Finding The Area Of A Pentagon Using Trigonometry

A regular pentagon, or any regular shape,

is one where all of the interior angles

have the same measure.

If you were on a pentagon shaped race track, to complete one lap, you would turn a total of 360 degrees.

These turns would each be the same angle.

Each would be 360 divided by 5 equals 72 degrees.

This 72 degrees is also the measure of an angle at each corner like so:

If the angle on the outside of a corner is 72 degrees,

then the angle on the inside is 180 - 72 = 108 degrees.

So for the whole pentagon we have:

Now slice the pentagon like a pizza from each corner to the center.

These slices cut our 108 degree angles in half.

Now slice the pentagon pizza from the center of each side

to the middle of the pentagon.

(these slice lines make right angles with the sides)

Now we have a pentagon pizza with ten right triangle slices.

In each triangle slice, two of the angles are 90 and 54,

so the third angle is 180 - 90 - 54 = 36.

(you might notice that the third angle measure (36) was half of the outside angle (72))

OK, say we have a regular pentagon with side lengths of 10 units.

That makes the crust/side edge of each slice 5 units.

Notice that for a pentagon of any size,

the angles will always have the same measure.

Let's look at one of these pizza slice right triangles.

We'll name the unknown side lengths "a" and "c".

From the point of view of the 54 degree angle, side "a" is the opposite side.

The crust side of length 5 is called the adjacent side.

(Side "c" is the hypotenuse.)

Trigonometry starts out by calculating the ratios

of the lengths of the sides of right triangles.

The trig ratio called the tangent is the ratio of

the side opposite an angle to the side adjacent the angle.

In equation talk for this triangle we have:

We can look up the tangent of 54 degrees in a book or get it from a calculator.

It is about 1.3764. Putting that in our equation we get:

Put that number into our right triangle pizza slice and we get:

The area of a triangle is one half times the base times the height, so:

There are ten of these right triangle pizza slices in our pentagon pizza,

so the area of the pentagon is:

Example:

Find the area of a regular pentagon with a side length of 14.

Slice it all up ...

Look at one slice ...

Put that value in the triangle ...

Caluclate the triangle area ...

Calculate the pentagon area ...

This strategy works with regular shapes of any number of sides,

you will just have different angle measures and number of triangles.