Research question

What are the general formulas that define the different relationships between a cone and the
different conic sections formed by different angles of cuts?

Introduction
When a circular cone is cut by a plane, the different angle of cuts allows different shapes to be
formed including circles, ellipses, parabolas and hyperbolas (see diagram 1). These shapes are
used to model out-of-this-world phenomenons such as the orbits of planets, meteorites and
satellites. Conic sections were recognised to be discovered by the Greek mathematician
Menaechmus who experimented with the different shapes produced when cutting a circular
cone with a plane. As a person who is developing my own connection to mathematics and
discovering my area of interest, exploring the correlation of the different angle of cuts on the
surface area of the conic sections, provides a mathematical approach towards developing
curiosity towards geometry. I will be investigating the effects of the angle of cut, from planes with
the same defined line of intersection and determine the different shapes produced when cutting
a circular cone and how the area or dimensions of the shapes can be expressed in relation to
the cone.

Aim and Methodology

In this investigation, I will be investigating the effects of the angle of cut, from planes with the
same defined line of intersection and determine the different shapes produced when cutting a
circular cone and how the area or dimensions of the shapes can be expressed in relation to the
cone. I aim to explore the different relationships between conic sections and the main body of
the cone that is cut to produce them. As there are no barriers or limitations to my exploration,
different formulas will be formed from relationships between different dimensions, areas or
angles in different dimensions that will be explored. All the challenges I have faced will be
documented to show my process of trial, error and evaluation.

Background
The area formed by the intersection of a plane and a circular cone, in my paper, will be defined
as “conic sections” whilst conic sections can also be categorized by the volume, surface area or
dimensions of the shape formed by a cone after it has been cut by a plane.

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 Diagram 1

The example above shows conic sections from a shape formed from a shape made by two
cones flipped upside down. A singular cone, used in this paper, will limit conic sections into
ellipses, circles, one triangle and hyperbolas. In my paper, I will limit the planes of cut to only
through one line of intersection, namely z = 1(on a three dimensional plane with axis x , y and
z), limiting my shapes to only ellipses, circles and triangles. The formulas I will be using are
proven as so:

Area of the ellipses

An ellipses is characterized as a closed curve formed by the intersection of a plane to the cone
that is not parallel to the base of the cone. From two fixed points that have an equal distance
from the center of the elllipses, both on the lines of symmetry of the shape have the same ratios
for distance traveled towards the edge of the shape. As shown in the two examples below, the
paths traveled from F1 to F2 all have the same distance ratio.

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Since an ellipses is not a one-to-one function, I can draw an ellipses by limiting the graph to one
quadrant defined by the area bounded by the ellipses’s two lines of symmetry. In this example,
the two lines of symmetry are x=0 and y=0. We can graph one quadrant of the ellipses with the
general formula:

The two significant figures used in calculating the area of an ellipses are the lengths of a and b.
By reorganizing the general formula we get:

Integrating this formula and multiplying it by 4 gives us the area for the ellipses:

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Circle
At the angle of 0 radians, a plane cuts the cone into a circle at any value of the z-axis.
Similarly, we can do the same for a circle with a constant length from the middle point from the
middle towards the edge of the circle. This length is called “a”, also referred to as the radius.
With a circle, a is equal to b with the same variables used in the ellipses.

Curved surface area of a cone:

A cone can be formed through a net made by the major sector of a circle made with angle theta.
When we wrap this net so the two radii at either sides of the sector are touching- but not
overlapping- we form the curved surface of a cone.

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The area of the major sector of the circle with radius r is defined by the formula:

This is also the curved surface area of the cone formed from the circular net, however using this
formula involves the disassembling of the cone to figure out angle theta and is not practical for
the exploration that I would want to conduct. Therefore, we can further use other aspects of the
circular net including arc length.

The arc length of the major sector much be equal to the circumference of the circular base of
the cone:

We substitute R into the formula involving the area of the major sector:

And r can be defined as the slant height of the cone:

Volume of elliptical base cone:

When an ellipsis is formed, we can use this ellipsis as the elliptical base for a cone using the tip
of the predetermined cone. The ellipses are the semi-major axis “b” and the semi-minor axis “a”.
The further we move up this cone, we will get ellipses that have the semi-major and semi-minor

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dimensions that are similar to the ellipses, in this example the ratio of A : a is equal to B : b, and
we can map this out using similar triangles.

By using the proven formula of the area of the ellipse, we can derive a formula for the volume of
this elliptical based cone using the integration of the area formula. We can prove for the formula
for the elliptical based cone as so:
It is important to remember the height of an elliptical based cone, here called “h” , is not in the
same line as the height of the circular based cone.

Limits

Due to the time restrictions, I would like to limit my variables so that the planes constructed to
cut through the cone will all have the same line of intersection (in the example used, this line is
z=1). This will limit the shapes produced with the intersection to only triangles, circles and
ellipses.
The formulas of triangle are will consists of:

and

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Furthermore, I will be limiting most of my calculations with theoretical cones that have a radius :
height ratio of 1 : 2, however, I do specify whenever this condition is used.

Exploration
The cone that I am going to be using is a circular based cone where the base circle has the
radius of 1, for simplicity I am not going to be mentioning the units used and each unit is defined
by one cube on the axis. The plane looks like this when constructed on the three axes x,y and z.

With this cone, the yellow lines represent the radius of the circular base and height of the cone
which are 1 and 2 respectively. As following the curved surface area formula previously proven
the curved surface area of the cone will be:

Triangle

Areas

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A containing the z-axis, cuts through the cones triangle and is produced with the intersection of
the cone and the plane. This triangle is an isosceles triangle with its two equal sides made up
of the slant height of the cone and the base is worth the diameter of the circular base as the
triangle cuts the cone straight down the middle from its tip. The area of the triangle is :

Now if we chose to keep the same ratios as the model cone, where the ratio of radius : height is
1: 2, we can obtain a table of results for the area of the triangle formed, when the cone is sliced
straight down from the middle, and the curved surface area of the cone.
We can use the formula for finding the Nth term to establish a general equation to predict (as
the CSA is rounded to two decimal places) how the CSA of the cone is connected to the area of
the triangle. The table below is produced from increasing radii starting from 1 leading to 10 and
the corresponding area of the triangle produced.

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Therefore the curved surface area of the cone can be defined with the following equation where
n is the area of the triangle formed.

It is important to point out that when n is 0, the curved surface area is a negative number, this
may be due to the numbers of decimals I have chosen in the calculatuations, specifically the
curved surface area is only taken to two decimal places. If more decimal places were used
instead, the closer a0 will get to 0.

We can double check the validity of this equation by graphing the curved surface area against
the area of the triangle. In order to increase the accuracy of the equation generated from this
graph, I will take the figures of the curved surface area rounded to 10 decimal places.

The equation shows that the value of x (area of the triangle) is predicted accurately.
We can say that the equation derived from the nth term formula is equal to a formula created
from rearranging the variables in the general equations of the triangle and curved surface area.

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However, I can only prove that this formula is correct with my sample of the first 10, positive
integer, radii with the radius, height ratio of 1:2. I cannot prove this formula correct for all the
cones that exist with the ratio of 1:2, not to mention all the cones with different radius height
ratios. This formula requires up to two independent variables including the radius of the circular
base and the height of the cone. and my mathematical knowledge and understanding is limited
and lacks the capability to universally prove this formula so my paper will be limited to this
much. However, the general formula I have proven seems to have no fault and considering that
I have only used general variables, therefore I will predict that this formula is proven true for
cones with the conditions of when the height is double the radius.

Angles
My next topic of exploration would be to find a general equation and identify the relationship
between the isosceles triangle formed when the cone is sliced through the middle with a plane
perpendicular to the z axis, and the angle formed by the major sector of the net of the cone,
these angles will be defined as theta (triangle) and var phi (cone net)

By rearranging the two area formula mentioned in my background section, we can define a
common variable that the two equations both have, the slant height.

The area of the triangle formed as a conic section with the height equal to double the radius (h =
2r) can be written as so using two area formulas:

The circumference of the cone is equal to the arc length of the major sector:

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Therefore, if we replace length “l” back into the area formula:

Again the limitations set for this formula is for a cone with radius : height ratio of 1 : 2 therefore,
the height can be expressed as 2r. As I was using the general formula for the area of the
triangle with height : base ratio 1 :2, the formula above is therefore proven true for all circular
base cones with radius : height ratio 1 : 2.

Circle

When a cone is cut with a plane parallel to the x-y plane, i.e 0 degrees from the x-axis, the
intersection forms a circle.

The ratio of the circle’s radius to the cone’s circular base’s radius is the circle’s height to the
cone’s height. The ratio can be expressed where r and h is the radius and height of the cone
respectively and r2 and h2 being the respective radius and height of the circle formed from the
intersection

The curved surface area of the cone, CSA1 is

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The relationship of the curved surface area of the cone with base of intersection circle conic
(CSA2) and the curved surface area of the cone (CSA1) can be proven as:

Since the general cone used in this paper has the planes passing through the line z = 1,
therefore the horizontal plane forming the circle will be limited to cutting the cone in half, height
wise. Therefore, due to ratios, the radius. of the circle formed, r, can be multiplied by 2 to
achieve the radius of the circular base of the cone, R.

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The area of the circle and curved surface area can be proven as follow:

Note that the ratio of radius : height is still 1 : 2. The previous method used when correlating the
area of a triangle and the CSA by obtaining a graph and determining the slope of the graph is
shown in the appendix, this formula has a similar conclusion as the relationship proven using
algebra above.

Now suppose the ratio of the radius : height ratio is given a variable 1 : n then the general
formula for the curved surface area of the cone will be:

Ellipses

From between the angles of 0° to 90°, the planes that are intersecting the cone create an
ellipsis. An ellipses is defined dominantly using its two lengths called the semi-minor axis (i.e the
distance from the center towards the closest edge of the ellipses) and the semi-major axis
(distance from center to furthest edge)

In the example above, I have used three planes that cut the plane at

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30°, 45° and 60° with the larger the angle, the closer the plane gets to 90°, however, due to
convenience of calculations later on, these angles will be defined as theta.

In the following images, I have shown the ellipse formed with the plane intersection at 60° at
different points of view.

Area

In the previous two shapes, I derived another formula for relating the area or angle of the shape
created with the intersection of the plane to the cone, however, this can only be done
algebraically if there is a common dimension between the curved surface area and the shape.
For the triangle, it was the base of the triangle written in terms of the diameter of the radius of
the cone. With the circles, the radius of the circle is expressed as a ratio with the radius of the
cone. However, for the ellipses, I cannot find the common dimension between the area of the
ellipses and the curved surface area of the cone.

Exploring a relationship between angles

However, after many months of inquiry and discussions, I discovered that I can express the
dimensions of the ellipses using angles that existed within the cone in 2 dimension, thus
connecting the ellipses with the cone through the common angles. Initially, through constructing
an image in my head, I assumed that the center of the ellipses will always match with the center
of the circular based cone thus the side of “b” on the ellipses will be made with two equal lines
from the center of the ellipses.

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Diagram a represents my imagination but in reality, as I have limited the fact that all my conic
sections to pass through a line from the z-axis that is one half of the main cone’s height the
center of the circular base will never be the same as the center of the ellipses, thus the length
“b” of my ellipses will always be unequal.

The angle of cut of the ellipses in relation to the radius of the circular based cone with the radius
“r” that is half of the main cone’s radius “R” is represented by angle beta. We can use angle beta
and half the angle of the tip of the cone, alpha, to determine an equation for length b1.

The two missing angles from the triangle formed with side b1 and the radius of the circular base
will be labeled y and x which can be determined using angles beta and alpha

and

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 Hence, using the sine rule, we can obtain length b1 in terms of alpha and beta

I did a similar process to find length b2 where the missing angles in the triangle are labeled “c”
and “d”, shown in the drawing below and the angle gamma is equal to beta as they are opposite
angles.

To find the other length needed on the ellipses, “a”, I needed to think differently.

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Unlike with length “b” I cannot define a common triangle between length a and any dimensions
or angles of the circular based cone. The closest dimension I could relate to length a was the
diameter of the circular based cone (cut at one half height of the main cone). The diameter of
the circular based cone will always cut the ellipses no matter the angle of cut, although it does
not equal length a the ellipses. However, the diameter is a length I can use to generate a
common formula for length a of the ellipses as it is always present.

It took me a long time to realize that my definition of an ellipses can be put into use to find
length a. As shown in the diagram above, since the ratio distance of c1 + c2 is the same, I can
define length a as the center of ellipses where the length on each side is one half of the total
distance c. Furthermore, I can connect this lengths c1 and c2 to the angles of beta and alpha to
link this to the angle of cut of the ellipses and an angle related to the circular based cone.

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I can define c1 and c2 using pythagoras theorem and rearrange the sum of the two lengths:

Finding limits

I tried using a website that could graph the length of “b1” in three dimensions to determine the

limits of my general formula.

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I entered my formula for the length of b1. The x-axis represents angle alpha, the angle of the
cone tip, and y-axis represents angle beta, the angle of cut. Finally, the z-axis represents the
length of b1. The domain and range I chose was between angles 0 and half pi, and the graph I
got is shown above. I deduced that the sudden slope to infinity is caused by the range I chose.
After visualizing the angle of cut on the cone, I realized that at a certain angle of cut, the conic
section is no longer an ellipsis but rather a parabola, therefore the length of “b” cannot be
defined as there is no longer a “center” to reference. I have never thought of this before as I
have always assumed an ellipses would be the conic section formed between the angles of cut
0 and half pi, however this is not the case as after one quarter pi, a parabola will be formed as
the ellipses has went out of the cone.

I wanted to do a further exploration and expanded my domain to angles from 0 to pi and

discovered an asymptote. This asymptote is most likely from x = pi/2 to y=pi/2 because then the
numerator is divided by 0 (sin(0)) and the function goes to infinity. It would be interesting to point
out how a cone would look like with these angle conditions. If angle alpha (the domain) is one
half pi then the angle of the tip of the cone is pi (in two dimensions) as alpha is half of the cone’s
tip angle in two dimensions, this would mean that the two sides of the cone will straighten and
form a straight line, therefore I would imagine a cone could not be formed when it no longer has
a “tip”.

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I have also set the domain (x) from 0 to half pi and range (y) from 0 to one quarter pi. I managed
to graph this surface. The z-intercept is equal to the length of the radius of the circular based
cone, in this case the radius is 1. It is interesting to find out that the smaller the angle of cut (y)
and the smaller the angle of the cone, the closer the length of b1 gets to the length of the radius
of the cone and the smaller the angle of the cone tip, the closer the the length of b1 gets to the
radius of the cone. With angle of beta, the smaller the angle of cut, the smaller the ellipses until
eventually at angle 0, the ellipses is now the circle with radius “r”. As both angle alpha and beta
increases, the length of b1 gets larger, however, b1 will increase more greatly with an increase
in angle beta, shown in the surface above as the slope of z increases at a steeper and larger
slope as y increases.

Volumes and areas

However, after some time of processing, I think I can attempt to define a relationship between
the height of the elliptical based cone and the height of the main body cone (circular based).
The height of the elliptical based cone, with the ellipses formed with the intersection of the plane
to a circular based cone, between 0 and one half radians. My thinking can be illustrated with this
diagram:

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In this drawing, the height of the circular base cone is “H” where the height of the elliptical base
cone is defined as “h” and their relationship is defined by:

It may be important to clarify that the height of the elliptical base cone is the distance from the
tip of the cone to the center of the ellipses, and in this scenario when the ellipses cone is
contained within the circular base cone, the height of these two cones do not lie on the same
line, i.e they do not have the same slope.

I can then work my way out to a formula that involves the volumes of these two cones, “V1”
being the volume of the circular base cone and “V2” being the volume of the elliptical base cone
(proved in background section). I will also be including the areas of the circular base, “A1” and
the area of the elliptical base “A2”. In this calculation, I will not be limiting any variables whether
that be the ratio of the semi-minor axis to the semi-major axis of the ellipses or the ratio of
radius to height of the circular base cone, these variables will be kept as general variables.

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By making this h value equal to the h value we derived from above, we can make H the subject
and then replace it back into the volume formula of the circular base cone:

And we get interesting formulas, or even ratios between the two volumes and areas.

This equation may have indicated that I have created a ratio formula for the ratios of volume
over area of a circular cone and the volume over area of an elliptical cone made from the
ellipses, a conic section of the circular cone, as the base.

Conclusion
In conclusion, I have attempted to explore the various relationships regarding conic sections
and the curved surface area of the cone. I have done so by formulating theoretical equations
that would link the general formula of these conic sections to the formula of the curved surface
area of the cone. These formulas have been restated in the table below.

Description Formula obtained

Triangle conic section

The relationship between the curved surface

area of the cone (CSA), the radius of the
circular base ( r ) and the area of the triangle
formed (A) with radius : height ratio 1 : 2

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 The relationship between the angle formed by
the net of the cone (varphi) and the angle
sses between the two equal sides of the
isosceles triangle (theta)

Circle conic section

Relationship between the curved surface

area of the cone with the conic circle base
(CSA2) and the curved surface area of the
cone (CSA1)

Relationship between the curved surface

area of the main body cone with the area of
the circle conic section with radius : height
ratio 1: n.

Ellipses conic section

The lengths that sum up to make the major

axis in an ellipses conic section with angle
beta representing the angle of cut and angle
alpha representing the angle between the two
equal sides of the cone in two dimensions.

The minor axis of the ellipses conic section

expressed in beta and alpha

Relationship between the volume of the main

body cone (V1), the area of the circular base
(A1) and the volume of the elliptical base cone
(V2) and the area of the ellipses base from
the conic section (A2)

I feel like the formulas I have calculated in the “circle” and “triangle”” area have answered my
research question to a lesser extent than the exploration I had done with the “ellipses”.
Especially in the circle section, I could have dived deeper into investigating more complex
relationships like I have done with the angle of the net in the triangle section or graphing
formulas as explicit surfaces as I did when exploring the ellipses. I also found my method of
concluding very vague as I have linked the arithmetic general equation to an algebraic equation
without much proving except for how they gave similar results was not the method that I had
initially hoped to do. I was more content with the different solutions that I have explored in the
ellipses section because I managed to find ways around the problems I faced of defining a

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length of the ellipses as a relationship with a length of the cone, instead I focused on angles and
finding limitations.

In terms of real-life applications of my exploration, I would like to hope that my findings could
foster more research and proving so that these formulas can be used to best maximize the
efficiency of the materials we use in our everyday life, such as paint, to cover curved surface
areas or used for engineers to decide manipulate the cones they want to produce just my
applying a formula to the net layout of the cone. However, regardless of its use, I found my
exploration journey to be very much enjoyable.

Appendix

Appendix 1:

data measured from cones with radius : height ratio of 1 : 2 and ellipses with semi-minor :

semi-major axis ratio of 1 : 2 where the radius : semi-major axis ratio is 1 : 1, used to calculate

the area of ellipses and CSA of cone.

Cone Ellipses
radius height a b Area ellipses CSA cone
2 4 1 2 6.283185307 28.09925892
4 8 2 4 25.13274123 112.3970357
6 12 3 6 56.54866776 252.8933303
8 16 4 8 100.5309649 449.5881428
10 20 5 10 157.0796327 702.4814731
12 24 6 12 226.1946711 1011.573321
14 28 7 14 307.8760801 1376.863687
16 32 8 16 402.1238597 1798.352571
18 36 9 18 508.9380099 2276.039973
20 40 10 20 628.3185307 2809.925892

Appendix 2:

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data measured from cones with radius : height ratio of 1 : 2 and intersecting circles with their

radius being one half of the radius of the cone (due to the limitation that the plane passes

through z = 1), used to form a relationship between the area of intersecting circle and CSA of

cone.

Cone Intersecting circle

height Radius Radius Area of circle CSA
4 2 1 3.141592654 28.09925892
8 4 2 12.56637061 112.3970357
12 6 3 28.27433388 252.8933303
16 8 4 50.26548246 449.5881428
20 10 5 78.53981634 702.4814731
24 12 6 113.0973355 1011.573321
28 14 7 153.9380400 1376.863687
32 16 8 201.0619298 1798.352571
36 18 9 254.4690049 2276.039973
40 20 10 314.1592654 2809.925892

Appendix 3:

Linear graph the forms the relationship between the area of the intersecting circle and the CSA

of the cone, this proves my general formula (with limitations) in the “circle” section correct

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Bibliography

Alvarez, Guillermo. “Volume of an Elliptical Cone HD.” Www.youtube.com, 8 Oct. 2014,

youtu.be/pe4pbnr_RW4?si=N1cugXP1vhFN8xk5. Accessed 27 Nov. 2023.

“Area of an Ellipse Proof for Area, Formula and Examples.” BYJUS,

byjus.com/maths/area-of-ellipse/.

Carson, Andy. “Proof That Curved Surface of Cone Is Pi R L.” Www.youtube.com, 7 May 2022,

youtu.be/f7Cu-SgbYD4?feature=shared. Accessed 27 Nov. 2023.

CUEMATH. “Perimeter of Ellipse - Formula, Definition, Examples.” Cuemath,

www.cuemath.com/measurement/perimeter-of-ellipse/.

L a T E X Mathematical Symbols.

The Editors of Encyclopedia Britannica. “Colin Maclaurin | Scottish Mathematician.”

Encyclopædia Britannica, 28 Jan. 2019, www.britannica.com/biography/Colin-Maclaurin.

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