Volume of a Hexagonal Pyramid Calculator

Enter the base edge length and height of a regular hexagonal pyramid to get its volume (V = (√3/2)a²h), base area, and surface area.

Quick Facts

Volume formula
V = (√3/2) × a² × h
Equivalent to (1/3) × base area × height, the general pyramid volume rule.
Base area formula
A = (3√3/2) × a² ≈ 2.598a²
Area of a regular hexagon with edge length a.
Apothem
m = (√3/2) × a
Distance from the hexagon's center to the midpoint of a side.
Slant height
l = √(h² + m²)
Used to find lateral and total surface area.

Your Results

Calculated
Volume
-
V = (√3/2)a²h
Hexagonal Base Area
-
A = (3√3/2)a²
Lateral Surface Area
-
6 triangular faces: 3 × a × slant height
Total Surface Area
-
Base area + lateral surface area

Ready

Enter a base edge length and height, then press Calculate.

Formula and Method for the Volume of a Hexagonal Pyramid

A hexagonal pyramid has a regular hexagon for its base and six triangular faces that meet at a single apex directly above the base's center. Like every pyramid, its volume is one-third of the base area times the height: V = (1/3) × A × h. Since the area of a regular hexagon with edge length a is A = (3√3/2)a², substituting gives the combined formula V = (√3/2) × a² × h ≈ 0.8660 × a² × h. This calculator also reports the base area, lateral surface area, and total surface area from the same two inputs.

How the calculation works

Enter the base edge length (a) and the vertical pyramid height (h) — the perpendicular distance from the apex down to the center of the hexagonal base — in the same unit. The calculator first finds the base area using A = (3√3/2)a², then multiplies by the height and divides by 3 to get the volume. For surface area, it computes the apothem m = (√3/2)a (the distance from the hexagon's center to the midpoint of a side), then the slant height l = √(h² + m²) — the distance from the apex to the midpoint of a base edge along a triangular face. The lateral surface area is the combined area of the six triangular faces, 3 × a × l, and the total surface area adds the base: A_total = A_base + 3al.

Common mistakes

  • Height vs. slant height: the volume formula needs the perpendicular height h, not the slant height l along a face. Using slant height in place of h overstates the volume.
  • Forgetting the 1/3 factor: a hexagonal prism with the same base and height has volume A × h; a pyramid holds exactly one-third of that, since the cross-section shrinks linearly to a point at the apex.
  • Edge length vs. apothem or circumradius: for a regular hexagon, the edge length equals the circumradius (center to vertex) but is different from the apothem (center to edge midpoint, m = a√3/2 ≈ 0.866a). Using the wrong one skews the base area.

Real-world applications

  • Architectural roofs, gazebos, and skylights with a hexagonal footprint use this volume for material and enclosed-space estimates.
  • Packaging and container design uses hexagonal pyramid volume for tapered caps, funnels, and hopper sections.
  • Crystallography and geology reference hexagonal pyramid geometry for crystal habit and mineral volume estimates.
  • Engineering drawings use the lateral and total surface area for cladding, sheet-metal, or paint-coverage calculations.

Frequently Asked Questions

What is the formula for the volume of a hexagonal pyramid?
The volume of any pyramid is one-third times the base area times the height: V = (1/3) × A × h. For a regular hexagonal pyramid with base edge length a, the base area is A = (3√3/2)a², so the volume simplifies to V = (√3/2) × a² × h ≈ 0.8660 × a² × h.
How do you find the base area of a regular hexagonal pyramid?
The base is a regular hexagon with six equal sides of length a. Its area is A = (3√3/2)a² ≈ 2.5981 × a². For example, a hexagon with 6-unit sides has a base area of about 2.5981 × 36 ≈ 93.53 square units.
What is the difference between the pyramid's height and its slant height?
The height h is the perpendicular distance from the apex straight down to the center of the base and is what the volume formula uses. The slant height l is measured along a triangular face, from the apex to the midpoint of a base edge, and is found with l = √(h² + m²), where m = (√3/2)a is the hexagon's apothem. Slant height is used to find the lateral and total surface area, not the volume.

Related calculators