Sectors, Segments, Apothems, Chords, Arcs, etc.
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).
(Based on the Central Angle and Radius)
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
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
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
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:
PI * Radius^2 * the angle.
Then divide by 360
For the area of the triangle:
Radius^2 * the sine of the angle. Then divide by 2
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:
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:
(area of the sector) (85.4513201778)
-- 66.757 (area of the triangle) (66.757237529)
18.694 (area of the segment)
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
then you might want to try this following method:
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
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
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
that to determine the opposite: Opposite = sin(angle)
x Hypotenuse. The sine of 34 degrees is
0.559 (0.559192903471); and
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
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,
that "Tan(angle) = Opposite
/ Adjacent. By re-arranging this to determine the
"adjacent = opposite /
tan(angle)"; and, in this
case, it is 6.7103 (6.71031484165)
divided by 0.67451
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)
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
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
3. Subtract the area of
the triangle from the area of the sector, which will give you the area
of the segment.
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,
"apothem" (adjacent) from the radius (hypotenuse).
In this example, it would result in 2.052.
If you want to know the apothem's
height, you already have it. It is the
"adjacent"; and, in this
It is what
we had determined
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).
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 segment (chord + arc) is 27.663
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
(radius x 2).is
24, and its circumference
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 (firstname.lastname@example.org)