Hexagonal Prism: Faces, Volume, and Surface Area
Hexagonal Prism: Faces, Volume, and Surface Area
A hexagonal prism is a 3-D solid with two parallel hexagonal bases joined by six rectangles, giving 8 faces, 18 edges, and 12 vertices. This article defines the hexagonal prism, derives its volume and surface area formulas, shows its net, and works through examples.
The Shape A Pencil Keeps So it Will Not Roll Off the Desk
A hexagonal prism is a three-dimensional solid with two identical, parallel hexagonal bases connected by six rectangular side faces. It belongs to the wider prisms family.
Faces, Edges, And Vertices
Counting the parts of a hexagonal prism:
- Faces: 8. Two hexagonal bases plus six rectangular sides.
- Vertices: 12. Each hexagon has 6 corners, and there are two hexagons.
- Edges: 18. Each hexagon contributes 6 edges, and 6 more vertical edges join the top corners to the bottom corners.
These satisfy Euler's formula for solids: ( V - E + F = 2 ) gives ( 12 - 18 + 8 = 2 ).
The Volume Formula And Where It Comes From
Every prism follows the same rule: volume equals base area times height, where the base area of a regular hexagon of side ( a ) is ( \frac{3\sqrt{3}}{2} a^2 ). Thus, the volume ( V = base\ area \times height ) gives:
[ V = \frac{3\sqrt{3}}{2} a^2 h ]
The Surface Area Formula
Surface area is the total of every face. It contributes:
- Two bases: ( 2 \times \frac{3\sqrt{3}}{2} a^2 = 3\sqrt{3} a^2 )
- Lateral surface area: ( 6ah )
Total surface area:
[ SA = 3\sqrt{3} a^2 + 6ah ]
Examples Of Hexagonal Prism
Example 1
Count the faces, edges, and vertices of a hexagonal prism.
- Faces: ( 2 + 6 = 8 )
- Vertices: ( 6 \times 2 = 12 )
- Edges: ( 12 + 6 = 18 )
Example 2
A regular hexagonal prism has base edge ( a=4 ) cm and height ( h=10 ) cm. Find its volume:
[ V = \frac{3\sqrt{3}}{2}(4)^2(10) \approx 415.7 \text{ cm}^3 ]
Example 3
Spot the error: Height and base edge cannot be treated as equal. Correctly using height gives:
[ V \approx 415.7 \text{ cm}^3 ]
Example 4
Find the total surface area with ( a=4 ) cm and ( h=10 ) cm:
[ SA = 83.1 + 240 = 323.1 \text{ cm}^2 ]
Example 5
Find only the lateral surface area with base edge 5 cm and height 12 cm:
[ Lateral \ SA = 6(5)(12) = 360 \text{ cm}^2 ]
Example 6
Estimate volume of a hexagonal pencil with base edge 3 mm and height 170 mm:
[ V \approx 3975 \text{ mm}^3 ]
Key Takeaways
- A hexagonal prism has two parallel hexagonal bases joined by six rectangles.
- It has 8 faces, 18 edges, and 12 vertices, satisfying ( V - E + F = 2 ).
- Volume is ( V = \frac{3\sqrt{3}}{2} a^2 h ).
- Surface area is ( SA = 3\sqrt{3} a^2 + 6ah ).
A practical next step
Practice these problems to solidify your understanding.