The shape in the image appears to be a rectangular block with a cylindrical hole passing through it. To find the volume of this shape, we need to calculate the volume of the rectangular block and subtract the volume of the cylindrical hole.

Step 1: Volume of the rectangular block

The dimensions of the block are:

• Length $L = 10 \, \text{cm}$
• Width $W = 6 \, \text{cm}$
• Height $H = 8 \, \text{cm}$

The volume of the block is given by: $V_{\text{block}} = L \times W \times H$ $V_{\text{block}} = 10 \, \text{cm} \times 6 \, \text{cm} \times 8 \, \text{cm} = 480 \, \text{cm}^3$

Step 2: Volume of the cylindrical hole

The cylinder has:

• Diameter $d = 6 \, \text{cm}$ (so radius $r = \frac{6}{2} = 3 \, \text{cm}$)
• Height (same as the width of the block) $h = 6 \, \text{cm}$

The volume of the cylinder is given by the formula: $V_{\text{cylinder}} = \pi r^2 h$ $V_{\text{cylinder}} = \pi (3 \, \text{cm})^2 \times 6 \, \text{cm}$ $V_{\text{cylinder}} = \pi \times 9 \times 6 = 54\pi \, \text{cm}^3 \approx 169.65 \, \text{cm}^3$

Step 3: Subtract the volume of the cylinder from the block

The volume of the shape is: $V_{\text{shape}} = V_{\text{block}} - V_{\text{cylinder}}$ $V_{\text{shape}} = 480 \, \text{cm}^3 - 169.65 \, \text{cm}^3 = 310.35 \, \text{cm}^3$

Final Answer:

The volume of the shape is approximately $310.35 \, \text{cm}^3$.

Would you like more details on any part of the solution? Here are some follow-up questions to consider:

1. How would the volume change if the cylindrical hole had a different radius?
2. How is the formula for the volume of a cylinder derived?
3. What if the hole passed vertically through the height instead of the width?
4. How would the volume calculation change if the block had two cylindrical holes?
5. What are other examples of composite shapes involving cylinders and rectangular prisms?

Tip: Always double-check whether the hole goes all the way through the object; partial holes require different volume calculations!