What Is a Hexagonal Pyramid?
A hexagonal pyramid is a pyramid whose base is a hexagon (a six-sided polygon) and whose six slanted faces are triangles that rise from the base edges and meet at a single top point called the apex. The base sits flat; the triangular faces lean inward to the peak.
Because it has seven flat faces in total, a hexagonal pyramid is also called a heptahedron. It belongs to the wider family of the pyramid, which is itself a type of polyhedron (a solid with flat polygon faces).
A pyramid is called regular when its base is a regular hexagon (all six sides and angles equal) and the apex sits directly above the base's centre. In that case the six triangular faces are congruent isosceles triangles.
For Example : A single point lifted above a six-sided base turns a flat hexagon into a pyramid.
That one move, pulling a peak up over a hexagon, is how honeycomb caps, some crystal tips, and many spire roofs get their shape. The rest is measuring what that shape holds and covers.
How Many Faces, Edges, and Vertices Does a Hexagonal Pyramid Have?
This is the single most-asked question about the shape, so here is the direct count:
Faces: 7 - one hexagonal base plus six triangular sides.
Edges: 12 - six around the hexagon, plus six lateral edges running from each base corner up to the apex.
Vertices: 7 - six at the base corners, plus the apex.
These three numbers obey Euler's formula for polyhedra, $F - E + V = 2$:
$$7 - 12 + 7 = 2$$
The check holds, which confirms the count is internally consistent.
What Are the Properties of a Hexagonal Pyramid?
A hexagonal pyramid has a compact set of defining properties:
7 faces, 12 edges, and 7 vertices, satisfying Euler's formula $F - E + V = 2$.
The base is a hexagon; the six lateral faces are triangles that share the apex.
It has one apex and, because it has seven flat faces, it is also called a heptahedron.
In a regular hexagonal pyramid, the six lateral faces are congruent isosceles triangles and the apex sits directly above the centre of the base.
Any cross-section parallel to the base is a hexagon similar to the base, shrinking to a point at the apex.
The lateral edges (base corner to apex) are all equal in length when the pyramid is regular.
What Are the Types of Hexagonal Pyramid?
Hexagonal pyramids are classified by where the apex sits and by the shape of the base:
Right (regular) hexagonal pyramid - the base is a regular hexagon and the apex lies directly above the base's centre, so the axis is perpendicular to the base. All six triangular faces are congruent. This is the type used in most formulas.
Oblique hexagonal pyramid - the apex is not above the centre, so the pyramid leans. The lateral faces are no longer all congruent, though the volume formula $V = \frac{1}{3} \times \text{base area} \times h$ still holds with $h$ the perpendicular height.
Irregular hexagonal pyramid - the base is an irregular hexagon (sides or angles unequal). The face-edge-vertex counts stay the same, but the base area must be found separately.
How Do You Calculate the Volume of a Hexagonal Pyramid?
Every pyramid, whatever its base, holds exactly one-third of the prism built on the same base and height. So the master formula is:
$$V = \frac{1}{3} \times \text{base area} \times h$$
Here $h$ is the perpendicular height from the apex straight down to the base's centre. All that changes from pyramid to pyramid is the base area.
For a regular hexagonal pyramid with base side length $a$, the area of a regular hexagon is $\frac{3\sqrt{3}}{2} a^2$. Substituting:
$$V = \frac{1}{3} \times \frac{3\sqrt{3}}{2} a^2 \times h = \frac{\sqrt{3}}{2} a^2 h$$
Variable glossary: $a$ is the base side length, $h$ is the vertical height, and the $\frac{1}{3}$ is the pyramid factor that separates a pyramid from a prism.
How Do You Find the Surface Area of a Hexagonal Pyramid?
Total surface area is the base plus the six triangular sides:
$$\text{Surface area} = \underbrace{\frac{3\sqrt{3}}{2} a^2}{\text{base}} + \underbrace{3 a s}{\text{six triangles}}$$
Each triangular face has area $\frac{1}{2} \times a \times s$, where $s$ is the slant height (apex to the midpoint of a base edge). Six of them give $6 \times \frac{1}{2} a s = 3as$, the lateral surface area.
Examples of Hexagonal Pyramid
Example 1
Find the volume of a regular hexagonal pyramid with base side $a = 6$ and height $h = 10$.
Base area $= \frac{3\sqrt{3}}{2} \times 6^2 = \frac{3\sqrt{3}}{2} \times 36 = 54\sqrt{3}$.
$$V = \frac{1}{3} \times 54\sqrt{3} \times 10 = 180\sqrt{3} \approx 311.8 \text{ cubic units}$$
Example 2
A hexagonal pyramid has volume $V = 200\sqrt{3}$ and base side $a = 4$. Find its height.
Wrong attempt first. A student reaches for the prism relation and writes $V = \text{base area} \times h$, then solves:
Base area $= \frac{3\sqrt{3}}{2} \times 16 = 24\sqrt{3}$, so $h = \frac{200\sqrt{3}}{24\sqrt{3}} \approx 8.3$.
That answer is too small, because it quietly dropped the $\frac{1}{3}$ that makes a pyramid a pyramid, not a prism.
Correct method. Start from $V = \frac{1}{3} \times \text{base area} \times h$:
$$200\sqrt{3} = \frac{1}{3} \times 24\sqrt{3} \times h = 8\sqrt{3}, h$$
$$h = \frac{200\sqrt{3}}{8\sqrt{3}} = 25 \text{ units}$$
The correct height is three times the wrong one, exactly as the missing factor predicts.
Example 3
Confirm the face, edge, and vertex count using Euler's formula.
With $F = 7$, $E = 12$, $V = 7$:
$$F - E + V = 7 - 12 + 7 = 2$$
The result is $2$, which every convex polyhedron must satisfy, so the count is valid.
Example 4
Find the total surface area of a regular hexagonal pyramid with base side $a = 4$ and slant height $s = 9$.
Base area $= \frac{3\sqrt{3}}{2} \times 16 = 24\sqrt{3} \approx 41.6$.
Lateral area $= 3 \times a \times s = 3 \times 4 \times 9 = 108$.
$$\text{Surface area} \approx 41.6 + 108 = 149.6 \text{ square units}$$
Example 5
A hexagonal pyramid has base side $a = 5$ and vertical height $h = 12$. Find its slant height $s$.
The slant height, the height, and the base's apothem form a right triangle. For a regular hexagon the apothem is $\frac{\sqrt{3}}{2} a = \frac{\sqrt{3}}{2} \times 5 \approx 4.33$.
$$s = \sqrt{h^2 + \text{apothem}^2} = \sqrt{12^2 + 4.33^2} = \sqrt{144 + 18.75} \approx 12.76 \text{ units}$$
The apothem is defined on the apothem page if you need the derivation.
Example 6
A tent is a regular hexagonal pyramid with base side $2$ m and height $3$ m. How much air does it enclose?
Base area $= \frac{3\sqrt{3}}{2} \times 2^2 = 6\sqrt{3} \approx 10.39$ m².
$$V = \frac{1}{3} \times 10.39 \times 3 \approx 10.39 \text{ m}^3$$
The tent holds roughly $10.4$ cubic metres of air.
Why Does the Hexagonal Pyramid Matter?
The value of a hexagonal pyramid is not the shape itself but the one-third rule it makes concrete: any pyramid holds a third of its matching prism. Once that clicks for a hexagon, it transfers to every base.
Roofing and spires. Six-sided turret roofs and pavilion tops are hexagonal pyramids; a builder sizing the covering material is computing lateral surface area.
Crystals and minerals. Quartz and beryl terminate in pyramid-like hexagonal points, so mineralogists use the same geometry to describe crystal habit.
Packaging and design. Faceted gift boxes, lamp shades, and pavilion canopies borrow the shape because six triangular panels fold neatly from one flat net.
The reason the $\frac{1}{3}$ factor exists at all traces back to how three congruent pyramids fill a prism, a result the ancient Greeks proved using Euclidean solid geometry. Understanding why the third appears is what stops it from being a formula you forget under pressure.
What Are the Most Common Mistakes With Hexagonal Pyramids?
Mistake 1: Dropping the one-third factor
Where it slips in: volume problems, especially right after studying prisms.
Don't do this: compute $V = \text{base area} \times h$ as if the pyramid were a prism.
The correct way: always include the pyramid factor, $V = \frac{1}{3} \times \text{base area} \times h$. The first instinct students reach for is the prism formula they just practised, and the missing third makes every answer exactly three times too large.
Mistake 2: Confusing slant height with vertical height
Where it slips in: surface-area problems that give the height instead of the slant height.
Don't do this: plug the vertical height $h$ straight into the lateral-area formula $3as$.
The correct way: the slant height $s$ runs along the sloping face, so recover it first with $s = \sqrt{h^2 + \text{apothem}^2}$. Using $h$ where $s$ belongs is the habit that a quick right-triangle sketch fixes on the spot.
Mistake 3: Miscounting edges and faces
Where it slips in: faces-edges-vertices questions under time pressure.
Don't do this: forget the six lateral edges and report $6$ edges instead of $12$.
The correct way: count both rings, the six base edges and the six edges climbing to the apex, for $12$ total.
Conclusion
A hexagonal pyramid has a hexagonal base and six triangular faces, giving 7 faces, 12 edges, and 7 vertices.
Its volume is $V = \frac{1}{3} \times \text{base area} \times h$, which becomes $\frac{\sqrt{3}}{2} a^2 h$ for a regular base.
Surface area is the hexagon base plus $3as$ for the six triangular faces.
The most common mistake is dropping the one-third factor and treating the pyramid like a prism.
Euler's formula, $F - E + V = 2$, confirms the shape's face, edge, and vertex count.
Practice These to Solidify Your Understanding
Work through three problems: find the volume of a regular hexagonal pyramid with $a = 8$, $h = 15$; find the slant height when $a = 6$ and $h = 8$; and confirm the surface area for $a = 3$, $s = 7$. If the one-third factor trips you up, return to the volume section above. To take three-dimensional geometry further with a teacher, explore Bhanzu's geometry tutor, a high school math tutor, or math classes online. Want a live trainer to walk through more solids? Book a free demo class.
Read More
Was this article helpful?
Your feedback helps us write better content