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Hexagonal Pyramid — Definition, Formula & Examples

A hexagonal pyramid is a three-dimensional solid with a hexagonal (six-sided) base and six triangular faces that rise from the base edges and meet at a single point called the apex.

A hexagonal pyramid is a polyhedron formed by connecting each vertex of a hexagonal base to a common point (the apex) not in the plane of the base, resulting in a solid with 7 faces (1 hexagonal base and 6 triangular lateral faces), 12 edges, and 7 vertices.

Key Formula

V=13BhV = \frac{1}{3}Bh
Where:
  • VV = Volume of the pyramid
  • BB = Area of the hexagonal base
  • hh = Height (perpendicular distance from the base to the apex)

Worked Example

Problem: Find the volume of a hexagonal pyramid whose regular hexagonal base has a side length of 4 cm and whose height is 9 cm.
Find the base area: The area of a regular hexagon with side length s is given by:
B=332s2=332(4)2=332(16)=243≈41.57 cm2B = \frac{3\sqrt{3}}{2}s^2 = \frac{3\sqrt{3}}{2}(4)^2 = \frac{3\sqrt{3}}{2}(16) = 24\sqrt{3} \approx 41.57 \text{ cm}^2
Apply the volume formula: Substitute B and h into the pyramid volume formula:
V=13(243)(9)=723≈124.71 cm3V = \frac{1}{3}(24\sqrt{3})(9) = 72\sqrt{3} \approx 124.71 \text{ cm}^3
Answer: The volume is 723≈124.7172\sqrt{3} \approx 124.71 cm³.

Why It Matters

Hexagonal pyramids appear in architecture, crystal structures, and design. Understanding them builds fluency with the general pyramid volume formula, which is tested frequently in geometry courses and standardized exams.

Common Mistakes

Mistake: Using the full hexagon side length as the apothem when computing the base area.
Correction: The apothem of a regular hexagon with side length s is s32\frac{s\sqrt{3}}{2}, not s. Using the formula B=332s2B = \frac{3\sqrt{3}}{2}s^2 avoids this error entirely.