Final answer:
The bases of a triangular prism and a cylinder with the same height and volume must have equal areas. The volume formulas for both shapes indicate that their base areas must be the same when their heights are identical.
Step-by-step explanation:
If a triangular prism and a cylinder have the same height and equal volumes, their bases must have equal areas. The volume of a prism is given by the area of its base times its height (V = base area × height), and the volume of a cylinder is given by the area of its circular base times its height (V = πr² × height, where r is the radius of the base).
To have the same volume, the equation V = base area (triangle) × height must equal πr² × height, which simplifies to base area (triangle) = πr² when both shapes have the same height. Therefore, we can say that the area of the triangular base must be equal to the area of the cylinder's circular base due to their volumes being equivalent and their heights being the same.