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Step-by-step solution for: Volume of Composite Rectangular Prism Worksheets (answers ...
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Step-by-step solution for: Volume of Composite Rectangular Prism Worksheets (answers ...
Let’s solve each problem step by step. We’ll find the volume of each composite figure by breaking it into two rectangular prisms, calculating their volumes separately, and then adding them together.
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Problem a:
The figure is made of two stacked boxes.
- Bottom box:
Length = 14 cm, Width = 3 cm, Height = 5 cm
Volume = 14 × 3 × 5 = 210 cm³
- Top box:
Same length and width (since it sits directly on top), height = 5 cm
Volume = 14 × 3 × 5 = 210 cm³
Total volume = 210 + 210 = 420 cm³
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Problem b:
This is an L-shaped figure. Let’s split it into two parts:
Option 1: Split vertically — left part and right part.
But easier: Think of it as a big bottom rectangle with a smaller one on top-right.
Actually, better to split horizontally or vertically? Let’s do this:
Split into:
- Left vertical prism: 4 in wide, 6 in tall, depth? Wait — look at dimensions.
Wait — let’s read carefully:
Figure b has:
- Total base length = 15 in
- Right side has a raised part: 7 in long, 3 in high
- The lower part goes full 15 in, but only 4 in deep? Wait — no.
Actually, looking again:
It says:
- Front view: total length 15 in
- On the right, there’s a block that is 7 in long (along the length) and 3 in high
- The whole thing is 4 in deep (front to back)
- The left part is 6 in high? Wait — label says “6 in” on the left side.
Actually, let me interpret:
The figure is like a step:
- Lower part: spans entire 15 in length, 4 in depth, and height = ? The left side says 6 in, but the right side has a bump up.
Wait — actually, the 6 in is the height of the left portion, and the right portion has a total height of 6 + 3 = 9? No — wait, the diagram shows:
On the right, there’s a small box on top: labeled 7 in (length), 3 in (height). And below it, the main body is 15 in long, 4 in deep, and the height on the left is 6 in — so the lower part must be 6 in high everywhere? But then the right part has an extra 3 in on top.
Yes! So:
We can split into:
1. Bottom prism: entire base — 15 in long, 4 in deep, 6 in high → Volume = 15 × 4 × 6 = 360 in³
2. Top prism (only on the right): 7 in long, 4 in deep, 3 in high → Volume = 7 × 4 × 3 = 84 in³
Total = 360 + 84 = 444 in³
Wait — but is the top prism sitting on top of the bottom one? Yes. And the bottom one is 6 in high, top adds 3 more on the right section.
Alternatively, we could think of the left part being taller? No — the diagram shows the left side is 6 in, and the right side has a step up — meaning the right side is higher. Actually, re-examining:
Labeling:
- On the left face: height = 6 in
- On the right, there’s a small box on top: 7 in (length along x), 3 in (height)
- Depth (into page) = 4 in for both
- Total length = 15 in
So yes — the bottom part is 15 in long, 4 in deep, 6 in high.
Then on top of the right 7 inches of that, we add another layer 3 in high.
So volumes:
Bottom: 15 × 4 × 6 = 360
Top: 7 × 4 × 3 = 84
Total: 444 in³ ✔️
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Problem c:
L-shaped again.
Dimensions:
- Bottom part: 10 cm long, 3 cm deep, 2 cm high
- Vertical part on right: 4 cm high, but how wide? It says 6 cm from left edge to start of vertical part? Wait.
Actually:
From diagram:
- Horizontal base: length 10 cm, depth 3 cm, height 2 cm
- On the right end, a vertical tower: it starts at 6 cm from left? So its length is 10 - 6 = 4 cm? But it says "4 cm" next to the vertical part — probably height.
Labels:
- Bottom: 10 cm (length), 3 cm (depth), 2 cm (height)
- Tower: positioned such that from left, 6 cm to where tower starts → so tower length = 10 - 6 = 4 cm? But it doesn’t say. Wait — it says “6 cm” arrow from left to start of tower, and tower itself is labeled “4 cm” vertically — that’s height.
Also, depth is same 3 cm.
Height of tower: total height from ground is 2 + 4 = 6 cm? But the label “4 cm” is next to the tower — likely the height of the tower above the base.
So:
Bottom prism: 10 × 3 × 2 = 60 cm³
Tower prism: since it sits on the right end, and assuming it’s 4 cm long (because 10 - 6 = 4), depth 3 cm, height 4 cm → Volume = 4 × 3 × 4 = 48 cm³
Total = 60 + 48 = 108 cm³
Wait — is the tower 4 cm long? The diagram shows “6 cm” from left to the start of the tower, and total length 10 cm, so yes, tower length = 4 cm.
And height of tower is 4 cm (as labeled), depth 3 cm.
Yes.
Alternative: maybe the 4 cm is the total height? But no — because the base is 2 cm, and tower sticks up 4 cm more? The label “4 cm” is placed next to the vertical side of the tower, so likely the height of the tower part alone.
In context, if base is 2 cm, and tower adds 4 cm, total height would be 6 cm — which makes sense.
So volumes:
Base: 10 × 3 × 2 = 60
Tower: 4 × 3 × 4 = 48
Total: 108 cm³ ✔️
---
Problem d:
Another composite shape.
Dimensions:
- Overall: 10 m long, 3 m deep
- Left part is taller: 12 m high
- Right part is shorter: 6 m high
- The top part on left is 8 m long? Label says “8 m” on top of the left block.
So, let’s split:
Option: Split vertically at the point where height changes.
Since the right part is 6 m high, and left part is 12 m high, and the top of the left part extends 8 m in length.
So:
Left prism: 8 m long, 3 m deep, 12 m high → Volume = 8 × 3 × 12 = 288 m³
Right prism: the remaining length is 10 - 8 = 2 m, depth 3 m, height 6 m → Volume = 2 × 3 × 6 = 36 m³
Total = 288 + 36 = 324 m³
Wait — is that correct?
Actually, the right part might be under the left part? No — the diagram shows the left part is taller and wider on top.
Actually, thinking again:
The entire base is 10 m long, 3 m deep.
The structure has:
- A lower section that is 10 m long, 3 m deep, 6 m high? But no — because the left part goes up to 12 m.
Better to think:
The figure consists of:
1. A large bottom prism: 10 m long, 3 m deep, 6 m high → Volume = 10 × 3 × 6 = 180 m³
2. An additional top prism on the left: 8 m long, 3 m deep, and height = 12 - 6 = 6 m → Volume = 8 × 3 × 6 = 144 m³
Total = 180 + 144 = 324 m³
Same answer! Good.
So either way, total volume is 324 m³.
---
Now, let’s write the final answers clearly.
Final Answer:
a. 420 cm³
b. 444 in³
c. 108 cm³
d. 324 m³
---
Problem a:
The figure is made of two stacked boxes.
- Bottom box:
Length = 14 cm, Width = 3 cm, Height = 5 cm
Volume = 14 × 3 × 5 = 210 cm³
- Top box:
Same length and width (since it sits directly on top), height = 5 cm
Volume = 14 × 3 × 5 = 210 cm³
Total volume = 210 + 210 = 420 cm³
---
Problem b:
This is an L-shaped figure. Let’s split it into two parts:
Option 1: Split vertically — left part and right part.
But easier: Think of it as a big bottom rectangle with a smaller one on top-right.
Actually, better to split horizontally or vertically? Let’s do this:
Split into:
- Left vertical prism: 4 in wide, 6 in tall, depth? Wait — look at dimensions.
Wait — let’s read carefully:
Figure b has:
- Total base length = 15 in
- Right side has a raised part: 7 in long, 3 in high
- The lower part goes full 15 in, but only 4 in deep? Wait — no.
Actually, looking again:
It says:
- Front view: total length 15 in
- On the right, there’s a block that is 7 in long (along the length) and 3 in high
- The whole thing is 4 in deep (front to back)
- The left part is 6 in high? Wait — label says “6 in” on the left side.
Actually, let me interpret:
The figure is like a step:
- Lower part: spans entire 15 in length, 4 in depth, and height = ? The left side says 6 in, but the right side has a bump up.
Wait — actually, the 6 in is the height of the left portion, and the right portion has a total height of 6 + 3 = 9? No — wait, the diagram shows:
On the right, there’s a small box on top: labeled 7 in (length), 3 in (height). And below it, the main body is 15 in long, 4 in deep, and the height on the left is 6 in — so the lower part must be 6 in high everywhere? But then the right part has an extra 3 in on top.
Yes! So:
We can split into:
1. Bottom prism: entire base — 15 in long, 4 in deep, 6 in high → Volume = 15 × 4 × 6 = 360 in³
2. Top prism (only on the right): 7 in long, 4 in deep, 3 in high → Volume = 7 × 4 × 3 = 84 in³
Total = 360 + 84 = 444 in³
Wait — but is the top prism sitting on top of the bottom one? Yes. And the bottom one is 6 in high, top adds 3 more on the right section.
Alternatively, we could think of the left part being taller? No — the diagram shows the left side is 6 in, and the right side has a step up — meaning the right side is higher. Actually, re-examining:
Labeling:
- On the left face: height = 6 in
- On the right, there’s a small box on top: 7 in (length along x), 3 in (height)
- Depth (into page) = 4 in for both
- Total length = 15 in
So yes — the bottom part is 15 in long, 4 in deep, 6 in high.
Then on top of the right 7 inches of that, we add another layer 3 in high.
So volumes:
Bottom: 15 × 4 × 6 = 360
Top: 7 × 4 × 3 = 84
Total: 444 in³ ✔️
---
Problem c:
L-shaped again.
Dimensions:
- Bottom part: 10 cm long, 3 cm deep, 2 cm high
- Vertical part on right: 4 cm high, but how wide? It says 6 cm from left edge to start of vertical part? Wait.
Actually:
From diagram:
- Horizontal base: length 10 cm, depth 3 cm, height 2 cm
- On the right end, a vertical tower: it starts at 6 cm from left? So its length is 10 - 6 = 4 cm? But it says "4 cm" next to the vertical part — probably height.
Labels:
- Bottom: 10 cm (length), 3 cm (depth), 2 cm (height)
- Tower: positioned such that from left, 6 cm to where tower starts → so tower length = 10 - 6 = 4 cm? But it doesn’t say. Wait — it says “6 cm” arrow from left to start of tower, and tower itself is labeled “4 cm” vertically — that’s height.
Also, depth is same 3 cm.
Height of tower: total height from ground is 2 + 4 = 6 cm? But the label “4 cm” is next to the tower — likely the height of the tower above the base.
So:
Bottom prism: 10 × 3 × 2 = 60 cm³
Tower prism: since it sits on the right end, and assuming it’s 4 cm long (because 10 - 6 = 4), depth 3 cm, height 4 cm → Volume = 4 × 3 × 4 = 48 cm³
Total = 60 + 48 = 108 cm³
Wait — is the tower 4 cm long? The diagram shows “6 cm” from left to the start of the tower, and total length 10 cm, so yes, tower length = 4 cm.
And height of tower is 4 cm (as labeled), depth 3 cm.
Yes.
Alternative: maybe the 4 cm is the total height? But no — because the base is 2 cm, and tower sticks up 4 cm more? The label “4 cm” is placed next to the vertical side of the tower, so likely the height of the tower part alone.
In context, if base is 2 cm, and tower adds 4 cm, total height would be 6 cm — which makes sense.
So volumes:
Base: 10 × 3 × 2 = 60
Tower: 4 × 3 × 4 = 48
Total: 108 cm³ ✔️
---
Problem d:
Another composite shape.
Dimensions:
- Overall: 10 m long, 3 m deep
- Left part is taller: 12 m high
- Right part is shorter: 6 m high
- The top part on left is 8 m long? Label says “8 m” on top of the left block.
So, let’s split:
Option: Split vertically at the point where height changes.
Since the right part is 6 m high, and left part is 12 m high, and the top of the left part extends 8 m in length.
So:
Left prism: 8 m long, 3 m deep, 12 m high → Volume = 8 × 3 × 12 = 288 m³
Right prism: the remaining length is 10 - 8 = 2 m, depth 3 m, height 6 m → Volume = 2 × 3 × 6 = 36 m³
Total = 288 + 36 = 324 m³
Wait — is that correct?
Actually, the right part might be under the left part? No — the diagram shows the left part is taller and wider on top.
Actually, thinking again:
The entire base is 10 m long, 3 m deep.
The structure has:
- A lower section that is 10 m long, 3 m deep, 6 m high? But no — because the left part goes up to 12 m.
Better to think:
The figure consists of:
1. A large bottom prism: 10 m long, 3 m deep, 6 m high → Volume = 10 × 3 × 6 = 180 m³
2. An additional top prism on the left: 8 m long, 3 m deep, and height = 12 - 6 = 6 m → Volume = 8 × 3 × 6 = 144 m³
Total = 180 + 144 = 324 m³
Same answer! Good.
So either way, total volume is 324 m³.
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Now, let’s write the final answers clearly.
Final Answer:
a. 420 cm³
b. 444 in³
c. 108 cm³
d. 324 m³
Parent Tip: Review the logic above to help your child master the concept of volume of a rectangular prism worksheet.
