Calculate volume, lateral surface area, and total surface area of a regular or irregular hexagonal pyramid. Enter base side length, height, and apothem.
The Hexagonal Pyramid Volume and Surface Area Calculator determines every geometric property of a pyramid whose base is a regular hexagon. A regular hexagonal pyramid has a flat hexagonal base with six equal sides and an apex directly above the center of the base. These structures appear in crystallography, architecture, game design, packaging, and decorative arts. The base of a regular hexagon with side length a has area A_base = (3√3/2)a² and apothem ap = (√3/2)a. The volume of the pyramid is V = (1/3)·A_base·h, where h is the perpendicular height from base to apex. The slant height from the midpoint of a base edge to the apex is l = √(h² + ap²), and each lateral triangular face has area (1/2)·a·l. The total lateral area is 6·(1/2)·a·l = 3·a·l, and the total surface area adds the hexagonal base. This calculator supports both regular mode (equal side lengths) and an irregular mode where you can enter a custom base area and perimeter. Eight preset configurations let you explore pyramids of various proportions, from squat and broad to tall and slender. The formulas reference table collects all equations, and visual bars compare the base dimensions against the pyramid height.
A hexagonal pyramid combines polygon geometry with 3D pyramid formulas, so it is easy to lose track of the base area, apothem, slant height, and final surface area. This calculator keeps those dependent measurements connected, which is useful when comparing tall versus squat designs, estimating cladding material, or checking whether a custom base should be treated as regular or irregular.
Base area A = (3√3/2)a² (regular). Apothem ap = (√3/2)a. Volume V = (1/3)·A·h. Slant height l = √(h² + ap²). Lateral area = 3·a·l. Total SA = A + 3·a·l.
Result: Volume ≈ 124.7077, lateral area ≈ 115.7247, total surface area ≈ 157.2939
For a = 4, h = 9: A = (3√3/2)·16 ≈ 41.57. ap = (√3/2)·4 ≈ 3.464. V = (1/3)·41.57·9 ≈ 124.71. l = √(81 + 12) ≈ 9.644. Lateral area = 3·4·9.644 ≈ 115.73. Total SA ≈ 157.30.
For a regular hexagonal pyramid, the base is not just any six-sided polygon. Its area and apothem come from the symmetry of a regular hexagon, which can be split into six equilateral triangles. That structure is what makes formulas like $A = rac{3sqrt{3}}{2}a^2$ and $ap = rac{sqrt{3}}{2}a$ possible, and those values feed directly into the pyramid's volume and slant-height calculations.
The total surface area is easier to understand if you separate it into the hexagonal base and the six triangular side faces. A taller pyramid increases slant height, which increases the lateral area even when the base stays fixed. That distinction matters in design problems because material use often depends on the outside faces, while storage capacity depends on the enclosed volume.
Not every six-sided base is perfectly regular. If the base area and perimeter are known from a drawing or CAD model, irregular mode lets you skip the regular-hexagon assumptions and still compute volume and surface measures consistently. This is especially helpful for architectural caps, decorative structures, or fabricated parts where the footprint is hexagonal in outline but not mathematically regular.
A hexagonal pyramid is a 3D solid with a hexagonal base and six triangular faces meeting at an apex point above the base. Use this as a practical reminder before finalizing the result.
The apothem is the perpendicular distance from the center to the midpoint of any side: ap = (√3/2)·a for side length a. Keep this note short and outcome-focused for reuse.
Pyramid height (h) is the vertical distance from base to apex. Slant height (l) is measured along the face from a base edge midpoint to the apex.
Yes — switch to Irregular mode and enter the base area and perimeter manually. Apply this check where your workflow is most sensitive.
V = (1/3) × base area × height. This works for any pyramid shape, not just hexagonal.
Crystal structures, architectural roof cappings, board game dice shapes, decorative obelisks, and certain packaging designs. Use this checkpoint when values look unexpected.