Volume, surface area and slant height of the pyramid

The formulas of the right regular pyramid revolve around one key element, the slant height of the pyramid.

Property — Formulas of the pyramid

• Volume: $V = \tfrac{1}{3}S_b\cdot h$.
• Lateral surface area (right regular pyramid): $S_{\text{lat}} = \tfrac{1}{2}\, p_b\cdot a$, where $a$ is the slant height of the pyramid, that is, the height of a triangular lateral face.
• Slant height: if $m$ is the apothem of the base and $h$ the height, then $a = \sqrt{m^2+h^2}$.
• Notable angles: the angle between an edge and the base is the one between the lateral edge and its projection onto the base; the angle between a face and the base is the dihedral angle, which is measured with the normal section.

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Topics: Synthetic geometry of space
Concepts: Dihedral angle · Apothem · Pyramid · Surface area · Volume
Skills: Using formulas