How to Find the Volume and Surface Area of a Hexagonal Prism?
The concept of hexagonal prism plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios.
What Is Hexagonal Prism?
A hexagonal prism is a 3D solid shape with two parallel, identical hexagonal bases and six rectangular side faces connecting them. This prism has 8 faces (2 hexagonal and 6 rectangular), 18 edges, and 12 vertices. You’ll find this concept applied in geometry, solid shapes, and areas such as polyhedra and nets.
Faces, Edges, and Vertices of Hexagonal Prism
| Property | Count | Description |
|---|---|---|
| Faces | 8 | 2 hexagonal (bases), 6 rectangular (sides) |
| Edges | 18 | Sides forming the polygons and joins |
| Vertices | 12 | Corner points where edges meet |
Key Formula for Hexagonal Prism
Here are the main formulas students need to remember while studying hexagonal prisms:
- Volume: \( V = \frac{3\sqrt{3}}{2} a^2 h \), where a = side of hexagon, h = height
- Surface Area (Total): \( SA = 6ah + 3\sqrt{3}a^2 \)
- Base Area: \( A_{base} = \frac{3\sqrt{3}}{2} a^2 \)
Step-by-Step Illustration
- To find the volume of a hexagonal prism with base edge 6 cm and height 12 cm:
Use the formula:
\( V = \frac{3\sqrt{3}}{2} a^2 h \ ) - Substitute values:
\( a = 6 \) cm, \( h = 12 \) cm - Calculate:
\( V = \frac{3\sqrt{3}}{2} \times (6)^2 \times 12 \) - Step by step:
\( (6)^2 = 36 \), so \( \frac{3\sqrt{3}}{2} \times 36 \times 12 \) - Final solution:
\( V \approx 3 \times 1.732 \times 18 \times 12 \approx 1122\, cm^3 \)
Net and Drawing of a Hexagonal Prism
The net of a hexagonal prism is what the 3D prism would look like if unfolded into a flat 2D shape. It has two hexagons (for bases) and six attached rectangles (for sides). Drawing a net helps visual learners understand the relationship between faces and edges and supports solid geometry problems in exams.
Real-Life Examples of Hexagonal Prism
- Pencil before sharpening (hexagonal body)
- Beehive cells (natural hexagonal prisms)
- Hexagonal nuts and bolts
- Some weights and scientific glassware
Speed Trick or Vedic Shortcut
To quickly calculate volume or surface area, first memorize the core hexagon formula (\( \frac{3\sqrt{3}}{2} a^2 \)). Substitute directly and multiply by the prism’s height for volume, or use \( 6ah \) for sides. Vedantu live sessions share such speed tricks to save time during exams.
Try These Yourself
- Find the surface area of a hexagonal prism with base edge 4 cm and height 10 cm.
- How many edges and vertices does a hexagonal prism have?
- List 2 objects around you shaped like a hexagonal prism.
- Sketch a net for a hexagonal prism on paper.
Frequent Errors and Misunderstandings
- Counting faces: Many confuse the number of faces or mix-up with vertices.
- Mistaking base area: Forgetting to use the correct formula for regular hexagon base.
- Missing the 3D aspect: Sometimes drawing only the base, not the full prism.
Relation to Other Concepts
Understanding hexagonal prisms helps with other solid shapes like cuboids, prisms of other polygons, and the general study of 3D shapes in geometry. It connects with nets, polyhedra, and Euler’s formula as discussed in polyhedron topics.
Classroom Tip
A simple way to remember: Every prism gets its name from the base shape (here, a hexagon), and has two bases and sides connecting them. Count “how many sides in the base” for quick total face calculation. Vedantu’s teachers often model prisms using real stationary like pencils and blocks for easy memory tricks.
We explored hexagonal prism — from definition, faces/edges/vertices, formulas, calculation methods, real-life examples, and classroom shortcuts. Keep practicing with Vedantu for confidence in geometry!
Explore More on Geometry and Prisms
FAQs on Hexagonal Prism – Definition, Properties, and Formulas
1. What is a hexagonal prism?
A hexagonal prism is a three-dimensional geometric shape with two parallel hexagonal bases and six rectangular lateral faces connecting the bases. It's a type of prism, a polyhedron with two congruent and parallel bases. Think of it like a stack of hexagons.
2. How many faces, edges, and vertices does a hexagonal prism have?
A hexagonal prism has a total of 8 faces (2 hexagonal bases and 6 rectangular lateral faces), 18 edges, and 12 vertices.
3. What is the formula for the surface area of a hexagonal prism?
The surface area of a hexagonal prism is calculated using the formula: Surface Area = 2 * (3√3/2 * a²) + 6 * a * h, where 'a' is the length of a side of the hexagonal base and 'h' is the height of the prism. This accounts for the area of the two hexagonal bases and the six rectangular sides.
4. What is the formula for the volume of a hexagonal prism?
The volume of a hexagonal prism is given by the formula: Volume = (3√3/2) * a² * h, where 'a' represents the side length of the hexagonal base and 'h' is the prism's height. This formula essentially multiplies the area of the hexagonal base by the prism's height.
5. How do you find the base area of a hexagonal prism?
The base area of a hexagonal prism is the area of one of its hexagonal bases. For a regular hexagon (where all sides are equal), the formula is: Base Area = (3√3/2) * a², with 'a' being the length of one side of the hexagon. For irregular hexagons, the area calculation is more complex and may require breaking the hexagon into smaller shapes.
6. What is the net of a hexagonal prism?
A net of a hexagonal prism is a two-dimensional representation showing how the faces of the 3D shape would appear if unfolded. It consists of two hexagons (the bases) and six rectangles arranged to form the lateral faces when folded. Different arrangements are possible.
7. What are some real-world examples of hexagonal prisms?
Many everyday objects approximate the shape of a hexagonal prism. Some examples include pencils (before sharpening), some types of nuts and bolts, honeycomb structures, and certain crystals.
8. What is a regular hexagonal prism?
A regular hexagonal prism is a hexagonal prism where the hexagonal bases are regular hexagons (all sides and angles are equal) and the lateral faces are perpendicular to the bases, forming right angles.
9. What is an oblique hexagonal prism?
An oblique hexagonal prism differs from a regular one because its lateral faces are not perpendicular to the bases. This means the lateral edges are not at right angles to the base, resulting in a slanted or tilted prism.
10. How can I calculate the lateral surface area of a hexagonal prism?
The lateral surface area is the total area of the six rectangular lateral faces. It's calculated as: Lateral Surface Area = 6 * a * h, where 'a' is the side length of the hexagonal base and 'h' is the prism's height.
11. What are the differences between a hexagonal prism and a hexagonal pyramid?
A hexagonal prism has two parallel hexagonal bases connected by rectangular faces. A hexagonal pyramid has only one hexagonal base and triangular lateral faces meeting at a single apex (point) above the base.
12. How is Euler's formula related to a hexagonal prism?
Euler's formula (V - E + F = 2), relating vertices (V), edges (E), and faces (F) of a polyhedron, holds true for a hexagonal prism. If you count the vertices, edges, and faces of a hexagonal prism and apply the formula, you'll find it equals 2, confirming its validity for this shape.
