In the context of a right square pyramid, what is the main difference between the $\mathrm{height}$ and the $\mathrm{slant} \mathrm{height}$ of the pyramid?

• A. There is no difference; both terms refer to the same measurement in a pyramid.
• B. The $\mathrm{height}$ is always longer than the $\mathrm{slant} \mathrm{height}$ in any pyramid.
• C. The $\mathrm{slant} \mathrm{height}$ is the distance from the apex to any corner of the base, while the $\mathrm{height}$ is the distance from the apex to the center of the base.
• D. The $\mathrm{height}$ is the perpendicular distance from the base to the apex, while the $\mathrm{slant} \mathrm{height}$ is the distance from the apex to the midpoint of a base edge along the face.

Answer: (D)

Verified step by step guidance

1. Understand that in a right square pyramid, the 'height' refers to the perpendicular distance from the apex (top point) straight down to the base plane, specifically to the center of the square base.
2. Recognize that the 'slant height' is measured along the triangular face of the pyramid, from the apex down to the midpoint of one of the base edges, not directly to the base center.
3. Visualize or draw the pyramid to see that the height forms a right angle with the base, while the slant height lies along the inclined face, creating a right triangle with the height and half the base edge as legs.
4. Use the Pythagorean theorem to relate the height, half the base edge length, and the slant height, since these three form a right triangle within the pyramid's face.
5. Conclude that the main difference is that the height is a vertical, perpendicular measurement to the base center, whereas the slant height is an inclined measurement along the face to the midpoint of a base edge.