Area of Composite Figures | Lecture notes Elementary Mathematics ... - Free Printable
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Step-by-step solution for: Area of Composite Figures | Lecture notes Elementary Mathematics ...
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Show Answer Key & Explanations
Step-by-step solution for: Area of Composite Figures | Lecture notes Elementary Mathematics ...
Let’s solve the first problem in the top-left corner.
We have a composite figure that looks like an “L” shape made of two rectangles.
Step 1: Break it into two rectangles.
Option A: Split vertically.
- Left rectangle: width = 8 - 5 = 3, height = 4 → area = 3 × 4 = 12
- Right rectangle: width = 5, height = 6 → area = 5 × 6 = 30
Total = 12 + 30 = 42 → but answer key says 36. So this split is wrong? Wait — let me check dimensions again.
Wait — looking at the figure:
Top side is labeled 8 (total width).
Right side is labeled 6 (total height).
Bottom part has width 5 and height 4 on the left vertical side.
Actually, better to split horizontally.
Split into:
- Top rectangle: width = 8, height = ?
The total height is 6, and the bottom part sticks down 4 units from the left, so the top rectangle’s height is 6 - 4 = 2? But then the right side is full height 6.
Wait — let's label all sides clearly.
From the drawing:
Imagine the shape:
It’s like a big rectangle 8 wide by 6 tall, but with a rectangle cut out from the bottom-left.
The missing part would be: width = 8 - 5 = 3, height = 4.
So area = big rectangle minus small rectangle.
Big rectangle: 8 × 6 = 48
Missing rectangle: 3 × 4 = 12
Area = 48 - 12 = 36 → matches answer key!
Alternatively, split into two rectangles:
Rectangle 1 (right part): 5 wide × 6 high = 30
Rectangle 2 (top-left part): width = 8 - 5 = 3, height = 6 - 4 = 2? No — wait, if we go up from the bottom, the left part only goes up 4, but the top extends across.
Better:
Horizontal split:
Top rectangle: spans full width 8, height = 6 - 4 = 2 → area = 8 × 2 = 16
Bottom rectangle: only under the right part, width = 5, height = 4 → area = 5 × 4 = 20
Total = 16 + 20 = 36 → correct.
Or vertical split:
Left rectangle: width = 3 (since 8 - 5), height = 4 → 3×4=12
Right rectangle: width = 5, height = 6 → 5×6=30
Total = 12+30=42 → too big. Why? Because the right rectangle includes the top part which overlaps with the left? No — actually, in vertical split, if I take left as 3x4, and right as 5x6, they don’t overlap, but the total height on the right is 6, while on the left it’s only 4 — so the top 2 units of the right rectangle are above the left part, which is fine. But 12+30=42 ≠36. Contradiction.
Ah — I see the mistake. In the vertical split, if I take the left rectangle as 3 units wide and 4 units high, that’s correct for the protruding part. Then the right part should be 5 units wide and 6 units high — but that would include the area directly above the left rectangle? No, because the left rectangle is only 4 high, and the right is 6 high, so together they form the L-shape without overlap.
But 3*4 + 5*6 = 12 + 30 = 42, but answer is 36. So something’s off.
Let me re-express the figure.
Label the coordinates mentally.
Assume bottom-left corner is (0,0).
Then:
- From (0,0) to (5,0) to (5,4) to (3,4)? No.
Standard way: the figure has outer dimensions.
Top edge: length 8.
Right edge: length 6.
Bottom edge: from left, there’s a segment of length 5, then it goes up 4, then left? No.
Looking at the labels:
On the left side, there’s a vertical segment labeled 4.
On the bottom, horizontal segment labeled 5.
On the top, horizontal segment labeled 8.
On the right, vertical segment labeled 6.
Also, implicitly, the inner corner.
So, starting from top-left:
Go right 8, down 6, left 5, up 4, left 3? Let's calculate the missing parts.
Total width 8, bottom part is 5, so the "step" on the bottom is 8 - 5 = 3 units to the left.
Total height 6, left part is 4, so the "step" on the left is 6 - 4 = 2 units up.
So the shape can be seen as:
- A rectangle on the right: 5 wide × 6 high = 30
- Plus a rectangle on the top-left: 3 wide × 2 high = 6
Total 36. Yes! That works.
Because the top-left rectangle is 3 units wide (8-5) and 2 units high (6-4), sitting on top of the left part, but since the left part only goes up 4, and total height is 6, the extra 2 is covered by this top rectangle.
In this case, the two rectangles do not overlap: one is 5x6 on the right, the other is 3x2 on the top-left, adjacent.
Area = 30 + 6 = 36.
Perfect.
So for this problem, area is 36 square units.
Now, just to confirm with another method: subtract the missing corner.
If it were a full rectangle 8x6=48.
The missing part is a rectangle at bottom-left: width 3 (8-5), height 4? But if I remove a 3x4 rectangle from bottom-left, then the remaining shape would have height 6 on the right, but on the left, from y=0 to y=4 is removed, so from y=4 to y=6 is still there, which is 2 units high, and width 3, so yes, that matches.
Missing area: 3 * 4 = 12
48 - 12 = 36.
Same answer.
So confidently, area is 36.
Final Answer: 36
We have a composite figure that looks like an “L” shape made of two rectangles.
Step 1: Break it into two rectangles.
Option A: Split vertically.
- Left rectangle: width = 8 - 5 = 3, height = 4 → area = 3 × 4 = 12
- Right rectangle: width = 5, height = 6 → area = 5 × 6 = 30
Total = 12 + 30 = 42 → but answer key says 36. So this split is wrong? Wait — let me check dimensions again.
Wait — looking at the figure:
Top side is labeled 8 (total width).
Right side is labeled 6 (total height).
Bottom part has width 5 and height 4 on the left vertical side.
Actually, better to split horizontally.
Split into:
- Top rectangle: width = 8, height = ?
The total height is 6, and the bottom part sticks down 4 units from the left, so the top rectangle’s height is 6 - 4 = 2? But then the right side is full height 6.
Wait — let's label all sides clearly.
From the drawing:
Imagine the shape:
It’s like a big rectangle 8 wide by 6 tall, but with a rectangle cut out from the bottom-left.
The missing part would be: width = 8 - 5 = 3, height = 4.
So area = big rectangle minus small rectangle.
Big rectangle: 8 × 6 = 48
Missing rectangle: 3 × 4 = 12
Area = 48 - 12 = 36 → matches answer key!
Alternatively, split into two rectangles:
Rectangle 1 (right part): 5 wide × 6 high = 30
Rectangle 2 (top-left part): width = 8 - 5 = 3, height = 6 - 4 = 2? No — wait, if we go up from the bottom, the left part only goes up 4, but the top extends across.
Better:
Horizontal split:
Top rectangle: spans full width 8, height = 6 - 4 = 2 → area = 8 × 2 = 16
Bottom rectangle: only under the right part, width = 5, height = 4 → area = 5 × 4 = 20
Total = 16 + 20 = 36 → correct.
Or vertical split:
Left rectangle: width = 3 (since 8 - 5), height = 4 → 3×4=12
Right rectangle: width = 5, height = 6 → 5×6=30
Total = 12+30=42 → too big. Why? Because the right rectangle includes the top part which overlaps with the left? No — actually, in vertical split, if I take left as 3x4, and right as 5x6, they don’t overlap, but the total height on the right is 6, while on the left it’s only 4 — so the top 2 units of the right rectangle are above the left part, which is fine. But 12+30=42 ≠36. Contradiction.
Ah — I see the mistake. In the vertical split, if I take the left rectangle as 3 units wide and 4 units high, that’s correct for the protruding part. Then the right part should be 5 units wide and 6 units high — but that would include the area directly above the left rectangle? No, because the left rectangle is only 4 high, and the right is 6 high, so together they form the L-shape without overlap.
But 3*4 + 5*6 = 12 + 30 = 42, but answer is 36. So something’s off.
Let me re-express the figure.
Label the coordinates mentally.
Assume bottom-left corner is (0,0).
Then:
- From (0,0) to (5,0) to (5,4) to (3,4)? No.
Standard way: the figure has outer dimensions.
Top edge: length 8.
Right edge: length 6.
Bottom edge: from left, there’s a segment of length 5, then it goes up 4, then left? No.
Looking at the labels:
On the left side, there’s a vertical segment labeled 4.
On the bottom, horizontal segment labeled 5.
On the top, horizontal segment labeled 8.
On the right, vertical segment labeled 6.
Also, implicitly, the inner corner.
So, starting from top-left:
Go right 8, down 6, left 5, up 4, left 3? Let's calculate the missing parts.
Total width 8, bottom part is 5, so the "step" on the bottom is 8 - 5 = 3 units to the left.
Total height 6, left part is 4, so the "step" on the left is 6 - 4 = 2 units up.
So the shape can be seen as:
- A rectangle on the right: 5 wide × 6 high = 30
- Plus a rectangle on the top-left: 3 wide × 2 high = 6
Total 36. Yes! That works.
Because the top-left rectangle is 3 units wide (8-5) and 2 units high (6-4), sitting on top of the left part, but since the left part only goes up 4, and total height is 6, the extra 2 is covered by this top rectangle.
In this case, the two rectangles do not overlap: one is 5x6 on the right, the other is 3x2 on the top-left, adjacent.
Area = 30 + 6 = 36.
Perfect.
So for this problem, area is 36 square units.
Now, just to confirm with another method: subtract the missing corner.
If it were a full rectangle 8x6=48.
The missing part is a rectangle at bottom-left: width 3 (8-5), height 4? But if I remove a 3x4 rectangle from bottom-left, then the remaining shape would have height 6 on the right, but on the left, from y=0 to y=4 is removed, so from y=4 to y=6 is still there, which is 2 units high, and width 3, so yes, that matches.
Missing area: 3 * 4 = 12
48 - 12 = 36.
Same answer.
So confidently, area is 36.
Final Answer: 36
Parent Tip: Review the logic above to help your child master the concept of area of composite figures worksheet.
