# 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.

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.

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.

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.

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.

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.

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