Worksheet for calculating the area of compound shapes with six labeled diagrams and dimensions.
A worksheet titled "Area of Compound Shapes" from Corbettmaths, featuring six compound shape diagrams labeled (a) through (f), each with dimensions in centimeters, and instructions to calculate their areas. Includes a video link, QR code, and buttons for examples and workout.
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Step-by-step solution for: Area of Compound Shapes Textbook Exercise - Corbettmaths
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Show Answer Key & Explanations
Step-by-step solution for: Area of Compound Shapes Textbook Exercise - Corbettmaths
Here are the step-by-step solutions for finding the area of each compound shape. To solve these, we split each complex shape into simpler rectangles, find the area of each rectangle, and then add them together.
Shape (a)
1. Split the L-shape into two vertical rectangles: a tall one on the left and a short one on the right.
2. Left Rectangle: The width is given as $9\text{ cm} - 7\text{ cm} = 2\text{ cm}$. The height is $8\text{ cm}$.
* Area = $2\text{ cm} \times 8\text{ cm} = 16\text{ cm}^2$.
3. Right Rectangle: The width is $7\text{ cm}$. The total height on the left is $8\text{ cm}$ and the bottom section height is $5\text{ cm}$, so the height of this top part is not needed if we split vertically. Let's split horizontally instead to match the labels easier.
* *Alternative Split (Horizontal):* Top rectangle and Bottom rectangle.
* Top Rectangle: Width is $9\text{ cm}$. Height is $8\text{ cm} - 5\text{ cm} = 3\text{ cm}$. Area = $9 \times 3 = 27\text{ cm}^2$.
* Bottom Rectangle: Width is $7\text{ cm}$. Height is $5\text{ cm}$. Area = $7 \times 5 = 35\text{ cm}^2$.
* Total Area = $27 + 35 = 62\text{ cm}^2$.
* *Let's re-verify with vertical split:*
* Left vertical part: Width $9-7=2$, Height $8$. Area $16$.
* Right vertical part: Width $7$, Height $5$. Area $35$.
* Wait, the diagram shows the left side is the long vertical part? No, the label "8cm" is for the top vertical edge. The label "5cm" is for the inner vertical edge. The label "9cm" is the top width. The label "7cm" is the bottom width.
* Let's look at shape (a) again carefully.
* Top width: $9\text{ cm}$.
* Right-side top vertical edge: $8\text{ cm}$.
* Inner vertical edge: $5\text{ cm}$.
* Bottom width: $7\text{ cm}$.
* This implies the shape is an inverted L or similar. Let's split it into a top rectangle and a bottom rectangle.
* Top Rectangle: The full width is $9\text{ cm}$. The height of the top part is the total height minus the bottom part? We don't have total height. We have the right vertical edge ($8\text{ cm}$) and the inner vertical edge ($5\text{ cm}$). This suggests the top block has height $8 - 5 = 3$? No, usually dimensions are for the segments shown.
* Let's assume standard orientation:
* Vertical left edge height = ?
* Horizontal top edge = $9\text{ cm}$.
* Vertical right edge (top part) = $8\text{ cm}$.
* Horizontal bottom edge = $7\text{ cm}$.
* Vertical inner edge = $5\text{ cm}$.
* Let's split into two rectangles:
1. Left/Top Block: Width = $9 - 7 = 2\text{ cm}$. Height = $8\text{ cm}$. Area = $2 \times 8 = 16\text{ cm}^2$.
2. Right/Bottom Block: Width = $7\text{ cm}$. Height = $8 - 5 = 3\text{ cm}$? No, the $5\text{ cm}$ is labeled on the inner vertical drop. If the total height on the right is determined by the left... this is ambiguous.
* Let's try another common interpretation:
* Rectangle 1 (Top): Width $9\text{ cm}$, Height $8\text{ cm}$? No, that would be a huge box.
* Let's look at the labels relative to the lines.
* Top horizontal: $9\text{ cm}$.
* Right vertical (upper segment): $8\text{ cm}$.
* Bottom horizontal: $7\text{ cm}$.
* Inner vertical: $5\text{ cm}$.
* This means the total height on the left is $8 + 5 = 13\text{ cm}$? Or is the $8\text{ cm}$ the total height? The arrow for $8\text{ cm}$ spans the top vertical segment. The arrow for $5\text{ cm}$ spans the inner vertical segment.
* So, we can split it into a top rectangle and a bottom rectangle.
* Top Rectangle: Width = $9\text{ cm}$. Height = $8\text{ cm}$. Area = $72\text{ cm}^2$.
* Bottom Rectangle: Width = $7\text{ cm}$. Height = $5\text{ cm}$. Area = $35\text{ cm}^2$.
* BUT, looking at the shape, the $9\text{ cm}$ is the top width. The $7\text{ cm}$ is the bottom width. The shape looks like an 'L' rotated.
* Actually, usually in these problems, if a dimension is missing, you calculate it.
* Let's split vertically into a left rectangle and a right rectangle.
* Left Rectangle: Width = $9 - 7 = 2\text{ cm}$. Height = $8 + 5 = 13\text{ cm}$? No, the $8$ and $5$ are on the right side.
* Let's assume the $8\text{ cm}$ is the height of the top protrusion and $5\text{ cm}$ is the height of the bottom base?
* Let's look at Shape (c) for comparison. It has clear outer dimensions. Shape (a) has inner dimensions.
* Okay, let's look at the arrows.
* Arrow $9\text{ cm}$: Top width.
* Arrow $8\text{ cm}$: Top-right vertical height.
* Arrow $5\text{ cm}$: Inner vertical height.
* Arrow $7\text{ cm}$: Bottom width.
* This creates a contradiction if we assume simple rectangles unless we deduce the left height.
* Left Height = Top Right Height ($8$) + Inner Height ($5$) = $13\text{ cm}$.
* So, Split into:
1. Left Vertical Rectangle: Width = $9 - 7 = 2\text{ cm}$. Height = $13\text{ cm}$. Area = $26\text{ cm}^2$.
2. Right Bottom Rectangle: Width = $7\text{ cm}$. Height = $5\text{ cm}$. Area = $35\text{ cm}^2$.
* Total = $26 + 35 = 61\text{ cm}^2$.
* *Alternative Split:*
1. Top Rectangle: Width = $9\text{ cm}$. Height = $8\text{ cm}$. Area = $72\text{ cm}^2$.
2. Bottom Rectangle: Width = $7\text{ cm}$. Height = $5\text{ cm}$. Area = $35\text{ cm}^2$.
* This assumes the $8\text{ cm}$ starts from the very top and goes down to the shelf. And the $5\text{ cm}$ is the height of the shelf part.
* Total = $72 + 35 = 107\text{ cm}^2$.
* Let's look closer at the image. The arrow for $8\text{ cm}$ is next to the top vertical edge. The arrow for $5\text{ cm}$ is next to the *inner* vertical edge. The arrow for $9\text{ cm}$ is the top width. The arrow for $7\text{ cm}$ is the bottom width.
* Usually, this layout implies:
* Top Rectangle: Width $9$, Height $8$.
* Bottom Rectangle attached below the right side? No, the bottom width is $7$, which is less than $9$. So the bottom part is narrower? No, the drawing shows the bottom part is the wider base? No, the top is wider ($9$) than the bottom ($7$).
* So it's an inverted L.
* Top part: Width $9$, Height $8$.
* Bottom part: Width $7$, Height $5$.
* Are they overlapping? No, they are stacked.
* Total Area = $(9 \times 8) + (7 \times 5) = 72 + 35 = 107\text{ cm}^2$.
* *Wait*, if the top width is $9$ and bottom is $7$, and it's an L-shape, the vertical line on the left would be straight. The vertical line on the right would step in.
* The diagram shows the left side is a single straight vertical line? No, it looks like the standard "L".
* Let's re-read the diagram (a).
* Top edge: $9$. Right edge (top part): $8$. Inner edge (vertical): $5$. Bottom edge: $7$.
* This implies the shape is composed of a top rectangle ($9 \times 8$) and a bottom rectangle ($7 \times 5$)?
* If so, the total height on the left would be $8+5=13$. The width on the bottom is $7$. The width on top is $9$. This means the left side is NOT straight. The right side is NOT straight.
* Actually, looking at the drawing, the left side is straight. The bottom is straight. The top is straight. The right side steps in.
* If the left side is straight, then the width at the bottom ($7$) must be the same as the width at the top ($9$) MINUS the cutout.
* Let's assume the standard "L" orientation (upright).
* Bottom width $7$. Top width $9$? That doesn't make sense for an upright L where the base is wide.
* Let's assume the shape is rotated.
* Let's try calculating based on the visible rectangular blocks.
* Block 1 (Top): Width $9$, Height $8$. Area $72$.
* Block 2 (Bottom): Width $7$, Height $5$. Area $35$.
* Do they share a boundary? Yes.
* Is there overlap? The drawing shows them connected end-to-end vertically.
* So Area = $72 + 35 = 107\text{ cm}^2$.
* *Self-Correction*: Let's look at shape (b). Top width $25$, Left height $12$, Inner height $10$, Bottom width $6$. This is a T-shape.
* Top Bar: Width $25$, Height $12-10=2$? Or is $12$ the total height? The arrow for $12$ covers the whole left side. The arrow for $10$ covers the stem height. So the top bar thickness is $12-10=2$. Area of top bar = $25 \times 2 = 50$. Stem width is $6$. Stem height is $10$. Area of stem = $6 \times 10 = 60$. Total = $110$.
* Let's apply this logic to (a).
* Arrow $8$ covers the top vertical segment. Arrow $5$ covers the inner vertical segment.
* This implies the total height on the left is $8+5=13$.
* Arrow $9$ is top width. Arrow $7$ is bottom width.
* If the left side is straight, the shape is a large rectangle ($9 \times 13$) with a chunk missing?
* Missing chunk width = $9 - 7 = 2$. Missing chunk height = $8$? No.
* Let's split into two rectangles:
1. Left Vertical Strip: Width = $9 - 7 = 2\text{ cm}$. Height = $8 + 5 = 13\text{ cm}$. Area = $26\text{ cm}^2$.
2. Right Bottom Block: Width = $7\text{ cm}$. Height = $5\text{ cm}$. Area = $35\text{ cm}^2$.
3. Total = $26 + 35 = 61\text{ cm}^2$.
* Let's check the other way:
1. Top Horizontal Block: Width = $9\text{ cm}$. Height = $8\text{ cm}$. Area = $72\text{ cm}^2$.
2. Bottom Right Block: Width = $7\text{ cm}$. Height = $5\text{ cm}$. Area = $35\text{ cm}^2$.
3. This creates a shape where the top is $9$ wide and the bottom is $7$ wide. The left side would be stepped. The diagram shows the LEFT side as the long continuous side? No, in (a), the left side has a corner. The RIGHT side has the long vertical drop?
4. Looking at (a): The left side goes down, then right, then up, then right, then down. It's a U shape? No.
5. It looks like a standard L-shape but mirrored horizontally.
6. Left side: Short vertical? No, the label $5\text{ cm}$ is inside. The label $8\text{ cm}$ is outside right.
7. Okay, let's look at the arrows specifically.
8. Top arrow ($9\text{ cm}$): Spans the entire top width.
9. Right arrow ($8\text{ cm}$): Spans the top vertical edge.
10. Inner arrow ($5\text{ cm}$): Spans the vertical edge of the "step".
11. Bottom arrow ($7\text{ cm}$): Spans the entire bottom width.
12. This geometry is impossible for a simple rectilinear polygon if the left side is a single straight line, because the top width ($9$) and bottom width ($7$) are different, meaning the left or right side must step.
13. The drawing shows the left side stepping out? Or the right side stepping in?
14. The drawing shows a shape like a backwards 'L'.
15. Top width $9$. Bottom width $7$.
16. This means the bottom part is narrower than the top part?
17. If the bottom is narrower, the "overhang" is on the left or right?
18. The vertical line on the right is broken into two segments: top ($8$) and bottom (implied).
19. The vertical line on the left is one segment?
20. If left is one segment, height = $8 + 5 = 13$.
21. Then we have a rectangle $9 \times 13$ with a piece missing at the bottom right?
22. Missing piece width = $9 - 7 = 2$. Height = $5$? No, the inner vertical is $5$.
23. Let's split it into:
* Top Rectangle: Width $9$, Height $8$. Area $72$.
* Bottom Rectangle: Width $7$, Height $5$. Area $35$.
* Total $107$.
* Does this match the visual? A wide top block sitting on a narrower bottom block? The diagram shows the bottom block extending to the left edge? Yes. The top block extends further to the right? Yes.
* So, Area = $107\text{ cm}^2$.
* *Wait, let me look at (c) to confirm my reading of arrows.*
* (c) Top width $4$. Right height $10$. Bottom width $13$. Left height $3$.
* This is an L-shape.
* Split into vertical left and vertical right?
* Left part: Width $13 - 4 = 9$. Height $3$. Area $27$.
* Right part: Width $4$. Height $10$. Area $40$.
* Total $67$.
* Alternatively: Bottom rectangle $13 \times 3 = 39$. Top rectangle $4 \times (10-3) = 4 \times 7 = 28$. Total $39+28=67$.
* This confirms the method: Split into non-overlapping rectangles.
* Back to (a):
* Top width $9$. Bottom width $7$.
* Right top height $8$. Inner height $5$.
* This implies the "Inner height" is the height of the bottom section?
* If the shape is a top block ($9 \times 8$) and a bottom block ($7 \times 5$), do they align to the left or right?
* The diagram shows the left edges aligned.
* So, Area = $(9 \times 8) + (7 \times 5) = 72 + 35 = 107\text{ cm}^2$.
Shape (b)
1. This is a T-shape.
2. Split into a top horizontal rectangle and a bottom vertical stem.
3. Top Rectangle:
* Width = $25\text{ cm}$.
* Height = Total height ($12\text{ cm}$) - Stem height ($10\text{ cm}$) = $2\text{ cm}$.
* Area = $25 \times 2 = 50\text{ cm}^2$.
4. Bottom Stem:
* Width = $6\text{ cm}$.
* Height = $10\text{ cm}$.
* Area = $6 \times 10 = 60\text{ cm}^2$.
5. Total Area = $50 + 60 = 110\text{ cm}^2$.
Shape (c)
1. This is an L-shape.
2. Split into a bottom horizontal rectangle and a top vertical rectangle.
3. Bottom Rectangle:
* Width = $13\text{ cm}$.
* Height = $3\text{ cm}$.
* Area = $13 \times 3 = 39\text{ cm}^2$.
4. Top Rectangle:
* Width = $4\text{ cm}$.
* Height = Total height ($10\text{ cm}$) - Bottom height ($3\text{ cm}$) = $7\text{ cm}$.
* Area = $4 \times 7 = 28\text{ cm}^2$.
5. Total Area = $39 + 28 = 67\text{ cm}^2$.
Shape (d)
1. This is an inverted T-shape (or podium shape).
2. Split into a bottom horizontal rectangle and a top vertical rectangle.
3. Bottom Rectangle:
* Width = $9\text{ cm}$.
* Height = $3\text{ cm}$.
* Area = $9 \times 3 = 27\text{ cm}^2$.
4. Top Rectangle:
* Width = $2\text{ cm}$.
* Height = $5\text{ cm}$.
* Area = $2 \times 5 = 10\text{ cm}^2$.
5. Total Area = $27 + 10 = 37\text{ cm}^2$.
* *Check:* The label $4\text{ cm}$ is the left shoulder width. The label $2\text{ cm}$ is the top width. The label $9\text{ cm}$ is total width. $4 + 2 + \text{right shoulder} = 9$. Right shoulder = $3$. This doesn't affect area calculation if we just take the two main blocks. The blocks are distinct.
Shape (e)
1. This is a wide rectangle with two rectangular notches cut out from the bottom.
2. Method: Calculate the area of the large outer rectangle and subtract the empty spaces, OR split into vertical strips. Subtraction is often easier here.
3. Large Outer Rectangle:
* Width = $24\text{ cm}$.
* Height = $20\text{ cm}$.
* Area = $24 \times 20 = 480\text{ cm}^2$.
4. Notches (Empty Spaces):
* There are two identical-looking notches, but let's check dimensions.
* The solid parts at the bottom are labeled $6\text{ cm}$, $6\text{ cm}$, and $2\text{ cm}$.
* Total width = $24\text{ cm}$.
* Sum of solid bottoms = $6 + 6 + 2 = 14\text{ cm}$.
* Total width of gaps = $24 - 14 = 10\text{ cm}$.
* There are two gaps. Are they equal? The drawing looks symmetric, but the bottom labels are $6$, $6$, $2$. The legs are $6$, $6$, $2$.
* Gap 1 is between the first $6$ and second $6$.
* Gap 2 is between the second $6$ and the $2$.
* Height of gaps = $8\text{ cm}$ (labeled).
* We need the widths of the gaps.
* Width of Gap 1 + Width of Gap 2 = $10\text{ cm}$.
* The diagram doesn't explicitly give individual gap widths. However, usually in these problems, if not specified, we might assume symmetry or look for clues.
* Let's look at the top. The top is a solid $24\text{ cm}$.
* Let's try adding the solid parts instead (Vertical Strips).
* Leg 1 (Left): Width $6\text{ cm}$. Height $20\text{ cm}$. Area = $120\text{ cm}^2$.
* Leg 2 (Middle): Width $6\text{ cm}$. Height $20\text{ cm}$. Area = $120\text{ cm}^2$.
* Leg 3 (Right): Width $2\text{ cm}$. Height $20\text{ cm}$. Area = $40\text{ cm}^2$.
* Top Connecting Bar? No, the height $20$ is the total height. The "legs" go all the way up?
* Look at the label $8\text{ cm}$. It indicates the height of the *cutout*.
* So the legs are not full height $20$. The top part is a solid bar connecting them.
* Let's split horizontally:
* Top Bar: Height = Total Height ($20$) - Cutout Height ($8$) = $12\text{ cm}$. Width = $24\text{ cm}$. Area = $24 \times 12 = 288\text{ cm}^2$.
* Bottom Legs:
* Leg 1: Width $6$, Height $8$. Area = $48$.
* Leg 2: Width $6$, Height $8$. Area = $48$.
* Leg 3: Width $2$, Height $8$. Area = $16$.
* Total Legs Area = $48 + 48 + 16 = 112\text{ cm}^2$.
* Total Area = $288 + 112 = 400\text{ cm}^2$.
Shape (f)
1. This is a square/rectangle with a rectangular notch cut out from the bottom.
2. Outer Rectangle:
* Width = $8\text{ cm}$.
* Height = $7\text{ cm}$.
* Area = $8 \times 7 = 56\text{ cm}^2$.
3. Notch (Empty Space):
* We need the width and height of the notch.
* Height of notch = $3\text{ cm}$ (labeled).
* Width of notch:
* Total width = $8\text{ cm}$.
* Solid bottom left = $2\text{ cm}$.
* Solid bottom right = $3\text{ cm}$.
* Notch Width = $8 - 2 - 3 = 3\text{ cm}$.
* Area of Notch = $3 \times 3 = 9\text{ cm}^2$.
4. Total Area = Outer Area - Notch Area = $56 - 9 = 47\text{ cm}^2$.
Final Answer:
(a) 107 cm²
(b) 110 cm²
(c) 67 cm²
(d) 37 cm²
(e) 400 cm²
(f) 47 cm²
Shape (a)
1. Split the L-shape into two vertical rectangles: a tall one on the left and a short one on the right.
2. Left Rectangle: The width is given as $9\text{ cm} - 7\text{ cm} = 2\text{ cm}$. The height is $8\text{ cm}$.
* Area = $2\text{ cm} \times 8\text{ cm} = 16\text{ cm}^2$.
3. Right Rectangle: The width is $7\text{ cm}$. The total height on the left is $8\text{ cm}$ and the bottom section height is $5\text{ cm}$, so the height of this top part is not needed if we split vertically. Let's split horizontally instead to match the labels easier.
* *Alternative Split (Horizontal):* Top rectangle and Bottom rectangle.
* Top Rectangle: Width is $9\text{ cm}$. Height is $8\text{ cm} - 5\text{ cm} = 3\text{ cm}$. Area = $9 \times 3 = 27\text{ cm}^2$.
* Bottom Rectangle: Width is $7\text{ cm}$. Height is $5\text{ cm}$. Area = $7 \times 5 = 35\text{ cm}^2$.
* Total Area = $27 + 35 = 62\text{ cm}^2$.
* *Let's re-verify with vertical split:*
* Left vertical part: Width $9-7=2$, Height $8$. Area $16$.
* Right vertical part: Width $7$, Height $5$. Area $35$.
* Wait, the diagram shows the left side is the long vertical part? No, the label "8cm" is for the top vertical edge. The label "5cm" is for the inner vertical edge. The label "9cm" is the top width. The label "7cm" is the bottom width.
* Let's look at shape (a) again carefully.
* Top width: $9\text{ cm}$.
* Right-side top vertical edge: $8\text{ cm}$.
* Inner vertical edge: $5\text{ cm}$.
* Bottom width: $7\text{ cm}$.
* This implies the shape is an inverted L or similar. Let's split it into a top rectangle and a bottom rectangle.
* Top Rectangle: The full width is $9\text{ cm}$. The height of the top part is the total height minus the bottom part? We don't have total height. We have the right vertical edge ($8\text{ cm}$) and the inner vertical edge ($5\text{ cm}$). This suggests the top block has height $8 - 5 = 3$? No, usually dimensions are for the segments shown.
* Let's assume standard orientation:
* Vertical left edge height = ?
* Horizontal top edge = $9\text{ cm}$.
* Vertical right edge (top part) = $8\text{ cm}$.
* Horizontal bottom edge = $7\text{ cm}$.
* Vertical inner edge = $5\text{ cm}$.
* Let's split into two rectangles:
1. Left/Top Block: Width = $9 - 7 = 2\text{ cm}$. Height = $8\text{ cm}$. Area = $2 \times 8 = 16\text{ cm}^2$.
2. Right/Bottom Block: Width = $7\text{ cm}$. Height = $8 - 5 = 3\text{ cm}$? No, the $5\text{ cm}$ is labeled on the inner vertical drop. If the total height on the right is determined by the left... this is ambiguous.
* Let's try another common interpretation:
* Rectangle 1 (Top): Width $9\text{ cm}$, Height $8\text{ cm}$? No, that would be a huge box.
* Let's look at the labels relative to the lines.
* Top horizontal: $9\text{ cm}$.
* Right vertical (upper segment): $8\text{ cm}$.
* Bottom horizontal: $7\text{ cm}$.
* Inner vertical: $5\text{ cm}$.
* This means the total height on the left is $8 + 5 = 13\text{ cm}$? Or is the $8\text{ cm}$ the total height? The arrow for $8\text{ cm}$ spans the top vertical segment. The arrow for $5\text{ cm}$ spans the inner vertical segment.
* So, we can split it into a top rectangle and a bottom rectangle.
* Top Rectangle: Width = $9\text{ cm}$. Height = $8\text{ cm}$. Area = $72\text{ cm}^2$.
* Bottom Rectangle: Width = $7\text{ cm}$. Height = $5\text{ cm}$. Area = $35\text{ cm}^2$.
* BUT, looking at the shape, the $9\text{ cm}$ is the top width. The $7\text{ cm}$ is the bottom width. The shape looks like an 'L' rotated.
* Actually, usually in these problems, if a dimension is missing, you calculate it.
* Let's split vertically into a left rectangle and a right rectangle.
* Left Rectangle: Width = $9 - 7 = 2\text{ cm}$. Height = $8 + 5 = 13\text{ cm}$? No, the $8$ and $5$ are on the right side.
* Let's assume the $8\text{ cm}$ is the height of the top protrusion and $5\text{ cm}$ is the height of the bottom base?
* Let's look at Shape (c) for comparison. It has clear outer dimensions. Shape (a) has inner dimensions.
* Okay, let's look at the arrows.
* Arrow $9\text{ cm}$: Top width.
* Arrow $8\text{ cm}$: Top-right vertical height.
* Arrow $5\text{ cm}$: Inner vertical height.
* Arrow $7\text{ cm}$: Bottom width.
* This creates a contradiction if we assume simple rectangles unless we deduce the left height.
* Left Height = Top Right Height ($8$) + Inner Height ($5$) = $13\text{ cm}$.
* So, Split into:
1. Left Vertical Rectangle: Width = $9 - 7 = 2\text{ cm}$. Height = $13\text{ cm}$. Area = $26\text{ cm}^2$.
2. Right Bottom Rectangle: Width = $7\text{ cm}$. Height = $5\text{ cm}$. Area = $35\text{ cm}^2$.
* Total = $26 + 35 = 61\text{ cm}^2$.
* *Alternative Split:*
1. Top Rectangle: Width = $9\text{ cm}$. Height = $8\text{ cm}$. Area = $72\text{ cm}^2$.
2. Bottom Rectangle: Width = $7\text{ cm}$. Height = $5\text{ cm}$. Area = $35\text{ cm}^2$.
* This assumes the $8\text{ cm}$ starts from the very top and goes down to the shelf. And the $5\text{ cm}$ is the height of the shelf part.
* Total = $72 + 35 = 107\text{ cm}^2$.
* Let's look closer at the image. The arrow for $8\text{ cm}$ is next to the top vertical edge. The arrow for $5\text{ cm}$ is next to the *inner* vertical edge. The arrow for $9\text{ cm}$ is the top width. The arrow for $7\text{ cm}$ is the bottom width.
* Usually, this layout implies:
* Top Rectangle: Width $9$, Height $8$.
* Bottom Rectangle attached below the right side? No, the bottom width is $7$, which is less than $9$. So the bottom part is narrower? No, the drawing shows the bottom part is the wider base? No, the top is wider ($9$) than the bottom ($7$).
* So it's an inverted L.
* Top part: Width $9$, Height $8$.
* Bottom part: Width $7$, Height $5$.
* Are they overlapping? No, they are stacked.
* Total Area = $(9 \times 8) + (7 \times 5) = 72 + 35 = 107\text{ cm}^2$.
* *Wait*, if the top width is $9$ and bottom is $7$, and it's an L-shape, the vertical line on the left would be straight. The vertical line on the right would step in.
* The diagram shows the left side is a single straight vertical line? No, it looks like the standard "L".
* Let's re-read the diagram (a).
* Top edge: $9$. Right edge (top part): $8$. Inner edge (vertical): $5$. Bottom edge: $7$.
* This implies the shape is composed of a top rectangle ($9 \times 8$) and a bottom rectangle ($7 \times 5$)?
* If so, the total height on the left would be $8+5=13$. The width on the bottom is $7$. The width on top is $9$. This means the left side is NOT straight. The right side is NOT straight.
* Actually, looking at the drawing, the left side is straight. The bottom is straight. The top is straight. The right side steps in.
* If the left side is straight, then the width at the bottom ($7$) must be the same as the width at the top ($9$) MINUS the cutout.
* Let's assume the standard "L" orientation (upright).
* Bottom width $7$. Top width $9$? That doesn't make sense for an upright L where the base is wide.
* Let's assume the shape is rotated.
* Let's try calculating based on the visible rectangular blocks.
* Block 1 (Top): Width $9$, Height $8$. Area $72$.
* Block 2 (Bottom): Width $7$, Height $5$. Area $35$.
* Do they share a boundary? Yes.
* Is there overlap? The drawing shows them connected end-to-end vertically.
* So Area = $72 + 35 = 107\text{ cm}^2$.
* *Self-Correction*: Let's look at shape (b). Top width $25$, Left height $12$, Inner height $10$, Bottom width $6$. This is a T-shape.
* Top Bar: Width $25$, Height $12-10=2$? Or is $12$ the total height? The arrow for $12$ covers the whole left side. The arrow for $10$ covers the stem height. So the top bar thickness is $12-10=2$. Area of top bar = $25 \times 2 = 50$. Stem width is $6$. Stem height is $10$. Area of stem = $6 \times 10 = 60$. Total = $110$.
* Let's apply this logic to (a).
* Arrow $8$ covers the top vertical segment. Arrow $5$ covers the inner vertical segment.
* This implies the total height on the left is $8+5=13$.
* Arrow $9$ is top width. Arrow $7$ is bottom width.
* If the left side is straight, the shape is a large rectangle ($9 \times 13$) with a chunk missing?
* Missing chunk width = $9 - 7 = 2$. Missing chunk height = $8$? No.
* Let's split into two rectangles:
1. Left Vertical Strip: Width = $9 - 7 = 2\text{ cm}$. Height = $8 + 5 = 13\text{ cm}$. Area = $26\text{ cm}^2$.
2. Right Bottom Block: Width = $7\text{ cm}$. Height = $5\text{ cm}$. Area = $35\text{ cm}^2$.
3. Total = $26 + 35 = 61\text{ cm}^2$.
* Let's check the other way:
1. Top Horizontal Block: Width = $9\text{ cm}$. Height = $8\text{ cm}$. Area = $72\text{ cm}^2$.
2. Bottom Right Block: Width = $7\text{ cm}$. Height = $5\text{ cm}$. Area = $35\text{ cm}^2$.
3. This creates a shape where the top is $9$ wide and the bottom is $7$ wide. The left side would be stepped. The diagram shows the LEFT side as the long continuous side? No, in (a), the left side has a corner. The RIGHT side has the long vertical drop?
4. Looking at (a): The left side goes down, then right, then up, then right, then down. It's a U shape? No.
5. It looks like a standard L-shape but mirrored horizontally.
6. Left side: Short vertical? No, the label $5\text{ cm}$ is inside. The label $8\text{ cm}$ is outside right.
7. Okay, let's look at the arrows specifically.
8. Top arrow ($9\text{ cm}$): Spans the entire top width.
9. Right arrow ($8\text{ cm}$): Spans the top vertical edge.
10. Inner arrow ($5\text{ cm}$): Spans the vertical edge of the "step".
11. Bottom arrow ($7\text{ cm}$): Spans the entire bottom width.
12. This geometry is impossible for a simple rectilinear polygon if the left side is a single straight line, because the top width ($9$) and bottom width ($7$) are different, meaning the left or right side must step.
13. The drawing shows the left side stepping out? Or the right side stepping in?
14. The drawing shows a shape like a backwards 'L'.
15. Top width $9$. Bottom width $7$.
16. This means the bottom part is narrower than the top part?
17. If the bottom is narrower, the "overhang" is on the left or right?
18. The vertical line on the right is broken into two segments: top ($8$) and bottom (implied).
19. The vertical line on the left is one segment?
20. If left is one segment, height = $8 + 5 = 13$.
21. Then we have a rectangle $9 \times 13$ with a piece missing at the bottom right?
22. Missing piece width = $9 - 7 = 2$. Height = $5$? No, the inner vertical is $5$.
23. Let's split it into:
* Top Rectangle: Width $9$, Height $8$. Area $72$.
* Bottom Rectangle: Width $7$, Height $5$. Area $35$.
* Total $107$.
* Does this match the visual? A wide top block sitting on a narrower bottom block? The diagram shows the bottom block extending to the left edge? Yes. The top block extends further to the right? Yes.
* So, Area = $107\text{ cm}^2$.
* *Wait, let me look at (c) to confirm my reading of arrows.*
* (c) Top width $4$. Right height $10$. Bottom width $13$. Left height $3$.
* This is an L-shape.
* Split into vertical left and vertical right?
* Left part: Width $13 - 4 = 9$. Height $3$. Area $27$.
* Right part: Width $4$. Height $10$. Area $40$.
* Total $67$.
* Alternatively: Bottom rectangle $13 \times 3 = 39$. Top rectangle $4 \times (10-3) = 4 \times 7 = 28$. Total $39+28=67$.
* This confirms the method: Split into non-overlapping rectangles.
* Back to (a):
* Top width $9$. Bottom width $7$.
* Right top height $8$. Inner height $5$.
* This implies the "Inner height" is the height of the bottom section?
* If the shape is a top block ($9 \times 8$) and a bottom block ($7 \times 5$), do they align to the left or right?
* The diagram shows the left edges aligned.
* So, Area = $(9 \times 8) + (7 \times 5) = 72 + 35 = 107\text{ cm}^2$.
Shape (b)
1. This is a T-shape.
2. Split into a top horizontal rectangle and a bottom vertical stem.
3. Top Rectangle:
* Width = $25\text{ cm}$.
* Height = Total height ($12\text{ cm}$) - Stem height ($10\text{ cm}$) = $2\text{ cm}$.
* Area = $25 \times 2 = 50\text{ cm}^2$.
4. Bottom Stem:
* Width = $6\text{ cm}$.
* Height = $10\text{ cm}$.
* Area = $6 \times 10 = 60\text{ cm}^2$.
5. Total Area = $50 + 60 = 110\text{ cm}^2$.
Shape (c)
1. This is an L-shape.
2. Split into a bottom horizontal rectangle and a top vertical rectangle.
3. Bottom Rectangle:
* Width = $13\text{ cm}$.
* Height = $3\text{ cm}$.
* Area = $13 \times 3 = 39\text{ cm}^2$.
4. Top Rectangle:
* Width = $4\text{ cm}$.
* Height = Total height ($10\text{ cm}$) - Bottom height ($3\text{ cm}$) = $7\text{ cm}$.
* Area = $4 \times 7 = 28\text{ cm}^2$.
5. Total Area = $39 + 28 = 67\text{ cm}^2$.
Shape (d)
1. This is an inverted T-shape (or podium shape).
2. Split into a bottom horizontal rectangle and a top vertical rectangle.
3. Bottom Rectangle:
* Width = $9\text{ cm}$.
* Height = $3\text{ cm}$.
* Area = $9 \times 3 = 27\text{ cm}^2$.
4. Top Rectangle:
* Width = $2\text{ cm}$.
* Height = $5\text{ cm}$.
* Area = $2 \times 5 = 10\text{ cm}^2$.
5. Total Area = $27 + 10 = 37\text{ cm}^2$.
* *Check:* The label $4\text{ cm}$ is the left shoulder width. The label $2\text{ cm}$ is the top width. The label $9\text{ cm}$ is total width. $4 + 2 + \text{right shoulder} = 9$. Right shoulder = $3$. This doesn't affect area calculation if we just take the two main blocks. The blocks are distinct.
Shape (e)
1. This is a wide rectangle with two rectangular notches cut out from the bottom.
2. Method: Calculate the area of the large outer rectangle and subtract the empty spaces, OR split into vertical strips. Subtraction is often easier here.
3. Large Outer Rectangle:
* Width = $24\text{ cm}$.
* Height = $20\text{ cm}$.
* Area = $24 \times 20 = 480\text{ cm}^2$.
4. Notches (Empty Spaces):
* There are two identical-looking notches, but let's check dimensions.
* The solid parts at the bottom are labeled $6\text{ cm}$, $6\text{ cm}$, and $2\text{ cm}$.
* Total width = $24\text{ cm}$.
* Sum of solid bottoms = $6 + 6 + 2 = 14\text{ cm}$.
* Total width of gaps = $24 - 14 = 10\text{ cm}$.
* There are two gaps. Are they equal? The drawing looks symmetric, but the bottom labels are $6$, $6$, $2$. The legs are $6$, $6$, $2$.
* Gap 1 is between the first $6$ and second $6$.
* Gap 2 is between the second $6$ and the $2$.
* Height of gaps = $8\text{ cm}$ (labeled).
* We need the widths of the gaps.
* Width of Gap 1 + Width of Gap 2 = $10\text{ cm}$.
* The diagram doesn't explicitly give individual gap widths. However, usually in these problems, if not specified, we might assume symmetry or look for clues.
* Let's look at the top. The top is a solid $24\text{ cm}$.
* Let's try adding the solid parts instead (Vertical Strips).
* Leg 1 (Left): Width $6\text{ cm}$. Height $20\text{ cm}$. Area = $120\text{ cm}^2$.
* Leg 2 (Middle): Width $6\text{ cm}$. Height $20\text{ cm}$. Area = $120\text{ cm}^2$.
* Leg 3 (Right): Width $2\text{ cm}$. Height $20\text{ cm}$. Area = $40\text{ cm}^2$.
* Top Connecting Bar? No, the height $20$ is the total height. The "legs" go all the way up?
* Look at the label $8\text{ cm}$. It indicates the height of the *cutout*.
* So the legs are not full height $20$. The top part is a solid bar connecting them.
* Let's split horizontally:
* Top Bar: Height = Total Height ($20$) - Cutout Height ($8$) = $12\text{ cm}$. Width = $24\text{ cm}$. Area = $24 \times 12 = 288\text{ cm}^2$.
* Bottom Legs:
* Leg 1: Width $6$, Height $8$. Area = $48$.
* Leg 2: Width $6$, Height $8$. Area = $48$.
* Leg 3: Width $2$, Height $8$. Area = $16$.
* Total Legs Area = $48 + 48 + 16 = 112\text{ cm}^2$.
* Total Area = $288 + 112 = 400\text{ cm}^2$.
Shape (f)
1. This is a square/rectangle with a rectangular notch cut out from the bottom.
2. Outer Rectangle:
* Width = $8\text{ cm}$.
* Height = $7\text{ cm}$.
* Area = $8 \times 7 = 56\text{ cm}^2$.
3. Notch (Empty Space):
* We need the width and height of the notch.
* Height of notch = $3\text{ cm}$ (labeled).
* Width of notch:
* Total width = $8\text{ cm}$.
* Solid bottom left = $2\text{ cm}$.
* Solid bottom right = $3\text{ cm}$.
* Notch Width = $8 - 2 - 3 = 3\text{ cm}$.
* Area of Notch = $3 \times 3 = 9\text{ cm}^2$.
4. Total Area = Outer Area - Notch Area = $56 - 9 = 47\text{ cm}^2$.
Final Answer:
(a) 107 cm²
(b) 110 cm²
(c) 67 cm²
(d) 37 cm²
(e) 400 cm²
(f) 47 cm²
Parent Tip: Review the logic above to help your child master the concept of area compound shapes worksheet.
