $16.8\text{ ft}$.
* Right height: $13.2\text{ ft}$.
* Bottom width: $12.4\text{ ft}$.
* Left side has two segments: Top part $8.1\text{ ft}$, bottom part $4.4\text{ ft}$? No, the label $4.4\text{ ft}$ is on the vertical drop. The label $5.1\text{ ft}$ is on the bottom horizontal of the small step.
* Let's decompose:
* Main Top Rectangle: Width $16.8$, Height $13.2 - (\text{something})$. This is tricky. Let's split vertically.
* Right Rectangle: Width = $16.8 - (\text{width of left part})$. We don't have the left width directly.
* Let's look at the bottom. Total width at bottom is not given as a single line. We have a segment $12.4\text{ ft}$ on the right bottom. And a segment $5.1\text{ ft}$ on the left bottom? No, the $5.1$ is the horizontal step.
* Let's assume the shape is composed of:
1. A large right rectangle: Width $12.4\text{ ft}$, Height $13.2\text{ ft}$. Area = $12.4 \times 13.2 = 163.68\text{ ft}^2$.
2. A smaller left rectangle attached to the side?
* The top width is $16.8$. The right part is $12.4$. So the left part width is $16.8 - 12.4 = 4.4\text{ ft}$.
* The height of this left part is labeled $8.1\text{ ft}$? Or is $8.1$ the top part of the left side?
* Let's look at the left side labels: $8.1\text{ ft}$ (top vertical), then a step in, then $4.4\text{ ft}$ (vertical drop), then $5.1\text{ ft}$ (horizontal). This doesn't match the width calculation ($4.4$ vs $5.1$).
* Alternative interpretation: The shape is one big rectangle $16.8 \times 13.2$ with a chunk missing from the bottom left?
* Missing chunk width: $16.8 - 12.4 = 4.4\text{ ft}$.
* Missing chunk height: $13.2 - 8.1 = 5.1\text{ ft}$.
* This matches the labels $4.4$ and $5.1$ perfectly if they represent the dimensions of the "cutout" or the remaining steps.
* Let's verify the labels on the drawing:
* Left side top vertical: $8.1\text{ ft}$.
* Inner vertical drop: $4.4\text{ ft}$. Total height = $8.1 + 4.4 = 12.5\text{ ft}$. But right side is $13.2\text{ ft}$. There is a discrepancy of $0.7\text{ ft}$.
* Let's re-read the diagram carefully.
* Top: $16.8$. Right: $13.2$. Bottom: $12.4$.
* Left side consists of a vertical segment $8.1$, a horizontal segment (indent), a vertical segment $4.4$, and a horizontal segment $5.1$.
* If the bottom horizontal is $12.4$, and the little foot is $5.1$, then the width of the main column is $12.4 - 5.1 = 7.3$? No.
* Let's try splitting into two vertical rectangles again.
* Rectangle 1 (Right): Width $12.4\text{ ft}$. Height $13.2\text{ ft}$. Area = $163.68\text{ ft}^2$.
* Rectangle 2 (Left): Width = Total Top ($16.8$) - Right Width ($12.4$) = $4.4\text{ ft}$. Height = $8.1\text{ ft}$. Area = $4.4 \times 8.1 = 35.64\text{ ft}^2$.
* Does this fit the other labels?
* If Left Height is $8.1$, and Right Height is $13.2$, the difference is $5.1\text{ ft}$.
* The label $4.4\text{ ft}$ is on the vertical segment connecting the lower level to the upper level? No, it looks like the vertical side of the lower block.
* The label $5.1\text{ ft}$ is the horizontal bottom of the lower block.
* If the left block has width $4.4$ (from $16.8-12.4$), why is the bottom labeled $5.1$?
* Maybe the $12.4$ is NOT the full width of the right section. Maybe $12.4$ is the bottom width of the RIGHT part, and $5.1$ is the bottom width of the LEFT part?
* If so, Total Width = $12.4 + 5.1 = 17.5\text{ ft}$. But Top Width is $16.8\text{ ft}$. Contradiction.
* Let's look at the labels again. $16.8$ top. $12.4$ bottom right. $5.1$ bottom left. $16.8 \neq 12.4 + 5.1$.
* Perhaps the $12.4$ refers to the entire bottom width excluding the little toe? No.
* Let's assume the standard "missing corner" approach.
* Full Box: $16.8 \times 13.2$.
* Cutout at bottom left.
* Cutout Width: $16.8 - 12.4 = 4.4\text{ ft}$.
* Cutout Height: $13.2 - 8.1 = 5.1\text{ ft}$.
* This creates a perfect match for the numbers $4.4$ and $5.1$ appearing in the diagram as the inner dimensions.
* So, Area = (Area of Big Box) - (Area of Cutout).
* Big Box Area: $16.8 \times 13.2 = 221.76\text{ ft}^2$.
* Cutout Area: $4.4 \times 5.1 = 22.44\text{ ft}^2$.
* Total Area: $221.76 - 22.44 = \mathbf{199.32\text{ ft}^2}$.

Perimeter Calculation:
For this type of shape (rectangle with a corner removed), the perimeter is equal to the perimeter of the bounding box.
* Perimeter = $2 \times (\text{Width} + \text{Height})$
* Perimeter = $2 \times (16.8 + 13.2)$
* Perimeter = $2 \times 30 = \mathbf{60\text{ ft}}$.
*(Check by adding segments: Top 16.8 + Right 13.2 + Bottom 12.4 + Inner Vert 5.1 + Inner Horz 4.4 + Left Top 8.1. Sum: $16.8+13.2+12.4+5.1+4.4+8.1 = 60$. Correct.)*

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Problem 5

Shape: A T-shape or stepped shape.
* Dimensions:
* Top part: Width $5.7\text{ in}$, Height unknown directly.
* Middle part: Width $6.5\text{ in}$? No, the label $6.5$ is on the horizontal shelf.
* Bottom part: Width $15.2\text{ in}$.
* Left side heights: $4.8\text{ in}$ (bottom block), and the rest is unknown.
* Right side total height: $17.8\text{ in}$.

Let's decompose into three vertical rectangles or horizontal strips.
Let's try horizontal strips from bottom to top.
1. Bottom Strip:
* Height = $4.8\text{ in}$.
* Width = $15.2\text{ in}$.
* Area = $15.2 \times 4.8 = 72.96\text{ in}^2$.
2. Middle Strip:
* This sits on top of the bottom strip.
* The label $6.5\text{ in}$ is the width of the protrusion on the left? Or the width of the middle section?
* Looking at the diagram: The bottom width is $15.2$. The top width is $5.7$. The middle shelf is $6.5$.
* This implies the shape widens in the middle? No, it looks like stairs going up to the right.
* Let's trace the widths:
* Bottom width: $15.2$.
* The left side goes up $4.8$. Then it steps IN to the right by some amount? Or OUT?
* The label $6.5$ is on the horizontal surface above the $4.8$ height.
* The label $5.7$ is the top width.
* The total height is $17.8$.
* Let's assume vertical slices.
* Slice 1 (Leftmost): Width? We know the shelf is $6.5$. If the total bottom is $15.2$, and the right part is... this is ambiguous.
* Let's look at the horizontal alignment.
* Top width $5.7$.
* Middle shelf $6.5$.
* Bottom width $15.2$.
* It seems the shape is composed of stacked blocks centered or aligned to the right? The right side is a straight vertical line of height $17.8$. This suggests all blocks are aligned to the right.
* Block 1 (Top): Width $5.7$. Height?
* Block 2 (Middle): Width $6.5$. Height?
* Block 3 (Bottom): Width $15.2$. Height $4.8$.
* If they are aligned to the right, the widths are cumulative from the right edge? No, widths are absolute.
* Let's find the heights of the top two blocks.
* We don't have explicit heights for the top sections. However, usually, in these problems, if heights aren't given, we might be able to deduce them or they are equal? No.
* Wait, look at the left side. There is only one vertical label $4.8$.
* Is it possible the $6.5$ and $5.7$ are horizontal distances from the left edge?
* Let's assume the standard "stacked boxes aligned right" model because the right side is a single straight line.
* We still need the heights of the top two sections.
* Is there information missing? Let's re-read the image.
* Ah, look at Problem 5 again.
* Left side: Vertical segment $4.8$. Horizontal segment $6.5$.
* Top: Width $5.7$.
* Right side: Total height $17.8$.
* Bottom: Width $15.2$.
* There is a vertical segment between the $6.5$ shelf and the $5.7$ top. Its height is not labeled.
* There is a vertical segment between the $4.8$ height and the bottom? No, $4.8$ is the height of the first step.
* Is it possible the vertical segments are equal? Or proportional? Unlikely.
* Let's look really closely at crop 5.
* There is a label "6.5 in" on the horizontal step.
* There is a label "5.7 in" on the top width.
* There is a label "4.8 in" on the lowest left vertical side.
* There is a label "15.2 in" on the bottom.
* There is a label "17.8 in" on the right.
* Crucial Insight: Look at the horizontal gaps.
* Bottom width = $15.2$.
* The first step in from the left is at width... wait. The label $6.5$ is the length of the horizontal shelf.
* If the shape is aligned to the right:
* Bottom Block Width = $15.2$.
* Middle Block Width = $6.5$? No, that would be very narrow compared to 15.2.
* Let's assume the labels indicate the width of that specific horizontal segment.
* Let's assume the blocks are aligned to the Left? No, the right side is straight.
* Let's calculate the "overhangs".
* Total Width at bottom = $15.2$.
* Top Width = $5.7$.
* Difference = $15.2 - 5.7 = 9.5$.
* This difference is distributed among the steps.
* We have a middle shelf of $6.5$.
* This geometry is confusing without explicit heights for the upper sections.
* Alternative Interpretation: Maybe the $6.5$ is the width of the *middle section* and the $5.7$ is the *top section*. And maybe the heights are derived?
* Let's look at the vertical space. Total height $17.8$. Bottom height $4.8$. Remaining height = $13.0$.
* If we assume the two remaining vertical sections are equal? $13 / 2 = 6.5$. Hey, the number $6.5$ appears again. Is the height of the middle section $6.5$?
* If Middle Height = $6.5$, then Top Height = $17.8 - 4.8 - 6.5 = 6.5$.
* This symmetry (two equal upper heights) is a very common pattern in such worksheets when a dimension is "missing" visually but a number is present elsewhere. Let's proceed with this assumption: The vertical segments above the bottom are equal in height, or the label 6.5 applies to height?
* Actually, looking at the placement, "6.5 in" is clearly horizontal.
* Is it possible the shape is defined by coordinates?
* Let's try another path. What if the $6.5$ is the height of the middle section? The text is horizontal, but sometimes labels are rotated. No, "in" is upright.
* Let's look at the horizontal widths again.
* Bottom: $15.2$.
* Top: $5.7$.
* Step: $6.5$.
* If we sum the horizontal parts from left to right?
* Let's assume the shape is composed of 3 rectangles stacked.
* Rect 1 (Bottom): $15.2 \times 4.8$.
* Rect 2 (Middle): Width? Height?
* Rect 3 (Top): Width $5.7$. Height?
* If the right side is flush, the widths are measured from the right edge.
* Width of Bottom = $15.2$.
* Width of Top = $5.7$.
* The label $6.5$ is on the intermediate horizontal surface. This usually means the width of that block is $6.5$.
* So we have widths: $15.2$ (bottom), $6.5$ (middle), $5.7$ (top).
* We have heights: $4.8$ (bottom). Total $17.8$.
* We are missing the split of the remaining height ($13.0$).
* However, often in these problems, if a dimension looks like it corresponds to another, it might be a square? No.
* Let's look at the previous problem (4). The "missing" dimensions were found by subtraction.
* Here, can we find heights by subtraction? No, no horizontal alignment markers.
* Wait! Look at the label "6.5 in" again. Is it possible it indicates the height of that section? In some poorly formatted worksheets, yes. But it's placed horizontally.
* Let's look at the label "5.7 in". It's on top.
* Let's look at the label "4.8 in". It's vertical.
* Let's look at the label "15.2 in". It's horizontal.
* Let's look at the label "17.8 in". It's vertical.
* There is a possibility that the heights of the top two sections are equal.
* If Height 2 = Height 3, then $H_2 = H_3 = (17.8 - 4.8) / 2 = 6.5\text{ in}$.
* This matches the number $6.5$ found in the diagram! It is highly likely that the height of the middle and top sections are both $6.5\text{ in}$, and the label "6.5 in" was mistakenly placed on the horizontal shelf, OR the horizontal shelf length is irrelevant/coincidental, OR the label "6.5 in" actually refers to the height of the middle block despite its position. Given the coincidence of the calculated height ($6.5$) and the label ($6.5$), this is the intended solution path.
* So:
* Bottom Rectangle: $15.2\text{ w} \times 4.8\text{ h}$. Area = $72.96$.
* Middle Rectangle: Width? If the label $6.5$ is the height, what is the width? The label is ON the horizontal segment. This implies the width IS $6.5$.
* If Width = $6.5$ and Height = $6.5$ (derived), then Area = $42.25$.
* Top Rectangle: Width $5.7$. Height? Remaining height = $17.8 - 4.8 - 6.5 = 6.5$. Area = $5.7 \times 6.5 = 37.05$.
* Let's check if this makes sense.
* Total Area = $72.96 + 42.25 + 37.05 = 152.26\text{ in}^2$.

* Alternative Theory: What if the label $6.5$ is the width, and the heights are determined differently? Without another label, the problem is unsolvable unless we assume symmetry or that the label applies to the height. Given the "Math Monks" style, simple integer/friendly number logic usually applies. The fact that $(17.8-4.8)/2 = 6.5$ exactly is too strong a coincidence to ignore. The label $6.5$ likely serves dual purpose or indicates the height of the middle tier, and the width is also $6.5$? Or perhaps the width is just $6.5$ and the height is something else? No, the height MUST be defined. The only way to define it with the given numbers is the symmetry argument yielding $6.5$.
* So, I will proceed with:
* Bottom: $15.2 \times 4.8$
* Middle: $6.5 \times 6.5$ (Width given as 6.5, Height derived as 6.5)
* Top: $5.7 \times 6.5$ (Width given as 5.7, Height derived as 6.5)

* Area:
* $A_1 = 72.96$
* $A_2 = 42.25$
* $A_3 = 37.05$
* Total = $\mathbf{152.26\text{ in}^2}$.

* Perimeter:
* Sum of all outer boundaries.
* Bottom: $15.2$
* Right: $17.8$
* Top: $5.7$
* Left Side Segments:
* Bottom vertical: $4.8$
* Middle vertical: $6.5$
* Top vertical: $6.5$
* Total Left Height = $4.8 + 6.5 + 6.5 = 17.8$. (Matches right side).
* Horizontal Steps (facing up):
* Step 1 (above bottom block): Width = $6.5$.
* Step 2 (above middle block): Width = $5.7$.
* Wait, the "steps" are the horizontal surfaces exposed to the air.
* The bottom block has width $15.2$. The middle sits on it. The exposed part is $15.2 - 6.5 = 8.7$?
* NO. The diagram shows the blocks getting SMALLER as they go up?
* Let's look at the shape again.
* Bottom is widest ($15.2$).
* Middle is narrower ($6.5$).
* Top is narrowest ($5.7$).
* They are aligned to the RIGHT.
* So, the horizontal "shelves" on the LEFT are:
* Shelf 1 (on top of bottom block): Length = $15.2 - 6.5 = 8.7\text{ in}$.
* Shelf 2 (on top of middle block): Length = $6.5 - 5.7 = 0.8\text{ in}$.
* BUT the label $6.5$ is written ON the shelf. Does it mean the shelf length is $6.5$?
* If Shelf 1 length is $6.5$, then Middle Width = $15.2 - 6.5 = 8.7$.
* If Shelf 2 length is... well, top width is $5.7$.
* This contradicts the "6.5 is the width" theory.
* Let's re-evaluate based on Label Placement.
* In technical drawings, a dimension line with arrows defines the extent. Here, there are no arrows, just text near lines.
* Text "6.5 in" is near the first horizontal step. It most likely defines the length of that step.
* Text "5.7 in" is near the top horizontal edge. It defines the top width.
* Text "15.2 in" is near the bottom. Defines bottom width.
* Text "4.8 in" is near the bottom-left vertical. Defines bottom-left height.
* Text "17.8 in" is near the right vertical. Defines total height.

New Calculation Path (Based on Step Lengths):
1. Bottom Rectangle:
* Width = $15.2$.
* Height = $4.8$.
* Area = $15.2 \times 4.8 = 72.96$.
2. Middle Rectangle:
* It sits on the bottom one.
* The step to its left is $6.5$ long.
* So, Middle Width = Total Bottom Width - Step Length = $15.2 - 6.5 = 8.7\text{ in}$.
* Height? We established the height coincidence earlier. Let's assume the vertical divisions are equal for the remaining $13.0$ height? Or is there another clue?
* If we don't assume equal heights, we can't solve it. But wait. Is the "6.5" the height?
* Let's look at the visual proportions. The step ($6.5$) looks longer than the bottom height ($4.8$). The middle height looks similar to the step length.
* Let's stick with the strongest mathematical clue: Remaining Height = 13.0. If we split it into two equal parts, we get 6.5. The number 6.5 is explicitly in the diagram. It is overwhelmingly likely that the height of the middle section is 6.5 and the height of the top section is 6.5. The label "6.5 in" is placed ambiguously, but given the calculation, it likely refers to the height, OR the width happens to be such that the step is also related?
* Actually, if Middle Height = $6.5$ and Top Height = $6.5$:
* And if the label "6.5 in" refers to the horizontal step length:
* Middle Width = $15.2 - 6.5 = 8.7$.
* Top Width = $5.7$ (given).
* Top Step Length = Middle Width - Top Width = $8.7 - 5.7 = 3.0$.
* This seems plausible.

Let's calculate Area with this model:
* Bottom: $15.2 \times 4.8 = 72.96$.
* Middle: Width $8.7$, Height $6.5$. Area = $8.7 \times 6.5 = 56.55$.
* Top: Width $5.7$, Height $6.5$. Area = $5.7 \times 6.5 = 37.05$.
* Total Area: $72.96 + 56.55 + 37.05 = \mathbf{166.56\text{ in}^2}$.

Let's calculate Perimeter with this model:
* Bottom: $15.2$
* Right: $17.8$
* Top: $5.7$
* Left Verticals: $4.8 + 6.5 + 6.5 = 17.8$.
* Horizontal Steps (exposed tops):
* Step 1: $6.5$ (given).
* Step 2: $3.0$ (calculated as $8.7 - 5.7$).
* Total Perimeter: $15.2 + 17.8 + 5.7 + 17.8 + 6.5 + 3.0 = \mathbf{66.0\text{ in}}$.

*Self-Correction/Verification:* Which interpretation of "6.5" is more standard? Usually, labels on a segment define that segment's length. The label is on the horizontal shelf. So Shelf = $6.5$. The height coincidence ($13/2 = 6.5$) explains the vertical dimensions. This feels like the intended "puzzle" solution.

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Problem 6

Shape: L-shape.
* Dimensions: Top width $18.8\text{ ft}$, Right height $25.5\text{ ft}$, Left top height $5.5\text{ ft}$, Inner horizontal $4.2\text{ ft}$.

Area Calculation:
Split into Top Horizontal and Bottom Vertical? Or Left Vertical and Right Vertical?
Let's use the "Cutout" method from a bounding box.
* Bounding Box Width = $18.8\text{ ft}$.
* Bounding Box Height = $25.5\text{ ft}$.
* The shape is an L. The empty space is at the bottom left.
* Empty Space Width: We need the width of the vertical leg.
* Top width is $18.8$.
* The label $4.2$ is the inner horizontal step. This is the width of the empty space? Or the width of the protruding part?
* Looking at diagram 6: The $4.2$ is the horizontal segment connecting the left wall to the right wall's inner face. So, the width of the "hole" is $4.2$? No, that's the thickness of the horizontal bar?
* Let's trace:
* Top edge: $18.8$.
* Right edge: $25.5$.
* Left edge (top part): $5.5$.
* Inner horizontal edge: $4.2$.
* This implies the vertical leg on the right has width = $18.8 - 4.2 = 14.6\text{ ft}$.
* The horizontal leg on top has height = $5.5\text{ ft}$.
* The remaining height of the vertical leg = $25.5 - 5.5 = 20.0\text{ ft}$.
* So we have two rectangles:
1. Top Horizontal Rectangle:
* Width = $18.8\text{ ft}$.
* Height = $5.5\text{ ft}$.
* Area = $18.8 \times 5.5 = 103.4\text{ ft}^2$.
2. Bottom Vertical Rectangle (the part below the top one):
* Width = Total Width ($18.8$) - Inner Step ($4.2$) = $14.6\text{ ft}$.
* Height = Total Height ($25.5$) - Top Height ($5.5$) = $20.0\text{ ft}$.
* Area = $14.6 \times 20.0 = 292.0\text{ ft}^2$.
* Total Area: $103.4 + 292.0 = \mathbf{395.4\text{ ft}^2}$.

Perimeter Calculation:
* Top: $18.8$
* Right: $25.5$
* Bottom: $14.6$ (calculated width of vertical leg)
* Inner Vertical: $20.0$ (calculated height of vertical leg part)
* Inner Horizontal: $4.2$ (given)
* Left Top: $5.5$
* Total Perimeter: $18.8 + 25.5 + 14.6 + 20.0 + 4.2 + 5.5 = \mathbf{88.6\text{ ft}}$.
*(Check: Bounding box perimeter is $2(18.8+25.5) = 88.6$. Since it's a simple L-shape with orthogonal corners, the perimeter equals the bounding box perimeter. Correct.)*

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Problem 7

Shape: C-shape or U-shape on its side.
* Dimensions:
* Total Height (Left): $16\text{ cm}$.
* Top Width: $9\text{ cm}$.
* Right side has two segments of $6.8\text{ cm}$ each? No, top right vertical is $6.8$, bottom right vertical is $6.8$.
* Inner horizontal depth: $5.5\text{ cm}$.

Area Calculation:
Split into three rectangles: Top, Bottom, and Back (Left).
1. Back (Left) Vertical Rectangle:
* Height = $16\text{ cm}$.
* Width = Total Width ($9$) - Inner Depth ($5.5$) = $3.5\text{ cm}$.
* Area = $16 \times 3.5 = 56\text{ cm}^2$.
2. Top Horizontal Rectangle (the arm):
* Width = Inner Depth = $5.5\text{ cm}$.
* Height = Right top vertical = $6.8\text{ cm}$? Wait. The label $6.8$ is on the vertical edge.
* So the top arm has height $6.8$?
* Let's check the bottom arm. Label $6.8$ on bottom right vertical.
* If Top Arm Height = $6.8$ and Bottom Arm Height = $6.8$, then the gap in the middle is $16 - 6.8 - 6.8 = 2.4\text{ cm}$.
* Area of Top Arm = $5.5 \times 6.8 = 37.4\text{ cm}^2$.
* Area of Bottom Arm = $5.5 \times 6.8 = 37.4\text{ cm}^2$.
* Total Area: $56 + 37.4 + 37.4 = \mathbf{130.8\text{ cm}^2}$.

Perimeter Calculation:
Add all outer edges.
* Left: $16$
* Top: $9$
* Right Top Vertical: $6.8$
* Inner Top Horizontal: $5.5$
* Inner Vertical (the back of the mouth): Height = $2.4$ (calculated above).
* Inner Bottom Horizontal: $5.5$
* Right Bottom Vertical: $6.8$
* Bottom: $9$
* Total Perimeter: $16 + 9 + 6.8 + 5.5 + 2.4 + 5.5 + 6.8 + 9 = \mathbf{61.0\text{ cm}}$.

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Problem 8

Shape: H-shape or I-beam shape.
* Dimensions:
* Total Height: $16.8\text{ m}$.
* Top Widths: Left $3.8$, Right $3.8$.
* Middle Gap Width: $8.8\text{ m}$? No, the label $8.8$ is on the horizontal bridge? Or the gap?
* Looking at diagram 8:
* Left vertical bar width: $3.8\text{ m}$.
* Right vertical bar width: $3.8\text{ m}$.
* The horizontal connection in the middle has a label $8.8\text{ m}$? No, $8.8$ is the width