Surface Area of a Pentagonal Prism

Surface Area of a Pentagonal Prism
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  • 0:04 Surface Area of a…
  • 1:00 The Area of the Prism's Sides
  • 1:39 Area of the Pentagons
  • 3:00 Total Surface Area of…
  • 4:46 A General Equation
  • 6:26 Lesson Summary
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Lesson Transcript
Instructor: Gerald Lemay

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

In this lesson, we determine the surface area of a pentagonal prism. By carefully calculating the surface area of a prism whose dimensions are given, you will understand how a general equation is derived.

Surface Area of a Pentagonal Prism

Your friend Richard needs some help - he wants to know how to find the surface area of a pentagonal prism. Before we go any further, let's review some key terms:

  • A prism is a solid object with the same ends and with flat sides.
  • If the ends are a 5-sided figure where each length is the same, we have a regular pentagon.
  • A prism that has regular pentagons for the end pieces is a pentagonal prism.

You realize pretty quickly that you might need a foldable to help Richard visualize all the parts of a pentagonal prism. Luckily, you happen to have this diagram on your desk.

Paper layout

You explain to Richard that the little tabs are left for taping the folded sides together. The first effort produces a reasonable 5-sided prism. With your 3D model in hand, Richard asks again how to calculate its total surface area. You explain that the details of this area computation are in finding the pentagon surface areas.

The Area of the Prism's Sides

The height of your model prism will be represented by the letter h, where h = 8 inches. When this prism is laid out we see h as the width of a rectangle.

There are five sides represented by the letter s, each of which has a length 6 inches (s = 6).

The total length of the rectangle is 5 x 6 = 30 inches; thus, the area of the sides of the prism is (length) x (width) = 30 x 8 = 240 square inches.

So far, all we've used is the formula for the area of a rectangle and some reasoning.

Layout for area calculation

Now comes the fun part; the area of the top and bottom surfaces.

Area of the Pentagons

The top and bottom surfaces are pentagons. Each of the sides of these pentagons has a length of 6 inches.

The end pentagon

If we draw a line from the center of the pentagon to the two vertices, we form a triangle whose base is 6 inches. There are five triangles like this in the pentagon. Thus, the subtended angle we see in the figure is 360o / 5 = 72o.

Let's play with one of these triangles.

One of the six triangles

If we cut the 72o angle in half with a vertical line, we've constructed a right-angled triangle.

The right-angled triangle

The base is 6 / 2 = 3 inches, and the angle at the top is (72o) / 2 = 36o.

If we knew the length of a, we could calculate the area of this triangle as (1/2) base x height = (1/2) 3a. There are 10 of these smaller triangles in the pentagon. Thus, the total area of the top pentagon would be 10 times this amount.

This is cool, but we know the side s; we don't know a. Enter fundamental trigonometry! The tangent of an angle is the opposite side divided by the adjacent side. So:

tan(36o) = 3 / a

And solving for a:

a = 3 / tan(36o)

We now have all the pieces to answer Richard's question.

Total Surface Area of the Prism

Let's review what we have so far:

  • Area of the side surfaces = 240 square inches
  • Area of a smaller triangle in the pentagon = (1/2) 3a square inches
  • The length a = 3 / tan(36o) inches
  • There are 10 of these smaller triangles in one pentagon
  • There are two pentagons (upper and lower) surfaces

Note the side a has a technical term - it's called the apothem.

The total area A is given by:

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