Volume of a Rectangular Prism Worksheets - Free Printable

To find the volume and surface area of these L-shaped figures, we need to break them down into simpler rectangular blocks (prisms).

Here is the step-by-step solution for each problem shown in the worksheet.

Problem 1 (Top Left)

Dimensions: The total height is 5, total width is 8. The top part has a height of 2 and width of 3. The bottom part has a height of 3 ($5-2$) and width of 8. The depth is 4.

1. Volume:
Split the shape into two blocks:
* Top Block: $3 \text{ (width)} \times 2 \text{ (height)} \times 4 \text{ (depth)} = 24$
* Bottom Block: $8 \text{ (width)} \times 3 \text{ (height)} \times 4 \text{ (depth)} = 96$
* Total Volume: $24 + 96 = \mathbf{120}$

2. Surface Area:
Calculate the area of all outer faces:
* Front & Back: The front area is $(3 \times 2) + (8 \times 3) = 6 + 24 = 30$. Since there is a back face too: $30 \times 2 = 60$.
* Top & Bottom: Top is $3 \times 4 = 12$. Bottom is $8 \times 4 = 32$. Total: $12 + 32 = 44$.
* Left & Right Sides: Left is $5 \times 4 = 20$. Right side consists of two parts: the lower right ($3 \times 4 = 12$) and the upper inner side ($2 \times 4 = 8$). Total Right side area is $12 + 8 = 20$. So, Left + Right = $20 + 20 = 40$.
* Total Surface Area: $60 + 44 + 40 = \mathbf{144}$

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Problem 2 (Top Right)

Dimensions: Total height 7, total width 5. Top block is $2 \times 5$. Bottom block is $3 \times 2$ (since width is 2 on the left side, implied by alignment, but looking at the diagram, the vertical stem is on the left. Let's assume the vertical stem is width 2 and height 7? No, standard L-shape decomposition: Vertical part on left is $2 \text{ wide} \times 7 \text{ high}$. Horizontal part sticking out is remaining width. Let's look closer. Actually, usually these are composed of two rectangles. Let's assume the vertical rectangle is $2 \times 7$ and the horizontal attachment is to the right. Wait, looking at Problem 1, the "L" was upright. Problem 2 looks like an inverted L or just rotated. Let's assume standard orientation: Vertical part on left is $2 \text{ units wide}$. Total width is 5, so the right part sticks out 3 units? Or is the top bar $5 \text{ wide} \times 2 \text{ high}$ and the stem is below it?
Let's re-read the diagram labels carefully.
* Top horizontal segment: Width 5, Height 2.
* Vertical segment below it: It aligns with the left edge. The label "2" is on the bottom width of the vertical stem. So the stem is $2 \text{ wide}$. The height of the stem *below* the top bar is $7 - 2 = 5$.
* Depth is 3.

1. Volume:
* Top Block: $5 \text{ (width)} \times 2 \text{ (height)} \times 3 \text{ (depth)} = 30$
* Bottom Stem: $2 \text{ (width)} \times 5 \text{ (height)} \times 3 \text{ (depth)} = 30$
* Total Volume: $30 + 30 = \mathbf{60}$

2. Surface Area:
* Front & Back: Front area = $(5 \times 2) + (2 \times 5) = 10 + 10 = 20$. Back is same. Total: $40$.
* Top & Bottom: Top is $5 \times 3 = 15$. Bottom is $2 \times 3 = 6$. There is also the underside of the overhang: $(5-2) \times 3 = 3 \times 3 = 9$. Total horizontal surfaces: $15 + 6 + 9 = 30$.
* Left & Right: Left is $7 \times 3 = 21$. Right side has two parts: top right end ($2 \times 3 = 6$) and lower right side of stem ($5 \times 3 = 15$). Total vertical sides: $21 + 6 + 15 = 42$.
* Total Surface Area: $40 + 30 + 42 = \mathbf{112}$

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Problem 3 (Middle Left)

Dimensions: This is a 3D view.
* Left block: Height 4, Width 3, Depth 4.
* Right block attached to side: Height 2, Width 2, Depth 4.
* Total Width = $3 + 2 = 5$. Total Height = 4.

1. Volume:
* Left Block: $3 \times 4 \times 4 = 48$
* Right Block: $2 \times 2 \times 4 = 16$
* Total Volume: $48 + 16 = \mathbf{64}$

2. Surface Area:
* Front & Back: Front area = $(3 \times 4) + (2 \times 2) = 12 + 4 = 16$. Back is same. Total: $32$.
* Top & Bottom: Top consists of two levels. Left top: $3 \times 4 = 12$. Right top: $2 \times 4 = 8$. Bottom is total width $5 \times 4 = 20$. Total horizontal: $12 + 8 + 20 = 40$.
* Left & Right: Left side: $4 \times 4 = 16$. Right side: The outer right face is $2 \times 4 = 8$. The inner vertical face connecting the two blocks is $(4-2) \times 4 = 2 \times 4 = 8$. Total vertical sides: $16 + 8 + 8 = 32$.
* Total Surface Area: $32 + 40 + 32 = \mathbf{104}$

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Problem 4 (Middle Right)

Dimensions:
* Top horizontal bar: Width 4, Height 2.
* Vertical stem: Width 2, Total Height 6. The stem is aligned to the left.
* Depth is 3.
* Note: The stem height *below* the top bar is $6 - 2 = 4$.

1. Volume:
* Top Block: $4 \times 2 \times 3 = 24$
* Bottom Stem: $2 \times 4 \times 3 = 24$
* Total Volume: $24 + 24 = \mathbf{48}$

2. Surface Area:
* Front & Back: Front area = $(4 \times 2) + (2 \times 4) = 8 + 8 = 16$. Back is same. Total: $32$.
* Top & Bottom: Top: $4 \times 3 = 12$. Bottom: $2 \times 3 = 6$. Underside of overhang: $(4-2) \times 3 = 6$. Total horizontal: $12 + 6 + 6 = 24$.
* Left & Right: Left: $6 \times 3 = 18$. Right side parts: Top end ($2 \times 3 = 6$) and lower stem side ($4 \times 3 = 12$). Total vertical sides: $18 + 6 + 12 = 36$.
* Total Surface Area: $32 + 24 + 36 = \mathbf{92}$

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Problem 5 (Bottom Left)

Dimensions:
* Top horizontal bar: Width 5, Height 2.
* Vertical stem: Width 2, Total Height 5. Aligned left.
* Depth is 4.
* Stem height below top bar: $5 - 2 = 3$.

1. Volume:
* Top Block: $5 \times 2 \times 4 = 40$
* Bottom Stem: $2 \times 3 \times 4 = 24$
* Total Volume: $40 + 24 = \mathbf{64}$

2. Surface Area:
* Front & Back: Front area = $(5 \times 2) + (2 \times 3) = 10 + 6 = 16$. Back is same. Total: $32$.
* Top & Bottom: Top: $5 \times 4 = 20$. Bottom: $2 \times 4 = 8$. Underside of overhang: $(5-2) \times 4 = 12$. Total horizontal: $20 + 8 + 12 = 40$.
* Left & Right: Left: $5 \times 4 = 20$. Right side parts: Top end ($2 \times 4 = 8$) and lower stem side ($3 \times 4 = 12$). Total vertical sides: $20 + 8 + 12 = 40$.
* Total Surface Area: $32 + 40 + 40 = \mathbf{112}$

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Problem 6 (Bottom Right)

Dimensions:
* Vertical stem on left: Width 2, Height 6.
* Horizontal foot on bottom right: Width 3 (extending from the stem), Height 2.
* Total Width = $2 + 3 = 5$. Total Height = 6.
* Depth is 3.
* Note: The horizontal part is at the bottom. The vertical part above the foot is $6 - 2 = 4$.

1. Volume:
* Vertical Stem (full height): $2 \times 6 \times 3 = 36$
* Horizontal Foot (remaining part): $3 \times 2 \times 3 = 18$
* Total Volume: $36 + 18 = \mathbf{54}$

2. Surface Area:
* Front & Back: Front area = $(2 \times 6) + (3 \times 2) = 12 + 6 = 18$. Back is same. Total: $36$.
* Top & Bottom: Top: $2 \times 3 = 6$. Bottom: Total width $5 \times 3 = 15$. Upper shelf (top of the foot): $3 \times 3 = 9$. Total horizontal: $6 + 15 + 9 = 30$.
* Left & Right: Left: $6 \times 3 = 18$. Right side parts: Bottom end ($2 \times 3 = 6$) and upper stem side ($4 \times 3 = 12$). Total vertical sides: $18 + 6 + 12 = 36$.
* Total Surface Area: $36 + 30 + 36 = \mathbf{102}$

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Final Answer:

1. Top Left
* Volume: 120
* Surface Area: 144

2. Top Right
* Volume: 60
* Surface Area: 112

3. Middle Left
* Volume: 64
* Surface Area: 104

4. Middle Right
* Volume: 48
* Surface Area: 92

5. Bottom Left
* Volume: 64
* Surface Area: 112

6. Bottom Right
* Volume: 54
* Surface Area: 102