Volume of Right Rectangular Prisms, Free PDF Download - Learn Bright - Free Printable
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Step-by-step solution for: Volume of Right Rectangular Prisms, Free PDF Download - Learn Bright
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Show Answer Key & Explanations
Step-by-step solution for: Volume of Right Rectangular Prisms, Free PDF Download - Learn Bright
Let’s solve each volume problem step by step.
---
Top Row: Rectangular Prisms
Volume of a rectangular prism = length × width × height
---
Problem 1 (left):
Dimensions:
Length = 3½ in. = 7/2
Width = 2⅔ in. = 8/3
Height = 6 in.
Multiply:
(7/2) × (8/3) × 6
First, multiply 7/2 × 8/3 = (7×8)/(2×3) = 56/6 = 28/3
Then, 28/3 × 6 = (28 × 6)/3 = 168/3 = 56 cubic inches
✔ Check: 3.5 × 2.666... × 6 ≈ 3.5 × 2.666 = ~9.333; 9.333 × 6 = 56 → Correct.
---
Problem 2 (middle):
Dimensions:
Length = 1⅕ = 6/5
Width = ¾
Height = ⅓
Multiply:
(6/5) × (3/4) × (1/3)
Step 1: 6/5 × 3/4 = (6×3)/(5×4) = 18/20 = 9/10
Step 2: 9/10 × 1/3 = 9/30 = 3/10
✔ Final answer: 3/10 cubic units
Check: 1.2 × 0.75 × 0.333… ≈ 1.2 × 0.75 = 0.9; 0.9 × 0.333 ≈ 0.3 → which is 3/10 → Correct.
---
Problem 3 (right):
Dimensions:
Length = 2¾ = 11/4
Width = ½
Height = ½
Multiply:
(11/4) × (1/2) × (1/2) = 11/(4×2×2) = 11/16
✔ Final answer: 11/16 cubic units
Check: 2.75 × 0.5 × 0.5 = 2.75 × 0.25 = 0.6875 → and 11 ÷ 16 = 0.6875 → Correct.
---
Bottom Row: Cubes made of smaller cubes
Each cube has edge length given. Volume of a cube = (edge length)³
But note: The diagrams show how many small cubes are along each edge — so we can also compute total volume as:
(Number of small cubes along one edge)³ × (volume of one small cube)
Wait — actually, the label says “Cube edge length = ___”, meaning the entire big cube’s edge is that fraction. So we just cube that number.
BUT — looking at the diagrams, they’re showing grids. For example, first one says “Cube edge length = 1/6” but shows a block that is 3 units wide, 4 tall, 2 deep? Wait — no, let’s re-read.
Actually, rereading: It says “Cube edge length = 1/6” — but the diagram is not a single cube — it’s a rectangular prism made of unit cubes? Or is the *small* cube’s edge length 1/6?
Looking again: The instruction says “Find the volume.” and labels say “Cube edge length = ___”
In context, this likely means: Each tiny cube in the grid has edge length equal to the given fraction. Then we count how many tiny cubes there are, and multiply by the volume of one tiny cube.
Let’s check the diagrams:
- First bottom left: Shows a block that is 3 cubes wide, 4 cubes high, 2 cubes deep → total 3×4×2 = 24 small cubes. Label: “Cube edge length = 1/6” → so each small cube has volume (1/6)³ = 1/216. Total volume = 24 × 1/216 = 24/216 = 1/9.
Wait — but the label says “Cube edge length = 1/6” — does that mean the whole object is a cube with edge 1/6? But the diagram isn’t a cube — it’s a rectangular prism. That doesn’t match.
Alternative interpretation: Perhaps “Cube edge length” refers to the edge length of each small cube used to build the figure. And the figure is built from those small cubes.
That makes sense with the diagrams.
So for each bottom problem:
→ Count number of small cubes in the figure.
→ Multiply by (given edge length)³ to get total volume.
Let’s do that.
---
Bottom Left:
Label: Cube edge length = 1/6 → so each small cube has volume (1/6)³ = 1/216
Figure: Let’s count the small cubes.
From the drawing: It looks like 3 columns (width), 4 rows (height), 2 layers (depth) → 3 × 4 × 2 = 24 small cubes.
Total volume = 24 × (1/216) = 24/216 = simplify: divide numerator and denominator by 24 → 1/9
✔ Answer: 1/9 cubic units
---
Bottom Middle:
Label: Cube edge length = 2/3 → each small cube volume = (2/3)³ = 8/27
Figure: Count small cubes.
Looks like 4 wide, 4 high, 3 deep? Let me see: From front view, 4 across, 4 up. Depth: you can see 3 layers going back.
So 4 × 4 × 3 = 48 small cubes.
Total volume = 48 × (8/27) = (48 × 8) / 27 = 384 / 27
Simplify: Divide numerator and denominator by 3 → 128 / 9
As mixed number: 14 and 2/9, but better as improper fraction unless specified.
✔ Answer: 128/9 cubic units
Check: 48 × 8 = 384; 384 ÷ 27 = 14.222... and 128 ÷ 9 = 14.222... → Correct.
---
Bottom Right:
Label: Cube edge length = 3/4 → each small cube volume = (3/4)³ = 27/64
Figure: Count small cubes.
Looks like 3 wide, 3 high, 3 deep → it’s a 3x3x3 cube → 27 small cubes.
Total volume = 27 × (27/64) = (27 × 27) / 64 = 729 / 64
✔ Answer: 729/64 cubic units
Check: 27 × 27 = 729; yes. 729 ÷ 64 = 11.390625 — reasonable.
---
Now, compiling all answers:
Top row:
1. 56 in³
2. 3/10
3. 11/16
Bottom row:
4. 1/9
5. 128/9
6. 729/64
All verified.
Final Answer:
56, 3/10, 11/16, 1/9, 128/9, 729/64
---
Top Row: Rectangular Prisms
Volume of a rectangular prism = length × width × height
---
Problem 1 (left):
Dimensions:
Length = 3½ in. = 7/2
Width = 2⅔ in. = 8/3
Height = 6 in.
Multiply:
(7/2) × (8/3) × 6
First, multiply 7/2 × 8/3 = (7×8)/(2×3) = 56/6 = 28/3
Then, 28/3 × 6 = (28 × 6)/3 = 168/3 = 56 cubic inches
✔ Check: 3.5 × 2.666... × 6 ≈ 3.5 × 2.666 = ~9.333; 9.333 × 6 = 56 → Correct.
---
Problem 2 (middle):
Dimensions:
Length = 1⅕ = 6/5
Width = ¾
Height = ⅓
Multiply:
(6/5) × (3/4) × (1/3)
Step 1: 6/5 × 3/4 = (6×3)/(5×4) = 18/20 = 9/10
Step 2: 9/10 × 1/3 = 9/30 = 3/10
✔ Final answer: 3/10 cubic units
Check: 1.2 × 0.75 × 0.333… ≈ 1.2 × 0.75 = 0.9; 0.9 × 0.333 ≈ 0.3 → which is 3/10 → Correct.
---
Problem 3 (right):
Dimensions:
Length = 2¾ = 11/4
Width = ½
Height = ½
Multiply:
(11/4) × (1/2) × (1/2) = 11/(4×2×2) = 11/16
✔ Final answer: 11/16 cubic units
Check: 2.75 × 0.5 × 0.5 = 2.75 × 0.25 = 0.6875 → and 11 ÷ 16 = 0.6875 → Correct.
---
Bottom Row: Cubes made of smaller cubes
Each cube has edge length given. Volume of a cube = (edge length)³
But note: The diagrams show how many small cubes are along each edge — so we can also compute total volume as:
(Number of small cubes along one edge)³ × (volume of one small cube)
Wait — actually, the label says “Cube edge length = ___”, meaning the entire big cube’s edge is that fraction. So we just cube that number.
BUT — looking at the diagrams, they’re showing grids. For example, first one says “Cube edge length = 1/6” but shows a block that is 3 units wide, 4 tall, 2 deep? Wait — no, let’s re-read.
Actually, rereading: It says “Cube edge length = 1/6” — but the diagram is not a single cube — it’s a rectangular prism made of unit cubes? Or is the *small* cube’s edge length 1/6?
Looking again: The instruction says “Find the volume.” and labels say “Cube edge length = ___”
In context, this likely means: Each tiny cube in the grid has edge length equal to the given fraction. Then we count how many tiny cubes there are, and multiply by the volume of one tiny cube.
Let’s check the diagrams:
- First bottom left: Shows a block that is 3 cubes wide, 4 cubes high, 2 cubes deep → total 3×4×2 = 24 small cubes. Label: “Cube edge length = 1/6” → so each small cube has volume (1/6)³ = 1/216. Total volume = 24 × 1/216 = 24/216 = 1/9.
Wait — but the label says “Cube edge length = 1/6” — does that mean the whole object is a cube with edge 1/6? But the diagram isn’t a cube — it’s a rectangular prism. That doesn’t match.
Alternative interpretation: Perhaps “Cube edge length” refers to the edge length of each small cube used to build the figure. And the figure is built from those small cubes.
That makes sense with the diagrams.
So for each bottom problem:
→ Count number of small cubes in the figure.
→ Multiply by (given edge length)³ to get total volume.
Let’s do that.
---
Bottom Left:
Label: Cube edge length = 1/6 → so each small cube has volume (1/6)³ = 1/216
Figure: Let’s count the small cubes.
From the drawing: It looks like 3 columns (width), 4 rows (height), 2 layers (depth) → 3 × 4 × 2 = 24 small cubes.
Total volume = 24 × (1/216) = 24/216 = simplify: divide numerator and denominator by 24 → 1/9
✔ Answer: 1/9 cubic units
---
Bottom Middle:
Label: Cube edge length = 2/3 → each small cube volume = (2/3)³ = 8/27
Figure: Count small cubes.
Looks like 4 wide, 4 high, 3 deep? Let me see: From front view, 4 across, 4 up. Depth: you can see 3 layers going back.
So 4 × 4 × 3 = 48 small cubes.
Total volume = 48 × (8/27) = (48 × 8) / 27 = 384 / 27
Simplify: Divide numerator and denominator by 3 → 128 / 9
As mixed number: 14 and 2/9, but better as improper fraction unless specified.
✔ Answer: 128/9 cubic units
Check: 48 × 8 = 384; 384 ÷ 27 = 14.222... and 128 ÷ 9 = 14.222... → Correct.
---
Bottom Right:
Label: Cube edge length = 3/4 → each small cube volume = (3/4)³ = 27/64
Figure: Count small cubes.
Looks like 3 wide, 3 high, 3 deep → it’s a 3x3x3 cube → 27 small cubes.
Total volume = 27 × (27/64) = (27 × 27) / 64 = 729 / 64
✔ Answer: 729/64 cubic units
Check: 27 × 27 = 729; yes. 729 ÷ 64 = 11.390625 — reasonable.
---
Now, compiling all answers:
Top row:
1. 56 in³
2. 3/10
3. 11/16
Bottom row:
4. 1/9
5. 128/9
6. 729/64
All verified.
Final Answer:
56, 3/10, 11/16, 1/9, 128/9, 729/64
Parent Tip: Review the logic above to help your child master the concept of volume of a rectangular prism worksheet.
