Question

which three - dimensional figure has the greatest number of faces? hexagonal prism octagonal pyramid rectangular prism pentagonal pyramid

Explanation:

To determine which three - dimensional figure has the greatest number of faces, we analyze each option:

1. Hexagonal prism: A prism has two congruent polygonal bases and lateral faces. A hexagonal prism has a hexagonal base. The number of faces of a prism is given by the formula \(F = 2 + n\), where \(n\) is the number of sides of the base. For a hexagonal prism, \(n = 6\), so the number of faces \(F=2 + 6=8\).
2. Octagonal pyramid: A pyramid has a polygonal base and triangular lateral faces. The number of faces of a pyramid is given by the formula \(F=1 + n\), where \(n\) is the number of sides of the base. For an octagonal pyramid, \(n = 8\), so the number of faces \(F = 1+8 = 9\).
3. Rectangular prism: A rectangular prism has a rectangular base. Using the prism face formula \(F=2 + n\), with \(n = 4\) (since a rectangle has 4 sides), the number of faces \(F=2 + 4=6\).
4. Pentagonal pyramid: Using the pyramid face formula \(F = 1 + n\), with \(n = 5\) (a pentagon has 5 sides), the number of faces \(F=1 + 5=6\).

By comparing the number of faces (8 for hexagonal prism, 9 for octagonal pyramid, 6 for rectangular prism, and 6 for pentagonal pyramid), we see that the octagonal pyramid has the greatest number of faces.

Answer:

B. octagonal pyramid