Hexagonal Pyramid Calculator

Calculate the volume, surface area, and slant height of a regular hexagonal pyramid from base side and height.

Enter Pyramid Dimensions

Result

Volume
311.769145
cubic units
Base Side Length 6 units
Height 10 units
Slant Height 10.897247 units
Base Area 93.530744 sq units
Lateral Surface Area 196.150493 sq units
Total Surface Area 289.681237 sq units
Base Perimeter 36 units
Base Apothem 5.196152 units

Step-by-Step Solution

V = (sqrt(3)/2) x s^2 x h

What Is a Hexagonal Pyramid?

A hexagonal pyramid is a three-dimensional solid with a regular hexagonal base and six triangular faces that converge at a single apex point above the base. A regular hexagonal pyramid has a regular hexagon as its base and its apex is directly above the center of the base, making all six triangular faces congruent isosceles triangles.

The hexagonal pyramid has 7 faces (1 hexagonal base + 6 triangular), 12 edges (6 base edges + 6 lateral edges), and 7 vertices (6 base vertices + 1 apex).

Hexagonal Pyramid Formulas

Volume

One-third of the base area times the height.

V = (sqrt(3)/2) x s2 x h

Base Area

Area of the regular hexagonal base.

Abase = (3sqrt(3)/2) x s2

Slant Height

Distance from apex to the midpoint of a base edge.

l = sqrt(h2 + a2)

Lateral Surface Area

Total area of the six triangular faces.

Alat = 3 x s x l

Total Surface Area

Base area plus lateral surface area.

Atotal = Abase + Alat

Lateral Edge

Distance from apex to a base vertex.

e = sqrt(h2 + s2)

Understanding the Formulas

Volume Derivation

The volume of any pyramid is V = (1/3) x Base Area x Height. Since the base area of a regular hexagon is (3sqrt(3)/2) x s2, we get:

V = (1/3) x (3sqrt(3)/2) x s2 x h = (sqrt(3)/2) x s2 x h

Slant Height Explained

The slant height (l) is not the lateral edge. It is the distance from the apex perpendicular to a base edge, measured along the triangular face. The apothem of the hexagonal base (a = s x sqrt(3)/2) forms a right triangle with the pyramid height (h) and the slant height (l). Therefore: l = sqrt(h2 + a2).

Step-by-Step Example

Find the volume and surface area of a hexagonal pyramid with base side s = 6 and height h = 10:

  1. Base area: A = (3sqrt(3)/2) x 62 = (3 x 1.7321 / 2) x 36 = 93.5307
  2. Volume: V = (1/3) x 93.5307 x 10 = 311.7691
  3. Apothem: a = 6 x sqrt(3) / 2 = 5.1962
  4. Slant height: l = sqrt(102 + 5.19622) = sqrt(127) = 10.8972
  5. Lateral area: Alat = 3 x 6 x 10.8972 = 196.1505
  6. Total surface area: Atotal = 93.5307 + 196.1505 = 289.6812

Practical Applications

  • Architectural design for pyramid-shaped roofs and structures
  • Crystallography, where hexagonal pyramidal crystal habits occur
  • Packaging design for containers and funnels
  • Civil engineering for calculating material volumes in hexagonal piers and foundations