Determining Sectors, Segments, Apothems, Chords, Arcs, etc.
(Based on the Central Angle and Radius)

A sector with a central angle of 45-degrees, in a circle, will  have the shape of a piece of pie and will be 1/8 of that circle (for 360 divided by 45 equals 8).  If you would draw a line between the points where the two radii meet the arc, that line is called a "chord"; and below it would be a triangle, while above it would be the "segment" that is between the chord and the arc.  Therefore, to determine the area of a segment, you need to merely subtract the area of the triangle from the area of the sector (which we will show below).  

As we think about that triangle in the sector, we note its symmetry; for its radii are of the same length.  That type of triangle is called an "isosceles triangle."   It doesn't have the advantage of having a right angle in it; but we can convert it into two right triangles by dividing it into two equal halves.  Then we'll be able to use the trig functions of the sine and tangent of the angle to determine side lengths.  And since both of these right triangles will be exact in angles and proportions, then we'll need to figure for only one; and can use our findings for the other, as well.

If you imagine that right triangle with the vertex of its angle in the center (of the circle), then its "opposite" side will be directly across from it; and since the right triangle is half of the original triangle, then that "opposite" side will also be 1/2 of the "chord" (which is the width of the triangle).

In addition, the right "leg," which stems from the vertex of the central angle to the vertex of the right angle, is the "adjacent" side of the triangle and corresponds with the "apothem."  By it, we'll determine the height of the triangle.   The "adjacent" or "apothem" does not go all the way to the arc, as the radius does; but just to the "chord."

The left "leg," which stems from the vertex of the central angle, goes all the way to the arc; so it is the radius of the circle.  And in our right triangle, it is also the "hypotenuse" (which is always the longest side) of the right triangle. 

We are actually going to build from that knowledge of the length of the radius and the degree of the angle in order to determine all the lengths and areas for these various parts.

Another one we'll be able to easily figure is the height of the segment.  For as you can probably guess, if you don't already know, we can determine the highest point in the segment by subtracting the "adjacent" (the "apothem") from the radius.

We will also be able to determine the arc's length, the chord's length, and the perimeters of the sector, triangle, and segment.  And even if we think of this sector as being removed from its circle -- like a piece of pie taken from out of the pie pan -- we can still determine the circumference, the diameter, the radius, and the area of that circle that the sector was in.

So by just starting out with the length of the radius and the degree of the angle, we'll be able to determine all these  things.

With that in mind, let's begin looking now at some formulas that will help us to do this.

First of all, if we merely want to know the area of a segment, we could determine it this way:

For the area of the sector:
Multiply PI * Radius^2 * the angle.  Then divide by 360

For the area of the triangle:

Multiply Radius^2 * the sine of the angle.  Then divide by 2

Then just use the following formula:
Sector - Triangle = Segment

If, for instance,  the central angle were 68 and the radius 12, then the following would be so:

85.451 (area of the sector)  (85.4513201778)
-- 66.757 (area of the triangle) 
     18.694 (area of the segment)  (18.6940826488)

But if you want to determine for not only the segment, the area of the sector, and the area of the triangle; but also for the height of the apothem, the height of the segment, the length of the arc,  the width of the chord, and more, then you might want to try this following method:

For this one, we'll find the sector the same way, but use a different method for the triangle.  Try the following example by starting with the same angle of 68 degrees and a radius of 12:

1) Find the area of the sector as before (PI x radius^2 x the angle.  Then divide by 360)

2)  Since the triangle in the sector forms an isosceles triangle, we can imagine dividing that in half in order to form two triangles with right angles.  The reason for doing this is so we can figure out the length of the other two sides with the sine and tangent of the angle.  This will also lead to our being able to know the height of the apothem, 1/2 the width of the chord, the area of the triangle, the height of the segment, the area of the segment, etc.

So now our angle is 34 degrees (1/2 of what it was), and we really need to figure only one of these right triangles (since they are both the same).   Picture this triangle with the vertex of the central angle at the bottom and the "opposite" (the "chord") at the top.  The left leg of the central angle, which is a radius to the circle, will, therefore, represent the "hypotenuse" of the right triangle; and the right leg is the "apothem," which stops at the chord and is, therefore, the triangle's height.  It is also the "adjacent" of the right triangle.

Now we already know the length of the hypotenuse (the radius).  So by using from the "Soh-Cah-Toa" mnemonic the "Soh" formula, which means "Sin(angle) = Opposite / Hypotenuse," we can re-arrange that to determine the opposite: Opposite = sin(angle)  x  Hypotenuse.  The sine of 34 degrees is 0.559 (0.559192903471); and that times 12 = 6.71 (6.71031484165) (for the opposite, which is also the chord, but just 1/2 of what the end result will be).

(NOTE: If you want to see more about the trig functions of the Sine, Cosine, and Tangent, here is an article with several examples for you to try:

Next,  we determine the "adjacent" side with the "TOA" formula, which says that "Tan(angle) = Opposite / Adjacent.  By re-arranging this to determine the adjacent, we have "adjacent = opposite / tan(angle)"; and, in this case, it is 6.7103 (6.71031484165) divided by 0.67451 (0.674508516842), which gives us 9.9485 (9.94845087067) units for the length of the adjacent  (the "Apothem").  So now we also know its height.  It extends from the vertex of the central angle to the chord (which is the base of the segment).

We can now multiply the opposite by the adjacent (and think of it as "width x height") to come up with the total triangle area in the sector (and which includes the area for the second right triangle, too).  For normally, as you know, the triangle's area is its width times its height divided by 2; but since we were working on just half the area with this right triangle, we want to double that size.  And the answer for the triangle's area is 66.757 (66.757237529).

3)  Lastly, we simply subtract the area of the triangle from the area of the sector to determine the area of the segment:

85.451 (sector area) (85.4513201778)
- 66.757 (triangle area) (66.757237529)
   18.694 (segment area) (18.6940826488)

So in summing it up:

1. Multiply PI x Radius^2 x Angle Degree.  Then divide by 360  for the area of the SECTOR.

2. Divide the angle degree  by 2 (because we're now working in one of two right triangles).
    Multiply the  sine of that angle by the radius for the "opposite" (which will be 1/2 the chord).
    Divide your answer by the tangent of the angle to determine the "adjacent" (the apothem or triangle's height)).
    Multiply "opposite x adjacent" (thinking of them as width and height).  The result will be the total area of both right triangles.

3.  Subtract the area of the triangle from the area of the sector, which will give you the area of the segment.

Determining Other Parts

Once you have figured the above, you can then also determine the following (which we will base on the above example of a radius of 12 and angle degree of 68):

If you want to determine the segment's height, just subtract the "apothem" (adjacent) from the radius (hypotenuse).   In this example, it would result in 2.052. (2.05154912933).

If you want to know the apothem's  height, you already have it.  It is the "adjacent"; and, in this case,  9.948  (9.94845087067)  It is what we had determined by dividing the "opposite" with the tangent of the central angle.

To determine the length of the chord (which separates the triangle from the segment), just multiply the "opposite" by 2 (since the opposite is representing 1/2 of the chord).  The answer will be 13.421  (13.4206296833).

To find the arc's length of the sector or segment, multiply the radius x the angle (in degrees) x PI.  Then divide by 180.  Your answer should be 14.242 (14.2418866963).

The perimeter of the sector (radius + radius + arc) is 38.242 (38.2418866963); the perimeter of the triangle (radius + chord + apothem) is 35.369  (35.369080554); and the perimeter of the segment (chord + arc) is 27.663 (27.6625163796).

If you need to know the entire area of the circle that this triangle, sector, and segment are formed from, just multiply PI x Radius^2.  It will show 452.389 (452.389342117).  Its diameter (radius x 2).is 24, and its circumference (PI x Radius x 2) is 75.398 (75.3982236862)

So we have seen in this that by just starting out with the angle degree and radius, we are able to determine many things pertaining to a sector.

-- Tom Edwards (