Calculate volume, surface area, and more for a regular hexagonal prism using edge length and height.
\[ \text{Base Area} = \frac{3\sqrt{3}}{2} a^2 = \frac{3\sqrt{3}}{2} \times (4\,\text{cm})^2 \approx 41.57\,\text{cm}^2 \]
\[ \text{Volume} = \frac{3\sqrt{3}}{2} a^2 h = \frac{3\sqrt{3}}{2} \times (4\,\text{cm})^2 \times 10\,\text{cm} \approx 415.69\,\text{cm}^3 \]
\[ \text{Lateral Surface Area} = 6ah = 6 \times 4\,\text{cm} \times 10\,\text{cm} = 240\,\text{cm}^2 \]
\[ \text{Total Surface Area} = 6ah + 2 \times \frac{3\sqrt{3}}{2} a^2 = 240\,\text{cm}^2 + 2 \times 41.57\,\text{cm}^2 \approx 323.14\,\text{cm}^2 \]
A hexagonal prism is a three-dimensional geometric solid with:
The diagram shows a 3D representation of a hexagonal prism. The top hexagon is visible with edges labeled 'a'. The height 'h' is shown as the vertical distance between the top and bottom hexagons. The rectangular side faces connect corresponding vertices of the two hexagons.
\[ \text{Base Area} = \frac{3\sqrt{3}}{2} a^2 \]
Where:
\[ \text{Volume} = \text{Base Area} \times \text{Height} = \frac{3\sqrt{3}}{2} a^2 h \]
Concept: Volume measures how much space the prism occupies. For any prism, volume = base area × height.
\[ \text{Lateral Surface Area} = 6 \times a \times h \]
Concept: This is the area of the six rectangular sides only. Each rectangle has dimensions a (width) × h (height).
\[ \text{Total Surface Area} = \text{Lateral Area} + 2 \times \text{Base Area} \]
\[ = 6ah + 2 \times \frac{3\sqrt{3}}{2} a^2 = 6ah + 3\sqrt{3} a^2 \]
Concept: All surfaces combined: six rectangles + two hexagons.
Let's calculate for a hexagonal prism with edge length a = 4 cm and height h = 10 cm:
Base Area = (3 × √3 ÷ 2) × a²
= (3 × 1.73205 ÷ 2) × (4 cm)²
= (5.19615 ÷ 2) × 16 cm²
= 2.598075 × 16 cm² = 41.5692 cm²
Volume = Base Area × Height
= 41.5692 cm² × 10 cm = 415.692 cm³
Lateral Area = 6 × a × h
= 6 × 4 cm × 10 cm = 240 cm²
Total Area = Lateral Area + 2 × Base Area
= 240 cm² + 2 × 41.5692 cm²
= 240 cm² + 83.1384 cm² = 323.1384 cm²
These formulas only work for regular hexagonal prisms where all base edges are equal. For irregular hexagons, you must calculate area differently.
Never mix units! If edge is in cm and height in m, convert both to the same unit before calculating.
Total surface area includes BOTH hexagonal bases. Remember: 2 × base area, not just base area.
√3 ≈ 1.73205. Common errors: using 1.73 (too low precision), forgetting √3 entirely, or placing it incorrectly in the formula.
High School Geometry: Hexagonal prisms often appear in surface area and volume problems. Remember the base area formula is derived from equilateral triangles.
SAT/ACT Math: These tests may include prism problems. Focus on understanding that volume = base area × height works for ALL prisms.
Engineering/Physics: Hexagonal prisms appear in structural engineering (bolts, nuts, columns) and material science (crystal structures).
This tool is designed for educational purposes to help understand geometric concepts. While calculations are accurate for regular hexagonal prisms, real-world applications may require adjustments for material thickness, manufacturing tolerances, or irregular shapes. Always verify critical calculations with multiple methods in academic or professional settings.
| Quantity | Formula | Variables |
|---|---|---|
| Base Area (Ab) | \[ A_b = \frac{3\sqrt{3}}{2}a^2 \] | a = edge length |
| Volume (V) | \[ V = A_b \times h = \frac{3\sqrt{3}}{2}a^2h \] | h = height |
| Lateral Surface Area (LSA) | \[ LSA = 6ah \] | |
| Total Surface Area (TSA) | \[ TSA = 6ah + 3\sqrt{3}a^2 \] |
Remember: These formulas apply only to regular hexagonal prisms where all base edges are equal and the hexagon is regular.