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Basic Geometry: Help & Review16 chapters | 109 lessons

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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.

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.

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 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.

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

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

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.

If we cut the 72o angle in half with a vertical line, we've constructed a 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) 3*a*. 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.

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) 3
*a*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:

A = area of the sides + 2(area of a pentagon) because there are 2 pentagons.

A = 240 + 2(10)(area of a small triangle) because there are 10 small triangles in one pentagon

A = 240 + 2(10)(1/2)(3)*a* because the area of one small triangle is (1/2)base x height = (1/2)(3)*a*

A = 240 + 2(10)(1/2)(3)(3) / tan(36o), using *a* = 3 / tan(36o)

We can simplify this: 2(10)(1/2)(3)(3) = 10(3)(3) = 90.

Thus, the total surface area A = 240 + 90 / tan(36o).

What about the tan(36o)? Using a calculator, tan(36o) â‰… .727; so, A = 240 + 90 / .727 â‰… 364 square inches. Richard now has the total surface area of the pentagonal prism.

What if other prisms are to be constructed with different dimensions? It would be nice to have a surface area equation that depended on the number of sides (*n*), the height (*h*), and the length of a side (*s*).

Remember the 72o? That 72o comes from 360o / 5 when *n* is 5, so 72o is 360o / *n*. In the equation A = 240 + 10(3)(3) / tan(36o), we have 36o, which is half of 360o / *n*, or 180o / *n*.

Other items:

- The 240 is 6(8)(5), or in terms of letters, 240 is
*s*(*h*)(*n*) - The 10(3)(3) is 2
*n*(*s*/2)(*s*/2), which is (*n*/2)*s*2.

Thus, a general equation for the total surface area is:

A = *s*(*h*)(*n*) + (*n*/2) *s*2 / tan(180o/*n*).

Checking our general formula:

A = *s*(*h*)(*n*) + (*n*/2) s2 / tan(180o/*n*)

= 6(8)(5) + (5/2) 62 / tan(180o/5)

â‰… 240 + 123.874

â‰… 364 square inches, which agrees with our earlier calculation.

A **prism** is a solid object with the same ends and with flat sides. The height of the prism is *h*. In this lesson, we find a general formula for the surface area of a **pentagonal prism**. This is a prism where the ends are regular pentagons. Regular pentagons have 5 sides of equal length. In developing the general equation for the surface area, we use the apothem. The **apothem** is the length of a line from the center of the pentagon to the side. This line bisects the side and creates a right-angle with the side. The general equation for the surface area A of the pentagonal prism is:

A = *s*(*h*)(*n*) + (*n*/2) *s*2 / tan(180o / *n*) where *s* is the length of the side of the prism, *h* is the height of the prism, and *n* is the number of sides.

For a pentagonal prism, *n* = 5, because pentagons have 5 sides.

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Basic Geometry: Help & Review16 chapters | 109 lessons

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