Hexagonal Pyramid Calculator
Calculate hexagonal pyramid volume, total surface area, lateral area, base area, slant height, edge length, and angles with 3D projection preview.
What is a Hexagonal Pyramid?
A regular hexagonal pyramid is a three-dimensional geometric polyhedron composed of a regular hexagonal base and six congruent isosceles triangle lateral faces that meet at a single shared vertex called the apex. It has \(7\) vertices (\(6\) at the base plus \(1\) apex), \(12\) edges (\(6\) base edges and \(6\) lateral edges), and \(7\) faces (\(1\) base and \(6\) triangular sides).
Key Geometric Formulas for a Regular Hexagonal Pyramid
Given a hexagonal pyramid with regular base edge length \(a\) and vertical height \(h\):
| Property | Formula | Description |
|---|---|---|
| Base Area (\(A_b\)) | \(A_b = \frac{3\sqrt{3}}{2} a^2\) | \(\approx 2.598076 \times a^2\) |
| Volume (\(V\)) | \(V = \frac{1}{3} A_b h = \frac{\sqrt{3}}{2} a^2 h\) | \(\approx 0.866025 \times a^2 \times h\) |
| Base Apothem (\(r_b\)) | \(r_b = \frac{\sqrt{3}}{2} a\) | Perpendicular distance from base center to edge |
| Slant Height (\(s\)) | \(s = \sqrt{h^2 + r_b^2} = \sqrt{h^2 + \frac{3}{4}a^2}\) | Height of each lateral triangular face |
| Lateral Edge (\(e\)) | \(e = \sqrt{h^2 + a^2}\) | Length of edge connecting base vertex to apex |
| Lateral Surface Area (\(A_l\)) | \(A_l = 3 a s\) | Sum of the 6 triangular lateral faces |
| Total Surface Area (\(A\)) | \(A = A_b + A_l = \frac{3\sqrt{3}}{2} a^2 + 3 a s\) | Base area plus lateral area |
Step-by-Step Example Calculation
Suppose a hexagonal pyramid has a base side length \(a = 4\text{ cm}\) and a vertical height \(h = 6\text{ cm}\).
- Calculate base area: $$A_b = \frac{3\sqrt{3}}{2} \times 4^2 = 24\sqrt{3} \approx 41.5692\text{ cm}^2$$
- Calculate volume: $$V = \frac{1}{3} \times A_b \times h = \frac{1}{3} \times 41.5692 \times 6 = 83.1384\text{ cm}^3$$
- Calculate base apothem: $$r_b = \frac{\sqrt{3}}{2} \times 4 = 2\sqrt{3} \approx 3.4641\text{ cm}$$
- Calculate slant height: $$s = \sqrt{6^2 + (2\sqrt{3})^2} = \sqrt{36 + 12} = \sqrt{48} = 4\sqrt{3} \approx 6.9282\text{ cm}$$
- Calculate total surface area: $$A = A_b + 3 a s = 24\sqrt{3} + 48\sqrt{3} = 72\sqrt{3} \approx 124.7077\text{ cm}^2$$
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Frequently Asked Questions
What is the difference between slant height and lateral edge?
Slant height (\(s\)) is the altitude of a lateral triangular face, measured along the surface from the midpoint of a base side straight to the apex. Lateral edge (\(e\)) is the corner line connecting a base vertex to the apex. In any pyramid, the lateral edge is always longer than the slant height because \(e = \sqrt{s^2 + (a/2)^2}\).
How do you find the volume of an oblique hexagonal pyramid?
According to Cavalieri's Principle, the volume of any pyramid (regular or oblique) depends only on the base area and perpendicular height: \(V = \frac{1}{3} A_b h\). If the base is a regular hexagon of side \(a\), the volume remains \(V = \frac{\sqrt{3}}{2} a^2 h\), where \(h\) is the perpendicular vertical distance from the apex to the plane containing the base.
How many faces, edges, and vertices does a hexagonal pyramid have?
A hexagonal pyramid has 7 faces (\(1\) hexagonal base + \(6\) triangular faces), 12 edges (\(6\) base edges + \(6\) lateral edges), and 7 vertices (\(6\) base vertices + \(1\) apex). This satisfies Euler's formula for polyhedra: \(V - E + F = 7 - 12 + 7 = 2\).
What is the dihedral angle of a hexagonal pyramid?
The dihedral angle between the hexagonal base and any lateral face is given by \(\theta = \arctan(h / r_b) = \arctan\left(\frac{2h}{\sqrt{3}a}\right)\), where \(r_b\) is the base apothem.
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