Area of Compound Shapes Textbook Exercise - Corbettmaths - Free Printable
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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
Let’s solve each shape one by one. We’ll break each compound shape into simpler rectangles, find the area of each rectangle, and then add them up.
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Shape (a)
This is an L-shape. Let’s split it vertically or horizontally.
Option: Split into two rectangles:
- Left vertical part: width = 5cm, height = 8cm → Area = 5 × 8 = 40 cm²
- Bottom horizontal part: but wait — if we do that, we might double-count or miss something.
Better way: Think of it as a big rectangle minus a missing piece? Or split cleanly.
Actually, look at dimensions:
Total width at bottom = 7cm
Top part width = 9cm → so the “step” on the right must be 9 - 7 = 2cm wide? Wait, no — let's draw mentally.
Actually, better to split into two rectangles:
Rectangle 1 (left): 5cm wide × 8cm tall → 5×8 = 40 cm²
But wait — the total height on left is 5cm + ? Actually, looking again:
The left side has a vertical segment labeled 5cm, and above it is another part going up to 8cm total height? Wait — the 8cm is the height of the top-right part.
Let me reorient:
From the diagram:
- The full height on the right is 8cm.
- The bottom part extends 7cm to the right.
- The top part extends 9cm to the right.
- The left side has a drop of 5cm from top to bottom of the lower step.
So, split into:
Rectangle A (top): 9cm wide × (8cm - 5cm) = 9 × 3 = 27 cm²
Rectangle B (bottom): 7cm wide × 5cm = 35 cm²
Total = 27 + 35 = 62 cm²
Wait — is that correct? Let’s check another way.
Alternative: Imagine full rectangle 9cm × 8cm = 72 cm²
Then subtract the missing rectangle on bottom right: width = 9 - 7 = 2cm, height = 5cm → area = 10 cm²
So 72 - 10 = 62 cm² → same answer. Good.
✔ Shape (a) = 62 cm²
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Shape (b)
T-shaped figure.
Split into top rectangle and bottom stem.
Top rectangle: width = 25cm, height = 12cm → 25 × 12 = 300 cm²
Bottom stem: width = 6cm, height = 10cm → 6 × 10 = 60 cm²
Total = 300 + 60 = 360 cm²
Check: Is there overlap? No — they are stacked. Correct.
✔ Shape (b) = 360 cm²
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Shape (c)
Another L-shape, mirrored.
Dimensions: total width = 13cm, total height = 10cm
Left part height = 3cm, top part width = 4cm
Split into:
Rectangle A (bottom): 13cm wide × 3cm high = 39 cm²
Rectangle B (right vertical): width = 4cm, height = 10 - 3 = 7cm → 4 × 7 = 28 cm²
Total = 39 + 28 = 67 cm²
Alternative: Full rectangle 13×10 = 130
Minus missing top-left rectangle: width = 13 - 4 = 9cm, height = 10 - 3 = 7cm → 9×7=63
130 - 63 = 67 → same.
✔ Shape (c) = 67 cm²
---
Shape (d)
Inverted T or plus-like shape.
Base: 9cm wide × 3cm high → 9×3 = 27 cm²
Middle column: 2cm wide × 5cm high → 2×5 = 10 cm²
But wait — does the middle column sit on top of the base? Yes.
Are they overlapping? The base is 3cm high, and the column starts from the top of the base? Actually, looking at labels:
The 5cm is the height of the vertical part above the base. So total height of column is 5cm, sitting on the 3cm base.
So areas don’t overlap — just add.
Total = 27 + 10 = 37 cm²
Wait — but the base is 9cm wide, and the column is centered? Doesn’t matter for area.
Yes, correct.
✔ Shape (d) = 37 cm²
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Shape (e)
Looks like a rectangle with two notches cut out from the bottom.
Full outer rectangle: width = 24cm, height = 20cm → 24×20 = 480 cm²
Now subtract the two notches.
Each notch: width = 6cm, height = 8cm → area per notch = 6×8 = 48 cm²
Two notches: 48 × 2 = 96 cm²
Also, there’s a small gap between notches? Look at bottom: segments are 6cm, 6cm, 2cm → total 6+6+2=14cm? But total width is 24cm — that doesn’t add up.
Wait — let’s read carefully.
At the bottom, it shows three segments: 6cm, 6cm, 2cm — but these are the widths of the *remaining* parts? Or the cuts?
Actually, looking at the diagram: the shape has two rectangular indentations (notches) at the bottom.
The distance between the notches is 2cm? And each notch is 6cm wide? Then total width covered by notches and gap: 6 + 2 + 6 = 14cm — but the whole width is 24cm, so the sides must have extra.
Actually, perhaps the 6cm, 6cm, 2cm are the widths of the *legs* at the bottom? That makes more sense.
Reinterpret:
The shape is like a table with two legs removed? No.
Actually, standard interpretation: it’s a large rectangle with two rectangular holes cut out from the bottom edge.
The height of each hole is 8cm (as labeled).
What is the width of each hole? From the bottom labels: the solid parts are 6cm, 6cm, and 2cm — meaning the gaps (holes) are between them.
Total width = 24cm
Solid parts at bottom: left leg 6cm, middle support 2cm, right leg 6cm → total solid = 6+2+6=14cm
Therefore, the two holes together take up 24 - 14 = 10cm → so each hole is 5cm wide? But that contradicts nothing labeled.
Wait — actually, in such diagrams, often the numbers under the shape indicate the widths of the protruding parts, and the cuts are implied.
But here, it says: below the shape, from left to right: 6cm, then a gap, then 6cm, then a gap, then 2cm? That doesn't make sense.
Looking back: the label "6cm" is under the first vertical leg, "6cm" under the second, "2cm" under the third — but there are only two notches, so probably three legs.
Actually, the shape has three downward projections? No — it looks like a rectangle with two upward notches? I think I misread.
Let me describe: the overall shape is 24cm wide, 20cm tall. At the bottom, there are two rectangular cut-outs (indentations) going upward 8cm. Between them is a 2cm wide strip, and on the sides are 6cm strips.
So, the material at the bottom consists of:
- Left rectangle: 6cm wide × 20cm high? No — the cut is only 8cm deep.
Actually, the entire shape is 20cm high. The notches go up 8cm from the bottom, so the remaining height above the notches is 20 - 8 = 12cm, which is solid across the full 24cm.
Then, below that, we have three separate rectangles at the bottom:
- Left: 6cm wide × 8cm high
- Middle: 2cm wide × 8cm high
- Right: 6cm wide × 8cm high
Is that right? But then the total width would be 6+2+6=14cm, but the top is 24cm — inconsistency.
I think I got it backwards.
Standard way: the shape is mostly solid, with two rectangular regions removed from the bottom.
The full rectangle is 24cm × 20cm = 480 cm²
Now, the two notches: each is a rectangle of width W and height 8cm.
From the bottom labels: the distances given are likely the widths of the *solid* parts between the notches.
Specifically: from left edge to first notch: 6cm? Or the notch starts after 6cm?
Actually, common convention: the numbers at the bottom indicate the lengths of the segments along the bottom edge.
So, starting from left:
- First segment: 6cm (solid)
- Then a notch (cut-out) — width unknown
- Then next solid segment: 6cm
- Then another notch
- Then last solid segment: 2cm
But that would mean two notches, and three solid parts totaling 6+6+2=14cm, so the two notches together are 24 - 14 = 10cm, so each notch is 5cm wide.
And each notch is 8cm high.
So area of each notch = 5 × 8 = 40 cm²
Two notches = 80 cm²
Total area = full rectangle - notches = 480 - 80 = 400 cm²
But let's verify with addition.
We can think of the shape as:
- Top part: 24cm wide × (20 - 8) = 24 × 12 = 288 cm²
- Bottom part: three rectangles:
- Left: 6cm × 8cm = 48
- Middle: 2cm × 8cm = 16
- Right: 6cm × 8cm = 48
Total bottom = 48+16+48 = 112 cm²
- Total = 288 + 112 = 400 cm² → matches.
Perfect.
✔ Shape (e) = 400 cm²
---
Shape (f)
Rectangle with a rectangular notch at the bottom center.
Overall: width = 8cm, height = 7cm → full area = 8×7 = 56 cm²
Notch: width = ? Height = 3cm
From bottom labels: left solid part 2cm, right solid part 3cm, so the notch width = 8 - 2 - 3 = 3cm
Height of notch = 3cm (given)
So area of notch = 3 × 3 = 9 cm²
Area of shape = 56 - 9 = 47 cm²
Addition method:
- Top part: 8cm wide × (7 - 3) = 8 × 4 = 32 cm²
- Bottom left: 2cm × 3cm = 6 cm²
- Bottom right: 3cm × 3cm = 9 cm²
- Total = 32 + 6 + 9 = 47 cm² → good.
✔ Shape (f) = 47 cm²
---
Final Answers:
(a) 62 cm²
(b) 360 cm²
(c) 67 cm²
(d) 37 cm²
(e) 400 cm²
(f) 47 cm²
──────────────────────────────────────
Final Answer:
(a) 62 cm²
(b) 360 cm²
(c) 67 cm²
(d) 37 cm²
(e) 400 cm²
(f) 47 cm²
---
Shape (a)
This is an L-shape. Let’s split it vertically or horizontally.
Option: Split into two rectangles:
- Left vertical part: width = 5cm, height = 8cm → Area = 5 × 8 = 40 cm²
- Bottom horizontal part: but wait — if we do that, we might double-count or miss something.
Better way: Think of it as a big rectangle minus a missing piece? Or split cleanly.
Actually, look at dimensions:
Total width at bottom = 7cm
Top part width = 9cm → so the “step” on the right must be 9 - 7 = 2cm wide? Wait, no — let's draw mentally.
Actually, better to split into two rectangles:
Rectangle 1 (left): 5cm wide × 8cm tall → 5×8 = 40 cm²
But wait — the total height on left is 5cm + ? Actually, looking again:
The left side has a vertical segment labeled 5cm, and above it is another part going up to 8cm total height? Wait — the 8cm is the height of the top-right part.
Let me reorient:
From the diagram:
- The full height on the right is 8cm.
- The bottom part extends 7cm to the right.
- The top part extends 9cm to the right.
- The left side has a drop of 5cm from top to bottom of the lower step.
So, split into:
Rectangle A (top): 9cm wide × (8cm - 5cm) = 9 × 3 = 27 cm²
Rectangle B (bottom): 7cm wide × 5cm = 35 cm²
Total = 27 + 35 = 62 cm²
Wait — is that correct? Let’s check another way.
Alternative: Imagine full rectangle 9cm × 8cm = 72 cm²
Then subtract the missing rectangle on bottom right: width = 9 - 7 = 2cm, height = 5cm → area = 10 cm²
So 72 - 10 = 62 cm² → same answer. Good.
✔ Shape (a) = 62 cm²
---
Shape (b)
T-shaped figure.
Split into top rectangle and bottom stem.
Top rectangle: width = 25cm, height = 12cm → 25 × 12 = 300 cm²
Bottom stem: width = 6cm, height = 10cm → 6 × 10 = 60 cm²
Total = 300 + 60 = 360 cm²
Check: Is there overlap? No — they are stacked. Correct.
✔ Shape (b) = 360 cm²
---
Shape (c)
Another L-shape, mirrored.
Dimensions: total width = 13cm, total height = 10cm
Left part height = 3cm, top part width = 4cm
Split into:
Rectangle A (bottom): 13cm wide × 3cm high = 39 cm²
Rectangle B (right vertical): width = 4cm, height = 10 - 3 = 7cm → 4 × 7 = 28 cm²
Total = 39 + 28 = 67 cm²
Alternative: Full rectangle 13×10 = 130
Minus missing top-left rectangle: width = 13 - 4 = 9cm, height = 10 - 3 = 7cm → 9×7=63
130 - 63 = 67 → same.
✔ Shape (c) = 67 cm²
---
Shape (d)
Inverted T or plus-like shape.
Base: 9cm wide × 3cm high → 9×3 = 27 cm²
Middle column: 2cm wide × 5cm high → 2×5 = 10 cm²
But wait — does the middle column sit on top of the base? Yes.
Are they overlapping? The base is 3cm high, and the column starts from the top of the base? Actually, looking at labels:
The 5cm is the height of the vertical part above the base. So total height of column is 5cm, sitting on the 3cm base.
So areas don’t overlap — just add.
Total = 27 + 10 = 37 cm²
Wait — but the base is 9cm wide, and the column is centered? Doesn’t matter for area.
Yes, correct.
✔ Shape (d) = 37 cm²
---
Shape (e)
Looks like a rectangle with two notches cut out from the bottom.
Full outer rectangle: width = 24cm, height = 20cm → 24×20 = 480 cm²
Now subtract the two notches.
Each notch: width = 6cm, height = 8cm → area per notch = 6×8 = 48 cm²
Two notches: 48 × 2 = 96 cm²
Also, there’s a small gap between notches? Look at bottom: segments are 6cm, 6cm, 2cm → total 6+6+2=14cm? But total width is 24cm — that doesn’t add up.
Wait — let’s read carefully.
At the bottom, it shows three segments: 6cm, 6cm, 2cm — but these are the widths of the *remaining* parts? Or the cuts?
Actually, looking at the diagram: the shape has two rectangular indentations (notches) at the bottom.
The distance between the notches is 2cm? And each notch is 6cm wide? Then total width covered by notches and gap: 6 + 2 + 6 = 14cm — but the whole width is 24cm, so the sides must have extra.
Actually, perhaps the 6cm, 6cm, 2cm are the widths of the *legs* at the bottom? That makes more sense.
Reinterpret:
The shape is like a table with two legs removed? No.
Actually, standard interpretation: it’s a large rectangle with two rectangular holes cut out from the bottom edge.
The height of each hole is 8cm (as labeled).
What is the width of each hole? From the bottom labels: the solid parts are 6cm, 6cm, and 2cm — meaning the gaps (holes) are between them.
Total width = 24cm
Solid parts at bottom: left leg 6cm, middle support 2cm, right leg 6cm → total solid = 6+2+6=14cm
Therefore, the two holes together take up 24 - 14 = 10cm → so each hole is 5cm wide? But that contradicts nothing labeled.
Wait — actually, in such diagrams, often the numbers under the shape indicate the widths of the protruding parts, and the cuts are implied.
But here, it says: below the shape, from left to right: 6cm, then a gap, then 6cm, then a gap, then 2cm? That doesn't make sense.
Looking back: the label "6cm" is under the first vertical leg, "6cm" under the second, "2cm" under the third — but there are only two notches, so probably three legs.
Actually, the shape has three downward projections? No — it looks like a rectangle with two upward notches? I think I misread.
Let me describe: the overall shape is 24cm wide, 20cm tall. At the bottom, there are two rectangular cut-outs (indentations) going upward 8cm. Between them is a 2cm wide strip, and on the sides are 6cm strips.
So, the material at the bottom consists of:
- Left rectangle: 6cm wide × 20cm high? No — the cut is only 8cm deep.
Actually, the entire shape is 20cm high. The notches go up 8cm from the bottom, so the remaining height above the notches is 20 - 8 = 12cm, which is solid across the full 24cm.
Then, below that, we have three separate rectangles at the bottom:
- Left: 6cm wide × 8cm high
- Middle: 2cm wide × 8cm high
- Right: 6cm wide × 8cm high
Is that right? But then the total width would be 6+2+6=14cm, but the top is 24cm — inconsistency.
I think I got it backwards.
Standard way: the shape is mostly solid, with two rectangular regions removed from the bottom.
The full rectangle is 24cm × 20cm = 480 cm²
Now, the two notches: each is a rectangle of width W and height 8cm.
From the bottom labels: the distances given are likely the widths of the *solid* parts between the notches.
Specifically: from left edge to first notch: 6cm? Or the notch starts after 6cm?
Actually, common convention: the numbers at the bottom indicate the lengths of the segments along the bottom edge.
So, starting from left:
- First segment: 6cm (solid)
- Then a notch (cut-out) — width unknown
- Then next solid segment: 6cm
- Then another notch
- Then last solid segment: 2cm
But that would mean two notches, and three solid parts totaling 6+6+2=14cm, so the two notches together are 24 - 14 = 10cm, so each notch is 5cm wide.
And each notch is 8cm high.
So area of each notch = 5 × 8 = 40 cm²
Two notches = 80 cm²
Total area = full rectangle - notches = 480 - 80 = 400 cm²
But let's verify with addition.
We can think of the shape as:
- Top part: 24cm wide × (20 - 8) = 24 × 12 = 288 cm²
- Bottom part: three rectangles:
- Left: 6cm × 8cm = 48
- Middle: 2cm × 8cm = 16
- Right: 6cm × 8cm = 48
Total bottom = 48+16+48 = 112 cm²
- Total = 288 + 112 = 400 cm² → matches.
Perfect.
✔ Shape (e) = 400 cm²
---
Shape (f)
Rectangle with a rectangular notch at the bottom center.
Overall: width = 8cm, height = 7cm → full area = 8×7 = 56 cm²
Notch: width = ? Height = 3cm
From bottom labels: left solid part 2cm, right solid part 3cm, so the notch width = 8 - 2 - 3 = 3cm
Height of notch = 3cm (given)
So area of notch = 3 × 3 = 9 cm²
Area of shape = 56 - 9 = 47 cm²
Addition method:
- Top part: 8cm wide × (7 - 3) = 8 × 4 = 32 cm²
- Bottom left: 2cm × 3cm = 6 cm²
- Bottom right: 3cm × 3cm = 9 cm²
- Total = 32 + 6 + 9 = 47 cm² → good.
✔ Shape (f) = 47 cm²
---
Final Answers:
(a) 62 cm²
(b) 360 cm²
(c) 67 cm²
(d) 37 cm²
(e) 400 cm²
(f) 47 cm²
──────────────────────────────────────
Final Answer:
(a) 62 cm²
(b) 360 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 of compound shapes worksheet.
