Surface Area of a Hexagonal Prism

Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

In this lesson, we will develop the equation for the surface area of a hexagonal prism and show how this relates to wax production in beehives. Using the equation, we will calculate the total surface area of a typical beehive cell.

Surface Area of a Hexagonal Prism

Honey storage in a beehive is in cells with a particular shape. These ''honey jars'' are built with wax produced by the bees themselves. As an avid candle maker, Fred wonders how much wax covers a typical cell.


A honeycomb
A_honeycomb


Each cell resembles a hexagonal prism. A prism is a solid figure with the same shape at the ends, and the side surfaces are parallel. In this lesson, we focus on a hexagonal prism having a regular hexagon (six equal sides) for each end.


A regular hexagon
A_regular_hexagon


This lesson's objective is understanding how the hexagonal prism area equation is obtained. Also, on the list is fueling Fred's ideas.

The Area of an Equilateral Triangle

Remember the special triangle called the 30-60-90 triangle? The leg:hypotenuse:leg ratio is 1:2:√3 where the first leg is the side opposite the 30o angle.


30-60-90 triangle
30_60_90_triangle


Having the ratio means we can extend this idea to other 30-60-90 triangles. What if the side opposite the 30o angle is not 1 but S/2. The letter ''S'' stands for ''side''. To get S/2 from 1, we multiply 1 by S/2. So, we multiply each number in the ratio by S/2. From 1:2:√3 we get 1(S/2):2(S/2):√3(S/2). The 2(S/2) is just S. The new triangle is still a 30-60-90 triangle, but the leg:hypotenuse:leg ratio is S/2:S:√3(S/2). Check out the triangle with the S's in it.


Triangle with S in it
triangle_with_Ss


Now, we put two of these triangles together back-to-back. The bottom side is S/2 + S/2 = S. This is the base of the triangle. We still have the √3(S/2), and we recognize it as the height of the triangle. The top angle is 30o + 30o = 60o. Hey, all three sides are equal to S and the interior angles are equal to 60o.


Not only is this an equilateral triangle , but we are really close to having the area.

In general, the area of a triangle is (1/2) base X height. Thus, for the equilateral triangle, the area

= (1/2) base X height

= (1/2) S(√3)(S/2); the base is S and the height is √3(S/2)

= (√3 S2)/4; the multiplying 2's in the denominator become 4

Fred is getting somewhat impatient as night approaches and he still does not have a candle to read by. We're almost there, Fred.

The Area of the Hexagon

Looking at the hexagon with lines from the vertices passing through the center, provides interesting observations.


Equilateral triangles in a hexagon
equilateral_triangles_in_a_hexagon


The 360o about the center is divided by 6 to give 360o /6 = 60o. We have our equilateral triangle again. In fact, a hexagon has 6 equilateral triangles in it.

We just figured out the area of one equilateral triangle as (√3 S2)/)/4. The hexagon has 6 equilateral triangles in it. Thus, the area of the hexagon is

6(√3 S2)/)/4 which simplifies to

3(√3 S2)/)/2.

Okay, time for some illumination on Fred's concerns.

How Much Wax

A sealed honeycomb cell has a hexagon for the two ends. The side surfaces have a height h and a width S. Thus, the total surface area A of this hexagonal prism is

6hS + 3(√3 S2)/2 + 3(√3 S2)/2 which simplifies to

6hS + 3√3 S2 because 3/2 + 3/2 = 6/2 = 3.


One hexagonal prism in the beehive
One_hexagonal_prism_in_the_beehive


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