5. The Area and Perimeter of Stars

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

uvsin(a)/2.

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

is

c2sin(a)cos(a)

or

c2sin(b)cos(b)

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

Definition 5.2: Let sj be the distance from Pj to Pj -1. sj-1 is the distance from Pj -1 to Pj -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(sj + sj-1).

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

2rcos(180k/n)sin(360/n)sec(180j/n)sec(180(j-2)/n).

Next we consider the area of the region.

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

sj2 sin(180j/n)cos(180j/n) + sj-12sin(180(j-2)/n)cos(180(j-2)/n)

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

sjsj-1 sin(360(j-1)/n)

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

2r2cos2(180k/n)sin2(180/n)sec(180j/n)tan(180(j-1)/n)sec(180(j-2)/n)

Note that s1 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 = sk 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 Pk 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 Po 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 Pk-1 points.

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

n[r2cos2(180k/n)sec2(180(k-1)/n)sin(180/n)cos(180/n) + s2sin(180k/n)cos(180k/n)]

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

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

n[r2sin(180/n)cos(180/n) - s2sin(180(k-1)/n)cos(180(k-1)/n)]

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

nr2sin(180/n)cos(180k/n)sec(180(k-1)/n)

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 ro, the perpendicular distance from the center of the figure to the Po points on each of the lines which make up the star. Recall that ro = 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

A = nsro

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 ro 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.

References