Interpreting the area between circles in polar coordinate
with a different conversion between rectangle and polar
coordinates (Part 1, abstract explanations)
Definition of circles
To study the intersections of circles, we must first make an agreement on
how we define circles. A circle in a plane, intuitively thinking, is the
collection or set of all points that have equal distance of a fixed value to a
same point in plane. We shall call this distance radius r, since the distance
formula tells us the distance from a point (x, y) in plane to another point (a,
b) in plane is 𝒓 = √(𝒙 − 𝒂)𝟐 + (𝒚 − 𝒃)𝟐, then we can express the definition
above as the set of points
{(𝒙, 𝒚)|𝒓 𝒊𝒔 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕 𝒇𝒐𝒓 𝒔𝒐𝒎𝒆 𝒇𝒊𝒙𝒆𝒅 𝒑𝒐𝒊𝒏𝒕 (𝒂, 𝒃)}. Therefore, the
equation for a circle centered at point (a, b) is 𝒓𝟐 = (𝒙 − 𝒂)𝟐 + (𝒚 − 𝒃)𝟐 for
some constant r. We shall also call a circle with r=1 a unit circle for study
convince.
Area between two circles
Step 1: deriving a better conversion between coordinate to
better interpret the problem
In polar coordinate, the property of the coordinate allows us to think area
between two unit circles as area between two curves as it is in rectangle
coordinates. Polar coordinate can be visualized as one holds the x-axis in
rectangle coordinates and squeeze it as a single point, while the y-axis
correspondingly radiates to every direction from that point. That is, if a circle
in polar coordinates in converted into rectangle coordinate in such way, the
circle would be curve in rectangle coordinates. In the special case of a circle
centered at the origin with radius r, the converted version of it in rectangle
coordinate would be a line with equation y=r; a straight line expanding an
infinite amount towards both directions of the x-axis that has a y value of r,
as the figure shown below.
As shown, polar coordinate’s axes are the “radiating lines”, which indicates
the value of 𝜃; and the “circles centered at the origin”, which indicates the
value of r, the circles are sets of points with some same values of r. Those
axes can correspond to axes in rectangle coordinate, where ‘radiating lines” is
the y-axis, the original in polar coordinates is the “squeezed” x-axis, and the
“circles” is lines with equation y=a with a being some constant; “circles” are
lines parallel to the x-axis. Note that some lines and circles are highlighted
because it is using the general type of conversion of the two coordinates,
where the two coordinates are viewed as two different ways of seeing or
studying a blank plane, but not alters the plane.
2
The figure shows a circle with equation r=2 in polar coordinates.
Using the transformation/conversion discussed above, the circle r=2 is a line
with equation y=2 in rectangle coordinates. As it is converted into polar
3
coordinate, the x-axis is curved and wrapped around into a circle until it is a
single point; meanwhile the y-axis is correspondingly curved, too. Since the
domain of y=2 is (−∞, ∞), means that it expands infinitely toward both
directions of x-axis, as the x-axis from (−∞, ∞) is all altered to be a point,
the line never break and thus form a closed geometric object with a constant
distance to the origin (origin is the altered entire x-axis, constant distance to
the origin is the constant value of y), namely a circle (centered at the origin).
Similarly thinking about a circle centers on the x-axis but not on the origin,
we can see that at any given point on the circle, the distance from such points
to the origin is not constant but different depends on the position of the point.
Recall that in the discussion above, we have stated that in the conversion
from polar to rectangle coordinates we discussed about, the origin is the xaxis and the distance from the original to any given point in the polar
coordinates is the value of y at some point of x. Therefore, since such circle
has various distance from the circle to the origin, in the equivalent graph of it
in rectangle coordinates, at some different given point of x, the value of y is
not constant but various. A sketch of an example of such circle is shown
below:
A circle with equation 𝑟 = 2 cos 𝜃 in polar coordinates.
4
(Not drawn in scale) Its corresponding conversion graph in rectangle
coordinates is in shape of a parabola facing downwards.
The graph passes point (0, 2) since the circle has the corresponding position
(𝑟, 𝜃) = (2, 0), where r corresponds\ to y, 𝜃 corresponds to x. Towards both
directions of x-axis from (0, 2), the parabola curves downwards, since for
circle in polar coordinates, the radius r of it gets smaller as 𝜃 gets bigger and
smaller. At some point of the graph, the parabola touches the x-axis and has a
value of y of 0, since the circle passes the origin, at where the distance from
the circle to the origin is 0, which correspondingly indicates that in its
conversion in rectangle coordinate, at some value of x, y has a corresponding
value of zero. At such point of the graph in rectangle coordinate, the graph
touches the x-axis and the conversion stops at this point. Note that in such
conversion method we discussed, it does not convert the part of the graph
below the x-axis in rectangle coordinate; but also note that rather than define
that part of the graph does not exist, the method just simply does not take that
part into account; the behavior of the graph under x-axis does not have affects
on the graph in polar coordinate. The conversion warps the x-axis around and
the part of graph under x-axis is squeezed as a single point and thus simply
5
banishes in the graph in polar coordinate. That is, such conversion only takes
care of or able to predict to the behavior of the graph above x-axis when
converting from polar coordinate; any part of the graph that is under the xaxis in rectangle coordinate is not shown when converted into polar
coordinate through such method. Therefore, this property, or flow, of such
method limits the use of it in only converting all polar graphs into rectangle
coordinate, but part of rectangle graphs into polar coordinate. In another
word, every polar graph has a corresponding graph in rectangle coordinate,
but not vice versa.
Step 2: apply the method, interpreting and converting
objects in both coordinate to approach the problem
With such understanding of polar and rectangle coordinates, we then shall
think about and interpret polar coordinates as if it is rectangle coordinates
since we have known the conversion. The difference, or the area between two
circles therefore shall be interpreted as the area between their corresponding
curves in rectangle coordinates in the corresponding interval of 0 ≤ 𝜃 ≤ 2𝜋,
we shall for now use some letters to represent the interval: 𝑎 ≤ 𝑥 ≤ 𝑏. The
𝑏
area A between the two curves is given by the formula 𝐴 = ∫𝑎 [𝑓(𝑥 ) −
𝑔(𝑥)] 𝑑𝑥, where f(x) and g(x) are the corresponding functions in rectangle
coordinate for the two circles in polar coordinate. According to the
conclusion we have from discussions above, to calculate the area between
two circles, we shall convert them into rectangle coordinate and calculate the
area between the corresponding graphs in the corresponding interval. Also,
since all graphs in polar coordinate are convertible into rectangle coordinate
and not vice versa, we are able to define two circles in polar coordinate and
convert them into rectangle coordinate in order to calculate the area. In order
to calculate the area numerically, we need to express the conversion in terms
of numerical relationship between the variables of two coordinates, “x, y”
and “r, 𝜃”, based on the conversion method we discussed above.
There are two ways in which we can interpret this problem. We shall either
solve it by applying the similar ideas to those circles in polar coordinate in
terms of variables r and 𝜃, or we shall numerically express the conversion
6
𝑏
method and use it to convert the formula 𝐴 = ∫𝑎 [𝑓(𝑥 ) − 𝑔(𝑥)] 𝑑𝑥 into its
corresponding polar coordinate form. We shall start with the first one first.
Recall that, the basic idea of calculating the area between lines is similar to
the idea of integration; that is, we approach the area by adding areas of small
simple geometric shapes such as rectangles. The area between curves is
essentially the absolute value of the difference of area under the the curves.
Thus, in calculating the area between curves, the basic idea of integration can
be interpreted as follows: within the domain of the function, for every value
of x or every point on the x-axis, there is a corresponding value of y, forms a
point on graph (x, y). we shall connect this point on x-axis and its
corresponding point (x, y), and call it the line on x=a, where a is the value of
x and is a constant; the area under the curve is the area swiped by the lines of
each value of x in a given interval. The area between curves is then simply
the difference in this swiped area of the two curves. The figure below
illustrates the idea.
For some function f integrated on a given interval in the figure.
7
This same idea can be applied to polar coordinate. For a function defined in
polar coordinate, within the domain of the function, for every value of 𝜃,
there is a corresponding value of r. In polar coordinate, the line on x=a is the
line connects the origin and the point (r, 𝜃), we shall then call it the line on
𝜃=m where m is some value for 𝜃. (Note that since for a same function, there
is a resultant value of r for an input value of 𝜃, thus we shall not mention the
value of r to identify the line since 𝜃 already indicates that value.) According
the conversion, the area under a curve in rectangle coordinate is interpreted as
the area of the closed graph in polar coordinate. In this case we are studying,
it is the area of the circle. Since the area under curves is convertible into area
of its corresponding circle, it indicates the domain of the graph in rectangle
coordinate. Since the area of the circle equals to the area under the curve, we
𝑏
have the following relationship: 𝜋𝑅 2 = ∫𝑎 𝑓 (𝑥 )𝑑𝑥, where f(x) is some
function y=f(x) over some interval [a, b], and R is the radius of the circle.
The values of a and b in the interval [a, b] are thus the values of them that
satisfy such relationship for the function y=f(x). Continue on the previous
problem, to integrate the area of a circle with the idea of integration in
rectangle coordinate, we slide the circle into sectors of the circle with very
small change in 𝜃. Thus, the area of one sector is 𝐴 = 𝜋𝑅 2 ∗
∆𝜃
2𝜋
; the value of
radius R can be calculated as follows: if we draw a diameter of the circle,
then its initial and terminal point must be on the circle; since the distance
between the two points is always the length of the diameter, we are able to
find such points that can satisfy some numerical relationships between points
on the circle involves radius R. There is a property of the diameter that if one
connects any other points on the circle to a fixed initial point on the circle,
the maximum length of such lines is the length of the diameter; that is, the
diameter line is the longest line exists that’s within a circle. It can be easily
proved using properties of triangles, as illustrated below:
8
As the figure above shows, line AD, AC, and AB are lines that connects
points B, D, C on circle to a same initial point A, forming two triangles ABC
and ACD. For triangle ACD, if side AD approaches side AC closer and
closer while side AC remains completely unchanged, forming a triangle with
less area, the triangle becomes an isosceles triangle if and only if when
AD=AC. At which instance, side CD=0. As side AD moves away from AC,
as two instances at different moment of the triangle is shown as triangle ACD
and ABC, the angle ACD within the triangle becomes smaller and smaller as
illustrated in the figure; since the degree of angle ACD is proportional to the
length of AD, as AD moves away from AC, the length of AD becomes
smaller; since when AD completes approaching to AC and the two lines
completely overlaps each other, AD=AC, thus as AD moves away from AC,
AD < AC. This proves that the diameter line is the longest line within the
circle. We shall use this fact to relate the diameter line to variable r in polar
coordinate. The idea is shown below in the figure:
9
the figure shows a circle C in plane. Let circle C be defined by some polar
function r=r(𝜃), and let point A = (r0, 𝜃0) be a point on circle where r0=r(𝜃0)
is the minimum value for r. Therefore, if r0 is represented as a line, the line is
normal to the tangent line of circle C at A. Similarly, we shall find the
maximum value of r, which, when considered as line, is a line intersect C at
point B. We can prove that the two lines of minimum and maximum of r are
on the same line and the difference between them is the length of diameter of
C as follows: suppose the line of max. r lands a point elsewhere on C rather
than point B, that is, a line connects the origin and some point on C except B;
then it must be smaller than the line of max. r according to the idea in our
previous discussion. Also, as the line lands on points away from B, the
segment of the line within the circle C is getting smaller and smaller, and this
property applies to all lines that lands elsewhere on the circle; thus, the
segment within C of line of max. r is the longest line of those of others, in
other words, it should be the longest line within the circle, therefore the
segment within C of line of max. r is the diameter of C. Similarly, since all
lines that lands elsewhere on C have line segments outside C longer than the
10
line of max. r, thus the line segment outside C of line of max. r is the shortest
line from the origin to the circle, which is the line of min. r. Therefore, the
line of max. r and the line of min. r are on the same line. Based on this
property we have deduced, we are able to express the radius R as 𝑅 = 𝑟𝑚𝑎𝑥 −
𝑟𝑚𝑖𝑛 .
Recall that we were integrating the area of a circle and have known that the
area of one sector is 𝐴 = 𝜋𝑅 2 ∗
∆𝜃
2𝜋
. From the above conclusion, we shall
rewrite this formula as 𝐴 = 𝜋(𝑟𝑚𝑎𝑥 − 𝑟𝑚𝑖𝑛 )2 ∗
2𝜋 (𝑟𝑚𝑎𝑥 −𝑟𝑚𝑖𝑛 )2
small value of ∆𝜃 is ∫0 (
2
∆𝜃
2𝜋
. The sum of all sectors with
) 𝑑𝜃 . According to the conversion, in
2𝜋 (𝑟𝑚𝑎𝑥 −𝑟𝑚𝑖𝑛 )2
its corresponding graph in rectangle coordinate, ∫0 (
2
) 𝑑𝜃 is the
area under the its graph. Recall that when calculating the area between two
curves, the ideas is to calculate the absolute value of the difference between
area under two curves; similarly apply it to circles, after converting, the
difference between area under curves becomes the difference between area of
the circles. Therefore, if r and R are two different polar functions of 𝜃, then
the difference D between two circles (in some cases the intersection of them)
2𝜋 (𝑟𝑚𝑎𝑥 −𝑟𝑚𝑖𝑛 )2
is 𝐷 = |∫0 (
2
2𝜋 (𝑅𝑚𝑎𝑥 −𝑅𝑚𝑖𝑛 )2
) 𝑑𝜃 − ∫0 (
2
) 𝑑𝜃 |.
This method has a very simple interpretation when it’s in some special cases.
A circle can be visualized as the trace of motion of a point with its behavior
of motion described in the relationship in its polar function. Depending on the
locations of the circles with respect to the origin, it may take various interval
of 𝜃 to complete one cycle of motion, as illustrated below:
11
12
As shown in the figure, the second circle takes longer interval than the first
one to complete one cycle of motion. Let Circle C1 and C2 be two off-center
circles that pass the origin (0, 0). Since this definition of such circles, such
circles always have an interval of [0, 𝜋] for one complete cycle of motion. It
can be proved as follows: as the figures shown above, such interval of one
complete cycle can be determined by the angle formed by two tangent line to
the circle from the origin; since the circle passes the origin, thus there is only
one tangent line from the origin to the circle, which separate the whole
interval of [0, 2𝜋] into two intervals of [0, 𝜋]. Because of this property of
such circles, when 𝜃 is outside of its interval of one complete motion, it
produces a negative value of r which start re-counting the cycle of motion
(the statement that such values of 𝜃 produces negative value of r can be
proved by the trigonometric identities sin(𝜃 + 𝜋) = − sin 𝜃 and
cos(𝜃 + 𝜋) = − cos 𝜃, which also proves that the negative values of r is not
arbitrary or different but follows the behavior of the motion performed by
positive r); when the interval of 𝜃 is [0, 2𝜋], it has completed two cycles of
motions. Since when calculating the intersection area or area between two
circles the interval took in account is the same for both circles regardless of
their individual interval of one complete cycle, and the interval is the interval
that covers the domain of both circle’s intervals of one complete cycle, area
of circles may be over-counted. Therefore, if we have already known the area
of the two circles through the previous method we discussed, then we shall
just calculate how many times the area of each circle is over-counted,
multiply this number to the area of the circle and subtract the circle on the top
to the circle on the bottom to get the area between two circles (note that a
circle on the top is the circle that’s further away from the origin, that is, the
circle that has a larger value of r at each value of 𝜃). For example, for the
circles shown in the figure below, one circle has an interval of one complete
cycle of [0, 2𝜋] and the other has that of [0, 𝜋]; then the latter circle is
counted twice for the same interval [0, 2𝜋] that’s used as the interval in
which we want to calculate the area between the circles, while the former
circle is counted once. If the area of the first circle is A and that of the second
circle is B, then the area between the two circles is B-2A, provided that the
latter circle is on top of the former one (note that this is always the case
where a off-center circle with smaller interval is on top of an centered circle,
13
since the off-center circle has a r larger than 0 by definition, while the
centered circle has r=0).
We shall also interpret the over-counting of the circle in terms of its
conversion in rectangle coordinate. Take the above circles as example to
study, we shall call the left circle C1 and the right circle C2 for studying
convenience.
When they are converted to rectangle coordinate, according to the previous
discussions, the centered circle becomes a line with equation y=c where c is
some constant, and the off-center circle that passes the origin becomes a
curve that’s concaving down and touches the x-axis at both ends at some
points on the x-axis that’s not the origin. Remember that we had the
conclusion that the domain of the curves/lines in rectangle coordinate
converted from polar coordinate varies depends on the domain of the
corresponding function/graph in polar coordinate, also, the two
corresponding domains in different coordinate of one function is proportional
to each other. Also, we found that according to such conversion method, the
behavior of the curve under the x-axis in rectangle coordinate is not taken
account in when converting. As we just discussed about the behavior of the
14
function when take values outside its domain, where we concluded that for a
circle that’s centered at the origin or a circle passes the origin, when taking
values outside its domain, it returns a negative value of r, nevertheless, this
value of r has the same behavior as the ones within the domain. Therefore,
this conclusion suggests that in the function’s conversion in rectangle
coordinate, outside the domain of the function, that is, when we are taking
values of x outside the domain, it should return a negative value of y, which
theoretically should also correspond to the behavior of y that’s within the
domain of the function. However, although as it is in polar coordinate that a
negative value of r can be interpreted as a positive value of r with the
opposite direction (which is a positive value of r with a difference of 𝜋 in the
value of 𝜃, when this idea is applied to its conversion in rectangle coordinate,
it indicates that when x is taken values outside its domain, it returns a positive
value of y elsewhere, at some value of x within the domain, and this value of
x has a difference of the corresponding value of 𝜋 in rectangle coordinate for
this particular function when converting variable 𝜃 to x (recall that the
relationship with domain in each coordinate varies with different functions);
also according to the previous discussion, this positive value of y at some
value of x within domain should be the same as the value of y calculated
taken that value of x within the domain.
15