A definitive mathematical reference sheet containing formulas for volume, surface area, slant height, apothem, and frustums across all polygon bases.
For any pyramid solid with base surface area $A_b$ and perpendicular altitude $h$:
This universal equation holds true regardless of whether the base polygon is regular, irregular, symmetric, or oblique.
| Pyramid Type | Base Area ($A_b$) | Volume Formula ($V$) | Slant Height ($s$) |
|---|---|---|---|
| Square Pyramid | $A_b = a^2$ | $V = \frac{1}{3} a^2 h$ | $s = \sqrt{h^2 + (a/2)^2}$ |
| Rectangular Pyramid | $A_b = l \times w$ | $V = \frac{1}{3} l w h$ | $s_l = \sqrt{h^2 + (w/2)^2}$ |
| Equilateral Triangle | $A_b = \frac{\sqrt{3}}{4} a^2$ | $V = \frac{\sqrt{3}}{12} a^2 h$ | $s = \sqrt{h^2 + (a\sqrt{3}/6)^2}$ |
| Regular Pentagon | $A_b \approx 1.7205 a^2$ | $V \approx 0.5735 a^2 h$ | $s = \sqrt{h^2 + r_{\text{in}}^2}$ |
| Regular Hexagon | $A_b = \frac{3\sqrt{3}}{2} a^2$ | $V = \frac{\sqrt{3}}{2} a^2 h$ | $s = \sqrt{h^2 + (a\sqrt{3}/2)^2}$ |
| Regular Tetrahedron | $A_b = \frac{\sqrt{3}}{4} a^2$ | $V = \frac{a^3}{6\sqrt{2}}$ | $s = \frac{\sqrt{3}}{2} a$ |
The total surface area $A_{\text{total}}$ of any right pyramid equals the sum of its base area $A_b$ and its lateral faces:
Where $P_{\text{base}}$ is the perimeter of the base polygon and $s$ is the perpendicular slant height along a triangular face. For a square pyramid with edge $a$, $P = 4a$, so $A_{\text{lateral}} = 2as$.
When a pyramid is sliced by a plane parallel to its base, removing the top apex, the remaining solid is a frustum. If the bottom base has area $A_1$, top base has area $A_2$, and vertical altitude is $h$:
This remarkable formula was known to ancient Egyptian mathematicians over 3,800 years ago as recorded in the Moscow Mathematical Papyrus. The term $\sqrt{A_1 A_2}$ represents the geometric mean of the two parallel bases.