The total surface area of a hexagonal prism is the sum of the areas of its two hexagonal bases and its six rectangular side faces. Which means a prism is a three-dimensional solid with two identical parallel bases connected by rectangular or parallelogram faces. So to calculate its total surface area, you need the area of one base, the perimeter of the base, and the height of the prism. This article explains the formula, shows how to derive it, and provides worked examples for different types of hexagonal prisms.
Real talk — this step gets skipped all the time.
Understanding the Parts of a Hexagonal Prism
A hexagonal prism has three important measurements:
- Base side length: The length of each side of the hexagonal base.
- Base area: The area of one of the two congruent hexagonal bases.
- Prism height: The perpendicular distance between the two bases.
The height of a prism is not the same as its total length or slant length. In a right hexagonal prism, the side faces are rectangles, and the height is the vertical distance from one base to the other Simple, but easy to overlook..
A hexagonal prism contains:
- Two hexagonal bases
- Six lateral faces
- Eight vertices
- Twelve edges
Imagine wrapping a hexagonal prism with paper. The paper covers the two hexagonal ends and all six rectangular sides. The total amount of paper needed is the object’s total surface area Simple as that..
The General Formula
The total surface area of a prism is found by adding the areas of the two bases to the lateral surface area:
[ \text{Total Surface Area} = 2(\text{Area of One Base}) + \text{Lateral Surface Area} ]
For a right prism, the lateral surface area equals the perimeter of the base multiplied by the height:
[ \text{Lateral Surface Area} = \text{Base Perimeter} \times \text{Height} ]
That's why, the general formula for the total surface area of a hexagonal prism is:
[ \boxed{\text{TSA} = 2B + Ph} ]
Where:
- (\text{TSA}) represents the total surface area.
- (B) represents the area of one hexagonal base.
- (P) represents the perimeter of the base.
- (h) represents the height of the prism.
This formula is useful when the area and perimeter of the hexagonal base are already known.
Formula for a Regular Hexagonal Prism
A regular hexagonal prism has a regular hexagon as its base. In a regular hexagon, all six sides have the same length, usually represented by (a) And that's really what it comes down to..
The area of one regular hexagon is:
[ B = \frac{3\sqrt{3}}{2}a^2 ]
The perimeter of the hexagonal base is:
[ P = 6a ]
The lateral surface area consists of six rectangles, each with dimensions (a \times h):
[ \text{Lateral Surface Area} = 6ah ]
Substituting the base area and lateral surface area into the general formula gives:
[ \text{TSA} = 2\left(\frac{3\sqrt{3}}{2}a^2\right) + 6ah ]
After simplifying:
[ \boxed{\text{TSA of a regular hexagonal prism} = 3\sqrt{3}a^2 + 6ah} ]
This formula applies when the hexagonal base is regular and the prism is right.
Worked Example 1: Finding the Total Surface Area
A regular hexagonal prism has a base side length of (5) cm and a height of (10) cm. Find its total surface area.
Step 1: Find the area of one base
[ B = \frac{3\sqrt{3}}{2}(5)^2 ]
[ B = \frac{25\sqrt{3}}{2} = 12.5\sqrt{3} ]
Step 2: Find the area of both bases
[ 2B = 25\sqrt{3} ]
Step 3: Find the lateral surface area
The perimeter of the base is:
[ P = 6(5) = 30\text{ cm} ]
Therefore:
[ \text{Lateral Surface Area} = 30(
