Identical Volume for Different Prisms and Cylinder

The question asks for the condition under which a cylinder, a square prism, and a triangular prism will have identical volumes. To answer this, we need to understand how the volume of these shapes is calculated.

Volume Formula for Prisms and Cylinders

The volume of any prism or a cylinder is generally calculated using a simple formula:

Volume = Base Area $\times$ Height

Let's consider each shape mentioned in the question:

• Cylinder: A cylinder can be thought of as a circular prism. Its base is a circle. If the radius of the base is \$r\$ and the height is \$h\$, the base area is \$\pi r^2\$. The volume of the cylinder is \$V_{cylinder} = (\pi r^2) \times h\$.
• Square Prism: A square prism has a square base. If the side length of the square base is \$s\$ and the height is \$h\$, the base area is \$s^2\$. The volume of the square prism is \$V_{square\_prism} = (s^2) \times h\$.
• Triangular Prism: A triangular prism has a triangular base. If the area of the triangular base is \$A_{base}\$ and the height is \$h\$, the volume of the triangular prism is \$V_{triangular\_prism} = A_{base} \times h\$.

Condition for Identical Volume

From the volume formulas above, we can see a common structure: Volume = Base Area $\times$ Height. For the volumes of the cylinder, square prism, and triangular prism to be identical, the product of their respective base area and height must be the same.

If we want their volumes to be identical, say \$V_{cylinder} = V_{square\_prism} = V_{triangular\_prism} = V\$, then based on the formula:

The simplest and most direct condition that ensures all three volumes are identical is when both the base areas AND the heights of all three containers are identical.

Let's assume the base area of the cylinder is \$A_1\$, the base area of the square prism is \$A_2\$, and the base area of the triangular prism is \$A_3\$. Let their respective heights be \$h_1, h_2, h_3\$. Their volumes are \$V_1 = A_1 h_1\$, \$V_2 = A_2 h_2\$, and \$V_3 = A_3 h_3\$.

For \$V_1 = V_2 = V_3\$, if we set \$A_1 = A_2 = A_3 = A\$ and \$h_1 = h_2 = h_3 = h\$, then \$V_1 = A \times h\$, \$V_2 = A \times h\$, and \$V_3 = A \times h\$. In this case, \$V_1 = V_2 = V_3\$.

Analyzing the Options

• Option 1: Their base areas and heights are identical

  As shown above, if the base area and the height are the same for all three shapes, their volumes will be identical according to the formula Volume = Base Area $\times$ Height. This condition guarantees identical volumes.

• Option 2: Their base areas are identical

  If only the base areas are identical (e.g., \$A_1 = A_2 = A_3 = A\$), but the heights are different (e.g., \$h_1 \neq h_2 \neq h_3\$), then the volumes would be \$V_1 = A h_1\$, \$V_2 = A h_2\$, and \$V_3 = A h_3\$. These volumes would only be identical if the heights happened to also be identical. So, this condition alone is not sufficient.

• Option 3: Their heights are identical

  If only the heights are identical (e.g., \$h_1 = h_2 = h_3 = h\$), but the base areas are different (e.g., \$A_1 \neq A_2 \neq A_3\$), then the volumes would be \$V_1 = A_1 h\$, \$V_2 = A_2 h\$, and \$V_3 = A_3 h\$. These volumes would only be identical if the base areas happened to also be identical. So, this condition alone is not sufficient.

• Option 4: Their base areas and heights are different

  If both base areas and heights are different, it is very unlikely that their volumes would be identical. While it might be possible for specific combinations of different base areas and heights to result in the same volume (e.g., \$A_1 h_1 = A_2 h_2\$), this is not a guaranteed condition based on them being different. This condition does not ensure identical volumes.

Therefore, the condition that will ensure they have identical volume is when their base areas and heights are identical.