As an application of the material developed in the previous section, we will compute the areas and perimeters of stars and the regions which they contain. First we will state a few well known results concerning the area of triangles in order that we may later be able to conveniently refer to them.

**Theorem 5.1a**:
(Well known) The area of the triangle in the figure

is

**Corollary 5.1b**:
The area of the isosceles triangle in the figure

is

or

We will first find the areas and perimeters of the quadrilateral regions.

**Definition 5.2**: Let s_{j} be the distance from
P_{j} to P_{j -1}. s_{j-1} is the distance
from P_{j -1} to P_{j -2}.

We first obtain formulas for the perimeter of these regions. The first result is obvious and is stated without proof.

**Theorem 5.3a**: The perimeter of the
quadrilateral region in the figure above is 2(s_{j} +
s_{j-1}).

**Theorem 5.3b**: The
perimeter of the quadrilateral region in the figure above is

Next we consider the area of the region.

**Theorem 5.4a**: The
area of the region in the figure above is

**Theorem 5.4b**: The
area of the region in the figure above is

**Theorem
5.4c**: The area of the region in the figure above is

Note that s_{1} is half of the length of an edge of the
central n-gon. Thus these last results also give us the area and
perimeter of the triangles we get when j = 2. The area and perimeter
formulas for the central n-gon will follow as a special case of the
formulas for general stars.

The area of the central n-gon is quite well known. As a result we can obtain the area of the entire star by simply adding up the areas of the regions.

**Theorem
5.5a**: The area of a {n/k} star is

**Theorem
5.5b**: The area of a {n/k} star is

**Theorem
5.5c**: The area of a {n/k} star is

For the remainder of the paper, we will be dealing with the entire
star. j will equal k, and s = s_{k} will denote the length of
each of the sides in the star.

**Theorem 5.6**: The
perimeter of a star is 2ns.

So we simply need to find s.

**Theorem
5.7a**: s = rcos(180k/n)[tan(180k/n) - tan(180(k-1)/n)]

This result gives us an indeterminate form in the case of the asterisk when k = n/2. One way to avoid the indeterminate form would be to simplify the formula as follows.

**Theorem
5.7b**: s = r[sin(180k/n) - cos(180k/n)tan(180(k-1)/n)]

There is a simpler form.

**Theorem
5.7c**: s = rsin(180/n)sec(180(k-1)/n)

In the case of the asterisk, we get that s = r in both Theorem 5.7b and 5.7c.

Notice that these formulas simplify to the perimeter for an n-gon
if k = 1. However, while Theorem 5.6 does not give
us indeterminate forms for the asterisk if we use the last two
formulas for s, they still do not quite give us the correct result.
If k = n/2, the formulas give us a result of 2nr while it looks like
a result of nr would be more appropriate. The reason for this is that
the formulas think that there are two rays forming the angle at the
P_{k} points, and in the asterisk it looks like there is only
one line from the point on the asterisk to the center of the circle.
It is not uncommon to get indeterminate forms when looking for a
point where two coincidental lines meet.

In the case of the discrete star it is important to realize that s
is the distance between a P_{o} point and itself which is 0.
As a result, the perimeter of a discrete star is 0.

There are many ways to approach the area of stars. The first
method which suggests itself after the last results is to add
triangles on to the n-gon generated by the P_{k-1} points.

**Theorem
5.8a**: The area of an {n/k} star inscribed in a circle of
radius r is

Another way of computing the area using the distance s would be to
chop triangles out of the n-gon generated by the P_{k}
points.

**Theorem
5.8b**: The area of an {n/k} star inscribed in a circle of
radius r is

**Theorem
5.8c**: The area of an {n/k} star inscribed in a circle of
radius r is

Now that we have established this last result, we can simplify it
by expressing it in terms of three very important invariants of the
star viz. n, the number of points, s, the length of each of the sides
of the star, and r_{o}, the perpendicular distance from the
center of the figure to the P_{o} points on each of the lines
which make up the star. Recall that r_{o} = rcos(180k/n) from
Theorem 4.2

The simplest formula for the area which we have found is the following.

**Theorem
5.8d **([3] p 38): The area
of an {n/k} star inscribed in a circle of radius r is

**Theorem
5.8e**: The area of an {n/k} star inscribed in a circle of
radius r is

where P is the perimeter of the star and r_{o} is the
apothem.

This is the famous formula for the area of a regular n-gon from Theorem 5.5 if k = 1. This generalizatioin is actually a special case of the well known fact that if a figure consists of lines which are all tangent to the same circle, then the area is one half the perimeter times the radius of the circle. All of the lines in a regular star polygon will be tangent to the inscribed circle of the central n-gon.

All of the formulas in Theorem 5.8 avoid indeterminate forms for the area of an asterisk, giving an answer of 0 in each case. The last two formula are the only ones which give the correct area of 0 for the discrete star.

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