math research paper, area between tangent circles
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Karan Patel
A Mathematical Study of the Areas Contained BetweenTangent Circles?
BY, Karan Patel
1
Karan Patel
Abstract
This paper is about how to find the area between tangent circles. There are various
regular polygons formed by connecting center of the tangent circles. The area of regular
polygons was used to find the area between tangent circles through the use of calculations
and charts, a formula was found. This formula was proven to work for all cases. There is
a relationship between the regular polygons and the area between tangent circles.
2
Karan Patel
Problem Statement:
Find the area contained between three circles of radius 10, each of which is externally
tangent to the other two, as shown.
Table #1
The area of the polygons and the area between tangent
circles
polygon # of circles Area of circle Area of
polygon
Area between
tangent circles
triangle 3 314.1592 86.60254 16.1254481
square 4 314.1592 400 85.8407346
pentagon 5 314.1592 688.19 216.9520619
hexagon 6 314.1592 1039.236 410.9119542
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Karan Patel
Solution:
To find the area, begin by drawing the diagram again with the given information put in.
After inputting the information about the radius of 10 into the diagram, use this
information to reconfigure the diagram, making an equilateral triangle out of the radii.
Since the constructed triangle is made by two radii of 10 on each side, the sides of
the triangle are all twenty, so the triangle is equilateral. To find the area contained
between the circles, we must find the area of the triangle. Construct a perpendicular
bisector that bisects any side of the triangle, as shown below. This will create two
triangles.
4
1010
10
20
2020
10 10
10
10
10
10
Karan Patel
Since the constructed line bisects the bottom side, the two sides are equal. Also, the
bisector formed two right angles. To find the area of the whole triangle, the area of the
two smaller triangles is needed. To find the area of the smaller triangles, you must find
the length of the missing side. You can find the missing side by using the Pythagorean
Theorem.
In the equation, a, and b are sides of the triangle, and c is the hypotenuse. Since a side is
missing from the triangle, the second equation must be used. Now, replace b with one
side, 10, and c with the hypotenuse, 20.
Now, find the area of the smaller triangles by multiplying the base times the height and
dividing by 2.
Now replace the base with 10, and the height with 17.32050808.
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Karan Patel
Since the area of one of the small triangles is 86.6025404, then the area of both of the
small triangles, or the whole triangle, is 173.2050808.
An alternative and easier way to find the area of the triangle is by using the formula for
the area of an equilateral triangle.
In the above equation, S is the length of one of the sides, and A is the area of the triangle.
If you plug in a side of 20, you get the following results.
Since the area of the triangle is now known, the area of the pieces of circle inside the
triangle must also be known.
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Karan Patel
60
6060
20
2020
Knowing that the triangle is equilateral, it can also be concluded that the triangle is
equiangular. Since there are 180º in a triangle, and there are three angles, and if all the
angles are equal, then each angle measures 60º. To find the area of the parts of the circles
that are inside the triangle, the area of the whole circle must first be found. The equation
for the area of a circle and the area of the circle is below.
As shown above, the area of each of the circles with a radius of 10 is 314.1592654. Since
the angles of the triangle are all 60 º and every circle has 360º, of each circle is
inside the triangle. Now the area of the circle is needed.
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Karan Patel
Since there are three circles, multiply the area of the circle by three.
Knowing that the area of the circles inside the triangle is 157.0796327, and that the area
of the triangle is 173.2050808, you can subtract the area of the circles inside of the
triangle from the area of the triangle.
The difference of the two numbers is 16.1254481, so the area contained by the three
circles is 16.1254481.
Case #1
Find the area contained between four circles of radius 10, each of which
is externally tangent to two others, as shown.
Solution:
To find the area, begin by drawing the diagram again with the given
information put in.
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Karan Patel
After inputting the information about the radius of 10 into the diagram, use this
information to reconfigure the diagram, making a square out of the radii.
9
10
10
10
10
10
10
10
10
10
10 10
10
Karan Patel
Since the constructed square is made by two radii of 10 on each side, the sides of
the square are all twenty. To find the area contained between the circles, we must find
the area of the square. Below is the formula for the area of a square.
In the above formula, S represents the length of one side of the square, and A
represents the total area. Plug in twenty as one side of the square, and then follow the
formula to find the area.
Since the area of the square is now known, the area of the pieces of circle inside
the square must also be known.
10
10
10
10
10
10
10 10
1090 90
90 90
Karan Patel
Since there are 360º in a square, and all four angles are equal in a square, then
each angle measures 90º. To find the area of the parts of the circles that are inside the
square, the area of the whole circle must first be found. The equation for the area of a
circle and the area of the circle is below.
As shown above, the area of each of the circles with a radius of 10 is 314.1592654.
Since the angles of the square are all 90º, and every circle has 360º, of each
circle is inside the triangle. Now the area of the circle is needed.
Since there are four circles, multiply the area of the circle by four.
Knowing that the area of the circles inside the triangle is 314.1592654, and that the area
of the square is 400, you can subtract the area of the circles inside of the square from the
area of the square.
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Karan Patel
The difference of the two numbers is 85.8407346, so the area contained by the three
circles is 85.8407346.
Case #2
Find the area contained between five circles of radius 10, each of which is externally
tangent to others, as shown.
Solution:
To find the area, begin by drawing the diagram again with the given information put in.
After inputting the information about the radius of 10 into the diagram, use this
information to reconfigure the diagram, making a regular pentagon out of the radii.
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Karan Patel
Since the constructed pentagon is made by two radii of 10 on each side, the sides of the
pentagon are all twenty, so the pentagon is regular pentagon. To find the area contained
between the circles, we must find the area of the pentagon. Below is the formula for the
area of a pentagon.
In the above formula, A represents the total area. Now needed is the apothem and
perimeter of the pentagon to find the area. To find the apothem, or a line segment from
the center of a polygon to the midpoint of one of the sides, one could use the equation for
the apothem of a polygon. The equation is shown below.
In the above equation, a represents the apothem, s represents the length of one side, tan
represents the tangent function of trigonometry, and n represents the number of sides in
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20 20
20
20
20
Karan Patel
the polygon. We can now substitute in the information that we now. Plug in 20 for the
length of one side, and 5 for the number of sides and solve.
Now, the perimeter is also needed. The formula for perimeter of a pentagon is below.
Now, plug in 20 instead of S, or length of side, to get the perimeter.
Plug in the apothem and the perimeter to find the area of the pentagon.
14
20 20
20
20
20
10 10 Apothem
Karan Patel
Since the area of the pentagon is now known, the area of the pieces of circle inside the
pentagon must also be known.
Since there are 540º in a Pentagon, and all five angles are equal in a pentagon with
congruent sides, then each angle measures 108º. To find the area of the parts of the
circles that are inside the pentagon, the area of the whole circle must first be found. The
equation for the area of a circle and the area of the circle is below.
As shown above, the area of each of the circles with a radius of 10 is 314.1592654. Since
the angles of the pentagon are all 108º, and every circle has 360º, of each
circle is inside the triangle. Now the area of the circle is needed.
15
20 20
20
20
20
108
108
108
108
108
Karan Patel
Since there are five circles, multiply the area of the circle by five.
Knowing that the area of the circles inside the pentagon is 471.2388981, and that the area
of the pentagon is , you can subtract the area of the circles inside of the square
from the area of the pentagon.
The difference of the two numbers is , so the area contained by the five
circles is .
Case #3
Find the area contained between six circles of radius 10, each of which is externally
tangent to two others, as shown.
Solution:
To find the area, begin by drawing the diagram again
with the given information put in.
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Karan Patel
After inputting the information about the radius of 10 into the diagram, use this
information to reconfigure the diagram, making a regular hexagon out of the radii, with
sides of 20.
Since the constructed hexagon is made by two radii of 10 on each side, the sides of the
pentagon are all twenty, so the hexagon is regular hexagon. To find the area contained
between the circles, we must find the area of the hexagon. Below is the formula for the
area of a hexagon.
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10
10
10
10
10
Karan Patel
In the above formula, A represents the total area. Now needed is the apothem and
perimeter of the pentagon to find the area. To find the apothem, or a line segment from
the center of a polygon to the midpoint of one of the sides, one could use the equation for
the apothem of a polygon. The equation is shown below.
In the above equation, a represents the apothem, s represents the length of one side, tan
represents the tangent function of trigonometry, and n represents the number of sides in
the polygon. We can now substitute in the information that we now. Plug in 20 for the
length of one side, and 6 for the number of sides and solve.
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20
20
20
20
20
20Apothem
Karan Patel
Now, the perimeter is also needed. The formula for perimeter of a hexagon is below.
Now, substitute in 20 for S, or length of side, to get the perimeter.
Substitute in the apothem and the perimeter to find the area of the hexagon.
Since the area of the hexagon is now known, the area of the pieces of circle inside the
hexagon must also be known.
Since there are 720º in a hexagon, and all six angles are equal in a hexagon with
congruent sides, then each angle measures 120º. To find the area of the parts of the
circles that are inside the hexagon, the area of the whole circle must first be found. The
equation for the area of a circle and the area of the circle is below.
19
20
20
20
20
20
20
120
120 120
120
120 120
Karan Patel
As shown above, the area of each of the circles with a radius of 10 is 314.1592654. Since
the angles of the hexagon are all 120º, and every circle has 360º, of each circle
is inside the triangle. Now the area of the circle is needed.
Since there are six circles, multiply the area of the circle by six.
Knowing that the area of the circles inside the hexagon is 628.3185308, and that the area
of the hexagon is , you can subtract the area of the circles inside of the
square from the area of the hexagon.
The difference of the two numbers is 410.9119542, so the area contained by the six
circles is 410.9119542.
Solving For N
Find the area contained between n circles of radius 10, each of which is externally
tangent to two others.
Solution:
To find the area contained between the circles, you must find the area of the regular
polygon, with sides of 20 (constructed by connecting two radii for each side) that is
formed by connecting the midpoints of the circles to each other and subtract the area of
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Karan Patel
the circles that rest inside that polygon. First needed is the area of the polygon. Below is
the formula for the area of a regular polygon.
In the above formula, A represents the total area. Now needed is the apothem and
perimeter of the pentagon to find the area. To find the apothem, or a line segment from
the center of a polygon to the midpoint of one of the sides, one could use the equation for
the apothem of a polygon. The equation is shown below.
In the above equation, a represents the apothem, s represents the length of one side, tan
represents the tangent function of trigonometry, and n represents the number of sides in
the polygon. We can now substitute in the information that we now. Plug in 20 for the
length of one side, and n for the number of sides.
Now, the perimeter is also needed. The formula for perimeter of a polygon is below.
In the above equation, s represents the length of a side of the polygon, and n is the
number of sides in the polygon. Now, plug in 20 instead of S, or length of one side, to
get the perimeter.
Plug in the apothem and the perimeter to find the area of the hexagon.
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Karan Patel
Since the area of the polygon is now known, the area of the pieces of circle inside the
polygon must also be known.
Since there are 180(n-2)º in a polygon, and all angles are equal in a regular polygon with
congruent sides, then each angle measures . To find the area of the parts of
the circles that are inside the polygon, the area of the whole circle must first be found.
The equation for the area of a circle is below.
As shown above, the area of each of the circles with a radius of 10 is 314.1592654. Since
the angles of the square are all , and every circle has 360º, of
each circle is inside the triangle. Now the area of the circle is needed.
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Karan Patel
Since there are n circles, multiply the area of the circle by n.
Knowing that the area of the circles inside the polygon is ,
and that the area of the polygon is , you can subtract the area of
the circles inside of the square from the area of the hexagon.
The area contained between n circles with a radius of 10 is
.
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Karan Patel
Table #2
# of Circles That Are
Tangent To Two Other
Circles
Are Contained Between Tangent Circles
Equation
3 16.125481
4 85.8407346
5 216.9520619
6 410.9119542
.
.
.n
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Karan Patel
Generalization
In the table above, its shows the number of circles that have two tangent circles next to
them and the area contained by that number of circles. It also shows the area contained
by n circles. The number of circles that are tangent to two other circles is related the area
between the entire tangent circles by the formula that is above. By plugging in the
number of circles for n, you can find the area between any numbers of circles that are
tangent to each other. The areas between different numbers of tangent circles are also
related.
Conclusion
To conclude, the area contained between tangent circles can be found by using the
formula
Area between n tangents circles =
There is a relationship between the areas contained between different numbers of
tangent circles. As the number of regular polygons (that are connected by the center of
the tangents circles) increases, the area of the polygons as well as area contained between
tangent circles increases. As for the area of the circle, it remains constant.
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Karan Patel
It doesn’t matter how many congruent circles you have, if they have the same radius their
area will be the same. This is why every case the circles area remains the same. As
observed from the chart #2, the formula was formed with “a” representing the apothem,
“s” representing the length of one side, “tan” representing the tangent function of
trigonometry, and “n” representing the number of sides in the polygon. These three
variables were used to find the relationship between the tangent circles. The length (the
side of a regular polygon) made from connecting consecutive center of the tangent circles
(trigonometric relationship in apothem). Trigonometry was a big help in solving and
forming the formula for the area contained between tangent circles. Since a tangent only
touches the circle at exactly one and only one point, that point must be perpendicular to a
radius. These are the formulas, relationships and pattern involved in this paper. One
pattern that is shown in this paper is how the regular polygon’s increase number of sides
would affect the increase of area between tangent circles. This problem can relate to real
life through a study of the friction that forms between tangent circles. For example two
tangent circular blades are spinning and they both come into contact, forming friction,
which can be measured. The measure of the friction will be related to the behavior
between the tangent circles in further study.
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Karan Patel
Bibliography
http://mathworld.wolfram.com/TangentCircles.html
http://mathworld.wolfram.com/Circle-CircleTangents.html
http://jwilson.coe.uga.edu/emt669/Student.Folders/
Kertscher.Jeff/Essay.3/Tangents.html
http://www.mathreference.com/geo,hex.html
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