Volume of Shapes Worksheet: Practice Problems for Geometry Students - Free Printable

This worksheet asks for the volume of six different compound 3D shapes. To solve this, I will break down each shape into simpler rectangular prisms (cuboids), calculate the volume of each part using the formula $V = l \times w \times h$, and then add or subtract these volumes as needed.

I will solve each problem one by one.

Problem 1

- First, look closely: The shape is an L-shape made of two rectangular prisms.
- The bottom part is a long cuboid with dimensions: length = 4 cm, width = 2 cm, height = 2 cm.
- The top part is a smaller cuboid sitting on the right side of the bottom part. Its dimensions are: length = 2 cm, width = 2 cm, height = 2 cm (since the total height is 4 cm and the bottom part is 2 cm tall).
- Next, find information: No external research is needed; I can calculate the volumes directly.
- Then, review the findings: I have all the necessary dimensions to calculate the volume.

Calculation:
- Volume of bottom part = $4 \text{ cm} \times 2 \text{ cm} \times 2 \text{ cm} = 16 \text{ cm}^3$
- Volume of top part = $2 \text{ cm} \times 2 \text{ cm} \times 2 \text{ cm} = 8 \text{ cm}^3$
- Total Volume = $16 \text{ cm}^3 + 8 \text{ cm}^3 = 24 \text{ cm}^3$

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Problem 2

- First, look closely: This is a U-shaped object. It can be thought of as one large rectangular prism with a smaller rectangular prism cut out from the middle.
- The outer dimensions are: length = 6 cm, width = 6 cm, height = 5 cm.
- The inner cut-out has dimensions: length = 4 cm, width = 6 cm, height = 3 cm (since the top part is 2 cm tall, the cut-out height is 5 cm - 2 cm = 3 cm).
- Next, find information: No external research is needed.
- Then, review the findings: I have all the necessary dimensions.

Calculation:
- Volume of the large outer prism = $6 \text{ cm} \times 6 \text{ cm} \times 5 \text{ cm} = 180 \text{ cm}^3$
- Volume of the inner cut-out = $4 \text{ cm} \times 6 \text{ cm} \times 3 \text{ cm} = 72 \text{ cm}^3$
- Total Volume = $180 \text{ cm}^3 - 72 \text{ cm}^3 = 108 \text{ cm}^3$

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Problem 3

- First, look closely: This is another L-shaped object, similar to problem 1 but with different dimensions.
- The bottom part is a long cuboid: length = 5 cm, width = 3 cm, height = 2 cm.
- The top part is a smaller cuboid on the left: length = 3 cm, width = 3 cm, height = 2 cm.
- Next, find information: No external research is needed.
- Then, review the findings: All dimensions are clear.

Calculation:
- Volume of bottom part = $5 \text{ cm} \times 3 \text{ cm} \times 2 \text{ cm} = 30 \text{ cm}^3$
- Volume of top part = $3 \text{ cm} \times 3 \text{ cm} \times 2 \text{ cm} = 18 \text{ cm}^3$
- Total Volume = $30 \text{ cm}^3 + 18 \text{ cm}^3 = 48 \text{ cm}^3$

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Problem 4

- First, look closely: This is a more complex shape. I can break it down into three parts: a bottom base, a middle step, and a top step.
- Bottom base: length = 3 cm, width = 4 cm, height = 2 cm.
- Middle step: length = 3 cm, width = 4 cm, height = 2 cm (the total height is 4 cm, so this step is 2 cm tall).
- Top step: length = 2 cm, width = 4 cm, height = 1 cm.
- Next, find information: No external research is needed.
- Then, review the findings: The dimensions are sufficient for calculation.

Calculation:
- Volume of bottom base = $3 \text{ cm} \times 4 \text{ cm} \times 2 \text{ cm} = 24 \text{ cm}^3$
- Volume of middle step = $3 \text{ cm} \times 4 \text{ cm} \times 2 \text{ cm} = 24 \text{ cm}^3$
- Volume of top step = $2 \text{ cm} \times 4 \text{ cm} \times 1 \text{ cm} = 8 \text{ cm}^3$
- Total Volume = $24 \text{ cm}^3 + 24 \text{ cm}^3 + 8 \text{ cm}^3 = 56 \text{ cm}^3$

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Problem 5

- First, look closely: This is a T-shaped object. It can be divided into two rectangular prisms: a vertical stem and a horizontal top bar.
- Vertical stem: length = 2 cm, width = 5 cm, height = 5 cm.
- Horizontal top bar: length = 7 cm, width = 3 cm, height = 1 cm.
- Next, find information: No external research is needed.
- Then, review the findings: The dimensions are clear, and the two parts do not overlap in volume.

Calculation:
- Volume of vertical stem = $2 \text{ cm} \times 5 \text{ cm} \times 5 \text{ cm} = 50 \text{ cm}^3$
- Volume of horizontal top bar = $7 \text{ cm} \times 3 \text{ cm} \times 1 \text{ cm} = 21 \text{ cm}^3$
- Total Volume = $50 \text{ cm}^3 + 21 \text{ cm}^3 = 71 \text{ cm}^3$

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Problem 6

- First, look closely: This is a large rectangular prism with a smaller rectangular prism cut out from its center.
- Outer dimensions: length = 8 cm, width = 4 cm, height = 6 cm.
- Inner cut-out dimensions: length = 4 cm, width = 2 cm, height = 4 cm (assuming the cut-out goes through the entire depth of 4 cm).
- Next, find information: No external research is needed.
- Then, review the findings: All dimensions are provided.

Calculation:
- Volume of the large outer prism = $8 \text{ cm} \times 4 \text{ cm} \times 6 \text{ cm} = 192 \text{ cm}^3$
- Volume of the inner cut-out = $4 \text{ cm} \times 2 \text{ cm} \times 4 \text{ cm} = 32 \text{ cm}^3$
- Total Volume = $192 \text{ cm}^3 - 32 \text{ cm}^3 = 160 \text{ cm}^3$

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Final Answers

Here are the volumes for all six problems:

1. $24 \text{ cm}^3$
2. $108 \text{ cm}^3$
3. $48 \text{ cm}^3$
4. $56 \text{ cm}^3$
5. $71 \text{ cm}^3$
6. $160 \text{ cm}^3$