Worksheet for calculating the area of irregular shapes by breaking them into rectangles.
A worksheet titled "Area of an Irregular Shape" showing four irregular shapes with dimensions, instructing students to find the area by dividing into rectangles and adding the areas. Includes an example and space for answers.
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Step-by-step solution for: Areas of Irregular Shapes (Rectilinear Figures): Worksheets
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Show Answer Key & Explanations
Step-by-step solution for: Areas of Irregular Shapes (Rectilinear Figures): Worksheets
Let’s solve each problem step by step. We’ll break each irregular shape into rectangles, find the area of each rectangle, and add them up.
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Problem 1:
The shape looks like a big rectangle with a smaller rectangle cut out from the bottom right.
We can split it into two parts:
- Top rectangle: width = 10 cm, height = 5 cm → Area = 10 × 5 = 50 cm²
- Bottom rectangle: width = 4 cm, height = (8 - 5) = 3 cm → Area = 4 × 3 = 12 cm²
Wait — actually, looking again: The total height is 8 cm, and the top part is 5 cm tall, so the bottom part must be 3 cm tall. But the bottom part only goes across 4 cm (since the left side has a 6 cm horizontal segment).
Actually, better way: Split vertically or horizontally? Let’s try splitting horizontally at the 5 cm mark.
So:
- Top rectangle: 10 cm wide × 5 cm high = 50 cm²
- Bottom rectangle: It starts after the 6 cm indent. So its width is 10 - 6 = 4 cm, and height is 8 - 5 = 3 cm → 4 × 3 = 12 cm²
Total area = 50 + 12 = 62 cm²
But wait — let me double-check by another method.
Alternative: Think of the whole outer rectangle minus the missing piece.
Outer rectangle would be 10 cm × 8 cm = 80 cm²
Missing piece: It’s a rectangle that’s 6 cm wide and 3 cm high (because from x=0 to x=6, y=5 to y=8 is missing? Wait no — actually, the shape has a “notch” on the bottom left? Let me visualize again.
Looking at the diagram description:
It says:
Left side: 5 cm vertical, then 6 cm horizontal inward, then 3 cm down, then 4 cm right, then 8 cm up? Hmm, maybe I should label coordinates.
Actually, simpler: The shape can be divided into:
- A large rectangle on the left: 6 cm wide × 5 cm high = 30 cm²
- Plus a rectangle on the right: 4 cm wide × 8 cm high = 32 cm²
Wait — but they overlap? No, if you look: From left, first 6 cm wide for 5 cm height, then from there, going right 4 cm (total width 10 cm), and that right part goes full 8 cm height.
Yes! That works.
So:
- Left rectangle: 6 cm × 5 cm = 30 cm²
- Right rectangle: 4 cm × 8 cm = 32 cm²
Total = 30 + 32 = 62 cm²
Same answer. Good.
✔ Problem 1 Answer: 62 cm²
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Problem 2:
This is a rectangle with two notches taken out from the top.
Total width = 12 mm, total height = 6 mm
There are two identical notches on top, each 4 mm wide and 3 mm deep.
So we can calculate:
Area of full rectangle: 12 × 6 = 72 mm²
Each notch is 4 mm × 3 mm = 12 mm², and there are two → 2 × 12 = 24 mm²
Subtract: 72 - 24 = 48 mm²
Alternatively, divide into three rectangles:
- Bottom rectangle: 12 mm wide × (6 - 3) = 3 mm high → 12 × 3 = 36 mm²
- Two side rectangles on top: each 4 mm wide × 3 mm high → 4 × 3 = 12 each → 12 + 12 = 24 mm²
Total = 36 + 24 = 60 mm²? Wait, that doesn’t match.
Wait — mistake!
If the notches are 3 mm deep, then the remaining height below the notches is 6 - 3 = 3 mm, yes.
But the bottom rectangle is full width 12 mm × 3 mm = 36 mm²
Then above that, between the notches, there’s a middle strip? Actually, no — the notches are on the sides? Let me read the dimensions.
Diagram says:
Top: two segments of 4 mm each, with a gap in between? And depth of notch is 3 mm.
Actually, looking at labels:
From left: 4 mm (top edge), then down 3 mm, then right 4 mm (this is the bottom of the notch?), then up 3 mm, then right 4 mm.
Wait — total width is 12 mm.
So: left 4 mm, then a dip down 3 mm, then across 4 mm (so this is the base of the notch), then up 3 mm, then right 4 mm.
That means the "notch" is actually a rectangular hole? Or is it indented?
Actually, it's an indentation — so the shape has a flat bottom, and on top, it dips down in the middle? No — the labels suggest:
Starting from top-left:
- Go right 4 mm
- Go down 3 mm
- Go right 4 mm
- Go up 3 mm
- Go right 4 mm → total 12 mm
And height overall is 6 mm.
So the shape is like a rectangle with a rectangular bite taken out of the top center.
Specifically, the missing part is 4 mm wide and 3 mm high, located in the middle of the top.
So:
Full rectangle: 12 × 6 = 72 mm²
Missing rectangle: 4 × 3 = 12 mm²
Area = 72 - 12 = 60 mm²
Earlier I thought two notches — but actually, it’s one central notch? Wait, the diagram shows two separate notches? Let me re-read.
User wrote: “2. [diagram] 4mm, 3mm, 4mm, 3mm, 4mm” along the top, and height 6mm, width 12mm.
Actually, standard interpretation: This is a rectangle with two square notches cut out from the top corners? Or one in the middle?
Wait — if you go: start at top-left, move right 4mm, then down 3mm, then right 4mm, then up 3mm, then right 4mm — that creates a single rectangular indentation in the middle of the top side.
Because: from left, 4mm solid, then a 4mm-wide section that is lowered by 3mm, then 4mm solid on the right.
So the missing area is a rectangle 4mm wide × 3mm high, sitting in the middle of the top.
Thus, area = full area minus that one rectangle: 12×6 - 4×3 = 72 - 12 = 60 mm²
To confirm by addition:
- Bottom part: full width 12mm × height (6-3)=3mm → 36 mm²
- Top-left block: 4mm × 3mm = 12 mm²
- Top-right block: 4mm × 3mm = 12 mm²
- Middle top? No, because the middle is indented — so nothing there.
Total: 36 + 12 + 12 = 60 mm²
Yes.
✔ Problem 2 Answer: 60 mm²
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Problem 3:
Shape looks like a cross or plus sign.
Dimensions given:
Vertical stem: 9m wide at bottom, 5m high on sides, then a 4m wide section going up 4m, then a 1m wide top.
Actually, let’s parse:
From bottom: 9m wide, 5m high (base)
Then above that, centered, a rectangle 4m wide and 4m high
Then on top of that, a rectangle 1m wide and ? — wait, it says “1m” at the very top, probably height.
Actually, labeling:
- Bottom rectangle: 9m wide × 5m high
- Middle rectangle: 4m wide × 4m high (centered on top of bottom)
- Top rectangle: 1m wide × ? — the diagram likely shows the top part is 1m wide and some height, but it says “1m” next to the top vertical segment.
Looking at text: “1m” is written beside the topmost vertical line, so probably the top rectangle is 1m wide and, say, h meters high — but no height given for top? Wait, perhaps the 4m includes up to the top?
Wait, let’s think differently.
Perhaps the entire shape can be seen as:
- A large vertical rectangle: but not really.
Better to split into three horizontal rectangles:
1. Bottom: 9m × 5m = 45 m²
2. Middle: 4m × 4m = 16 m² (sitting on top of bottom, centered)
3. Top: 1m × ? — what’s the height? The diagram might imply that the top part is 1m wide and the same height as the middle? But no.
Wait — looking back at user input: “3. [diagram] 1m, 4m, 4m, 5m, 9m”
Probably:
- The bottom part is 9m wide and 5m high.
- Above that, a section that is 4m wide and extends upward 4m (so total height from bottom to here is 5+4=9m)
- Then on top of that, a 1m wide section — but how high? The “1m” is likely the width, and the height is not specified? That can’t be.
Perhaps the “1m” is the height of the top rectangle.
Assume:
- Bottom rectangle: 9m × 5m = 45 m²
- Middle rectangle: 4m × 4m = 16 m² (placed on top of bottom, centered)
- Top rectangle: 1m × h — but what is h? If the middle is 4m high, and top is additional, but no dimension given.
Wait — perhaps the 4m is the total height from bottom of middle to top of top? Unlikely.
Another approach: The shape is symmetric. Total height: from bottom to top.
Bottom: 5m
Then above that, a 4m high section that is 4m wide
Then above that, a 1m high section that is 1m wide? But the label “1m” is next to the top vertical, so likely height.
I think it’s safe to assume:
- Bottom: 9m × 5m = 45 m²
- Middle: 4m × 4m = 16 m²
- Top: 1m × 1m = 1 m²? But why 1m height? The diagram might show the top part is 1m in both directions.
Perhaps the “1m” refers to the width of the top, and the height is implied to be the same as the protrusion.
To avoid confusion, let’s use the fact that the shape can be divided vertically.
Notice that the shape has a central column.
From left to right:
- Left arm: from x=0 to x=(9-4)/2 = 2.5m? But measurements are integers, so probably not.
Better: Since it's symmetric, and bottom is 9m, middle is 4m, top is 1m, all centered.
So:
- The bottom rectangle: 9m × 5m = 45 m²
- The middle rectangle: sits on top of bottom, centered, so it adds area: 4m × 4m = 16 m²
- The top rectangle: sits on top of middle, centered, 1m × ? — if we assume the height of the top part is 1m (as labeled), then 1m × 1m = 1 m²
Total = 45 + 16 + 1 = 62 m²
But is the top height 1m? The label “1m” is placed next to the top vertical segment, which typically indicates the length of that segment, so yes, height is 1m.
Similarly, the “4m” next to the middle vertical segment likely means the height of that section is 4m.
So:
- Bottom: 9 × 5 = 45
- Middle: 4 × 4 = 16
- Top: 1 × 1 = 1
Sum: 62 m²
✔ Problem 3 Answer: 62 m²
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Problem 4:
Similar to problem 2, but different dimensions.
Shape: rectangle with a notch in the middle of the top.
Total width: 9 cm
Total height: 6 cm
Notch: from top, down 2 cm, and the notch is 2 cm wide? Labels: “4cm, 2cm, 3cm” on top, and “2cm” for the depth.
Specifically:
Start at top-left: go right 4 cm, then down 2 cm, then right 2 cm, then up 2 cm, then right 3 cm.
Total width: 4 + 2 + 3 = 9 cm, good.
Height: 6 cm overall.
So, the missing part is a rectangle 2 cm wide and 2 cm high, located in the middle of the top.
Thus:
Full rectangle area: 9 × 6 = 54 cm²
Missing rectangle: 2 × 2 = 4 cm²
Area = 54 - 4 = 50 cm²
By addition:
- Bottom part: full width 9 cm × height (6 - 2) = 4 cm → 9 × 4 = 36 cm²
- Left top block: 4 cm × 2 cm = 8 cm²
- Right top block: 3 cm × 2 cm = 6 cm²
- Middle top? Missing, so nothing.
Total: 36 + 8 + 6 = 50 cm²
Perfect.
✔ Problem 4 Answer: 50 cm²
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Final Answer:
1. 62 cm²
2. 60 mm²
3. 62 m²
4. 50 cm²
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Problem 1:
The shape looks like a big rectangle with a smaller rectangle cut out from the bottom right.
We can split it into two parts:
- Top rectangle: width = 10 cm, height = 5 cm → Area = 10 × 5 = 50 cm²
- Bottom rectangle: width = 4 cm, height = (8 - 5) = 3 cm → Area = 4 × 3 = 12 cm²
Wait — actually, looking again: The total height is 8 cm, and the top part is 5 cm tall, so the bottom part must be 3 cm tall. But the bottom part only goes across 4 cm (since the left side has a 6 cm horizontal segment).
Actually, better way: Split vertically or horizontally? Let’s try splitting horizontally at the 5 cm mark.
So:
- Top rectangle: 10 cm wide × 5 cm high = 50 cm²
- Bottom rectangle: It starts after the 6 cm indent. So its width is 10 - 6 = 4 cm, and height is 8 - 5 = 3 cm → 4 × 3 = 12 cm²
Total area = 50 + 12 = 62 cm²
But wait — let me double-check by another method.
Alternative: Think of the whole outer rectangle minus the missing piece.
Outer rectangle would be 10 cm × 8 cm = 80 cm²
Missing piece: It’s a rectangle that’s 6 cm wide and 3 cm high (because from x=0 to x=6, y=5 to y=8 is missing? Wait no — actually, the shape has a “notch” on the bottom left? Let me visualize again.
Looking at the diagram description:
It says:
Left side: 5 cm vertical, then 6 cm horizontal inward, then 3 cm down, then 4 cm right, then 8 cm up? Hmm, maybe I should label coordinates.
Actually, simpler: The shape can be divided into:
- A large rectangle on the left: 6 cm wide × 5 cm high = 30 cm²
- Plus a rectangle on the right: 4 cm wide × 8 cm high = 32 cm²
Wait — but they overlap? No, if you look: From left, first 6 cm wide for 5 cm height, then from there, going right 4 cm (total width 10 cm), and that right part goes full 8 cm height.
Yes! That works.
So:
- Left rectangle: 6 cm × 5 cm = 30 cm²
- Right rectangle: 4 cm × 8 cm = 32 cm²
Total = 30 + 32 = 62 cm²
Same answer. Good.
✔ Problem 1 Answer: 62 cm²
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Problem 2:
This is a rectangle with two notches taken out from the top.
Total width = 12 mm, total height = 6 mm
There are two identical notches on top, each 4 mm wide and 3 mm deep.
So we can calculate:
Area of full rectangle: 12 × 6 = 72 mm²
Each notch is 4 mm × 3 mm = 12 mm², and there are two → 2 × 12 = 24 mm²
Subtract: 72 - 24 = 48 mm²
Alternatively, divide into three rectangles:
- Bottom rectangle: 12 mm wide × (6 - 3) = 3 mm high → 12 × 3 = 36 mm²
- Two side rectangles on top: each 4 mm wide × 3 mm high → 4 × 3 = 12 each → 12 + 12 = 24 mm²
Total = 36 + 24 = 60 mm²? Wait, that doesn’t match.
Wait — mistake!
If the notches are 3 mm deep, then the remaining height below the notches is 6 - 3 = 3 mm, yes.
But the bottom rectangle is full width 12 mm × 3 mm = 36 mm²
Then above that, between the notches, there’s a middle strip? Actually, no — the notches are on the sides? Let me read the dimensions.
Diagram says:
Top: two segments of 4 mm each, with a gap in between? And depth of notch is 3 mm.
Actually, looking at labels:
From left: 4 mm (top edge), then down 3 mm, then right 4 mm (this is the bottom of the notch?), then up 3 mm, then right 4 mm.
Wait — total width is 12 mm.
So: left 4 mm, then a dip down 3 mm, then across 4 mm (so this is the base of the notch), then up 3 mm, then right 4 mm.
That means the "notch" is actually a rectangular hole? Or is it indented?
Actually, it's an indentation — so the shape has a flat bottom, and on top, it dips down in the middle? No — the labels suggest:
Starting from top-left:
- Go right 4 mm
- Go down 3 mm
- Go right 4 mm
- Go up 3 mm
- Go right 4 mm → total 12 mm
And height overall is 6 mm.
So the shape is like a rectangle with a rectangular bite taken out of the top center.
Specifically, the missing part is 4 mm wide and 3 mm high, located in the middle of the top.
So:
Full rectangle: 12 × 6 = 72 mm²
Missing rectangle: 4 × 3 = 12 mm²
Area = 72 - 12 = 60 mm²
Earlier I thought two notches — but actually, it’s one central notch? Wait, the diagram shows two separate notches? Let me re-read.
User wrote: “2. [diagram] 4mm, 3mm, 4mm, 3mm, 4mm” along the top, and height 6mm, width 12mm.
Actually, standard interpretation: This is a rectangle with two square notches cut out from the top corners? Or one in the middle?
Wait — if you go: start at top-left, move right 4mm, then down 3mm, then right 4mm, then up 3mm, then right 4mm — that creates a single rectangular indentation in the middle of the top side.
Because: from left, 4mm solid, then a 4mm-wide section that is lowered by 3mm, then 4mm solid on the right.
So the missing area is a rectangle 4mm wide × 3mm high, sitting in the middle of the top.
Thus, area = full area minus that one rectangle: 12×6 - 4×3 = 72 - 12 = 60 mm²
To confirm by addition:
- Bottom part: full width 12mm × height (6-3)=3mm → 36 mm²
- Top-left block: 4mm × 3mm = 12 mm²
- Top-right block: 4mm × 3mm = 12 mm²
- Middle top? No, because the middle is indented — so nothing there.
Total: 36 + 12 + 12 = 60 mm²
Yes.
✔ Problem 2 Answer: 60 mm²
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Problem 3:
Shape looks like a cross or plus sign.
Dimensions given:
Vertical stem: 9m wide at bottom, 5m high on sides, then a 4m wide section going up 4m, then a 1m wide top.
Actually, let’s parse:
From bottom: 9m wide, 5m high (base)
Then above that, centered, a rectangle 4m wide and 4m high
Then on top of that, a rectangle 1m wide and ? — wait, it says “1m” at the very top, probably height.
Actually, labeling:
- Bottom rectangle: 9m wide × 5m high
- Middle rectangle: 4m wide × 4m high (centered on top of bottom)
- Top rectangle: 1m wide × ? — the diagram likely shows the top part is 1m wide and some height, but it says “1m” next to the top vertical segment.
Looking at text: “1m” is written beside the topmost vertical line, so probably the top rectangle is 1m wide and, say, h meters high — but no height given for top? Wait, perhaps the 4m includes up to the top?
Wait, let’s think differently.
Perhaps the entire shape can be seen as:
- A large vertical rectangle: but not really.
Better to split into three horizontal rectangles:
1. Bottom: 9m × 5m = 45 m²
2. Middle: 4m × 4m = 16 m² (sitting on top of bottom, centered)
3. Top: 1m × ? — what’s the height? The diagram might imply that the top part is 1m wide and the same height as the middle? But no.
Wait — looking back at user input: “3. [diagram] 1m, 4m, 4m, 5m, 9m”
Probably:
- The bottom part is 9m wide and 5m high.
- Above that, a section that is 4m wide and extends upward 4m (so total height from bottom to here is 5+4=9m)
- Then on top of that, a 1m wide section — but how high? The “1m” is likely the width, and the height is not specified? That can’t be.
Perhaps the “1m” is the height of the top rectangle.
Assume:
- Bottom rectangle: 9m × 5m = 45 m²
- Middle rectangle: 4m × 4m = 16 m² (placed on top of bottom, centered)
- Top rectangle: 1m × h — but what is h? If the middle is 4m high, and top is additional, but no dimension given.
Wait — perhaps the 4m is the total height from bottom of middle to top of top? Unlikely.
Another approach: The shape is symmetric. Total height: from bottom to top.
Bottom: 5m
Then above that, a 4m high section that is 4m wide
Then above that, a 1m high section that is 1m wide? But the label “1m” is next to the top vertical, so likely height.
I think it’s safe to assume:
- Bottom: 9m × 5m = 45 m²
- Middle: 4m × 4m = 16 m²
- Top: 1m × 1m = 1 m²? But why 1m height? The diagram might show the top part is 1m in both directions.
Perhaps the “1m” refers to the width of the top, and the height is implied to be the same as the protrusion.
To avoid confusion, let’s use the fact that the shape can be divided vertically.
Notice that the shape has a central column.
From left to right:
- Left arm: from x=0 to x=(9-4)/2 = 2.5m? But measurements are integers, so probably not.
Better: Since it's symmetric, and bottom is 9m, middle is 4m, top is 1m, all centered.
So:
- The bottom rectangle: 9m × 5m = 45 m²
- The middle rectangle: sits on top of bottom, centered, so it adds area: 4m × 4m = 16 m²
- The top rectangle: sits on top of middle, centered, 1m × ? — if we assume the height of the top part is 1m (as labeled), then 1m × 1m = 1 m²
Total = 45 + 16 + 1 = 62 m²
But is the top height 1m? The label “1m” is placed next to the top vertical segment, which typically indicates the length of that segment, so yes, height is 1m.
Similarly, the “4m” next to the middle vertical segment likely means the height of that section is 4m.
So:
- Bottom: 9 × 5 = 45
- Middle: 4 × 4 = 16
- Top: 1 × 1 = 1
Sum: 62 m²
✔ Problem 3 Answer: 62 m²
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Problem 4:
Similar to problem 2, but different dimensions.
Shape: rectangle with a notch in the middle of the top.
Total width: 9 cm
Total height: 6 cm
Notch: from top, down 2 cm, and the notch is 2 cm wide? Labels: “4cm, 2cm, 3cm” on top, and “2cm” for the depth.
Specifically:
Start at top-left: go right 4 cm, then down 2 cm, then right 2 cm, then up 2 cm, then right 3 cm.
Total width: 4 + 2 + 3 = 9 cm, good.
Height: 6 cm overall.
So, the missing part is a rectangle 2 cm wide and 2 cm high, located in the middle of the top.
Thus:
Full rectangle area: 9 × 6 = 54 cm²
Missing rectangle: 2 × 2 = 4 cm²
Area = 54 - 4 = 50 cm²
By addition:
- Bottom part: full width 9 cm × height (6 - 2) = 4 cm → 9 × 4 = 36 cm²
- Left top block: 4 cm × 2 cm = 8 cm²
- Right top block: 3 cm × 2 cm = 6 cm²
- Middle top? Missing, so nothing.
Total: 36 + 8 + 6 = 50 cm²
Perfect.
✔ Problem 4 Answer: 50 cm²
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Final Answer:
1. 62 cm²
2. 60 mm²
3. 62 m²
4. 50 cm²
Parent Tip: Review the logic above to help your child master the concept of finding area of irregular shapes worksheet.
