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How do I calculate the volume and surface area of a cone?

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The Cone

Thursday, March 23rd, 2023

The cone is a primitive three-dimensional shape that appears in nature and has a wide range of applications in construction, aerodynamics, and various industrial fields. Understanding the properties of cones is essential to professionals and enthusiasts alike, as cones have relationships with other 3D objects such as pyramids, cylinders, and spheres. In this article, we will explore cones and their variants and learn how to determine the properties of cones, including surface area and volume. Through step-by-step examples and clear explanations, you will gain a thorough understanding of this fundamental shape and its practical uses.
Examples of cones in real world

The Right Circular Cone

The cone can be described as a 3D shape that tapers smoothly from a flat circular base to an apex. The principle parameters that are used to describe a cone are the radius of the base, the perpendicular height from the base to the apex, and slant length (the distance from a point on the edge of the base to the apex).
The right circular cone - If the axis perpendicular to the base of the cone through the center directly passes through the apex, which can also be stated as the line connecting the apex and center of the base being perpendicular to the surface of the base, if this is true then the cone is considered as a right circular cone.
The right circular cone

Other Variations of the conventional cone

  1. A truncated cone can be described as a 3D object that has a flat base that is tapered smoothly to another flat face. It’s easy to imagine this as slicing a cone with a plane parallel to its base across a given height and taking away the smaller cone formed above the slice.
    The truncated cone with labels of surfaces and dimensions
  2. The oblique cone - On contrary to the right circular cone, if the line connecting the apex and center of the base is at an angle other than 90 degrees, then it is an oblique cone.
    The oblique cone
  3. The elliptical cone is another variant of the cone, where the distinction is in the shape of the base. Rather than the circular base of the conventional cone, the elliptical cone, as its name suggests, has an ellipse as its base.

The area of a cone

A cone’s surface is made up of two major surfaces, a flat circular surface at the base, and a curved surface. Finding the area of the flat surface can be done easily by applying the formula for the area of a circle, but the curved surface is more challenging.
A right circular cone 3D depiction with surfaces colored
Since it is difficult to work with curved surfaces it’s quite common to cut and flatten out the curved surface when figuring out the area. The diagram below shows both surfaces of a right circular cone laid out on a 2D plane.
Surfaces of a right circular cone 2D and 3D depiction
You may notice that when the curved surface is flattened out (shown by the blue shape) it is a sector of a circle of radius equal to the slant length of the cone, and the perimeter is equivalent to the circumference of the base (2πr2\pi\text{r}). Let's walkthrough how to calculate the total surface area of a cone step by step using an example.
Example
Example 1. Determine the surface area of a right circular cone in terms of π\pi
An Example for calculating surface area of a cone
Answer:
Step 1: Calculate the area of the base.
The base is a circle of radius 5cm, hence:
Area=π×52\text{Area} = \pi\times\text{5}^2
Area=25π cm2\text{Area} = 25\pi\text{ cm}^2
Step 2: Calculate the area of the curved surface.
As shown previous the 2D net of athe curved surface is a sector of radius l and perimeter 2πr2\pi\text{r}. First we need to build an equation that relates the measures r and l to the area of the sector. We can use ratios for this.
Area of SectorArea of circle of radius l=Circumference of SectorCircumference of whole circle\frac{\text{Area of Sector}}{\text{Area of circle of radius l}} = \frac{\text{Circumference of Sector}}{\text{Circumference of whole circle}}
Aπl2=2πr2πl\frac{\text{A}}{\pi\text{l}^2} = \frac{2\pi\text{r}}{2\pi\text{l}}
Simplifying this we get:
A=πrl\text{A} = \pi\text{r}\text{l}
Substitute r & l in the equation to obtain the curved surface area
Acurved=π×5×13\text{A}_\text{curved} = \pi\times\text{5}\times\text{13}
Acurved=65π cm2\text{A}_\text{curved} = 65\pi \text{ cm}^2
Step 3: Calculate the total surface area.
Surface Area=Curved surface area +Base area\text{Surface Area} = \text{Curved surface area } + \text{Base area}
Substitute results from Step 1 and 2.
Surface Area=65π+25π\text{Surface Area} = 65\pi + 25\pi
Surface Area=90πcm2\text{Surface Area} = 90\pi \text{cm}^2

How to find the volume of a cone

The formula for the volume of a cone is given by 13πr2h\frac{1}{3}\pi\text{r}^2\text{h}, where r is the radius of the base and h is the perpendicular height from the base to the apex. As with any other formula, memorizing this may be challenging but it may help to think of the volume of the cone as a ratio to the volume of a cylinder. Since the cylinder is a 3D object with a uniform circular cross-section across its height, its volume is given by the Area of cross section×height\text{Area of cross section}\times\text{height}, i.e πr2h\pi\text{r}^2\text{h}. A cone with an identical base and altitude would have a volume that is 13\frac{1}{3} of that of the cylinder.
Example
Example 2. Determine the volume of a right circular cone
An Example for applying volume of a cone formula
Answer:
Step 1: Identify the measures of the cone The two measures that can be extracted from the diagram are the radius of the base(r) and the perpendicular height (h).
h=9cm\text{h} = 9\text{cm}
r=5cm\text{r} = 5\text{cm}
Step 2: Apply the formula Since we have all the parameters required we can dirctly plug it in the formula for formula for volume of a right circular cone.
V=13πr2h\text{V} = \frac{1}{3}\pi\text{r}^2\text{h}
V=13π×52×9\text{V} = \frac{1}{3}\pi\times 5^2\times 9
Simplifying this we get:
V=75π cm3V = 75\pi \text{ cm}^3
V=235.62 cm3(nearest 2 d.p)V = 235.62\text{ cm}^3 (\text{nearest }2 \text{ d.p})

The volume of a truncated cone

A labeled truncated cone
Apart from the usual measures of a cone, the truncated cone has an extra measure which is the radius of its top surface which is essential to determine the volume of the truncated cone. The formula that can be used is:
Volume=13π×(r2+rR+R2)h\text{Volume} = \frac{1}{3}\pi\times\left(\text{r}^2 + \text{r}\cdot \text{R} + \text{R}²\right)\cdot\text{h}
Note that the height measured here is the perpendicular distance from the bottom surface to the top surface.
Question
Question 1. A metal manufacturing company is designing a cone shaped machine part. The cone is made up of two layers of metal, metal A and metal B.Metal A will fill up a third of the height and the rest by Metal B. What is the volume of each metal required to manufacture a single machine part?
Diagram of machine part with dimensions marked
Answer:
First let's focus on finding the volume of metal A required. The shape formed by metal A is a truncated cone.
Step 1: Determine the measures of the truncated cone.
The diagram below illustrates the truncated cone fromed by metal A.
2D section of cone-shaped toy
Upon inspection we can determine the following measures:
R=6 cm\text{R} = 6\text{ cm}
The height (h) of the truncated cone is 13\frac{1}{3} of the height of the toy, hence:
h=13×12 cm\text{h} = \frac{1}{3}\times 12\text{ cm}
h=4 cm\text{h} = 4 \text{ cm}
We can use proportions of lengths in a shape to work out the radius r.Just like the heights, the ratio between R and r would be 3:1, therefore:
r=13×Rr = \frac{1}{3}\times\text{R}
r=13×6r = \frac{1}{3}\times\text{6}
r=2 cmr = 2 \text{ cm}
Now we know all the measures required to calculate the volume of the truncated cone.
Step 2: Calculate volume of metal A
Volume of Truncated Cone=13π×(r2+rR+R2)h\text{Volume of Truncated Cone} = \frac{1}{3}\pi\times\left(\text{r}^2 + \text{r}\cdot \text{R} + \text{R}²\right)\cdot\text{h}
VA=13π×(22+26+62)4\text{V}_A = \frac{1}{3}\pi\times\left(\text{2}^2 + \text{2}\cdot \text{6} + \text{6}²\right)\cdot\text{4}
VA=217.82cm3\text{V}_A = 217.82\text{cm}^3
Step 3: Calculate the volume of Metal B
Here we can either directly apply the formula for volume of a right circular cone to the small cone formed by metal B, or deterine the volume of the entire toy and subtract it by the result obtained in Step 2. let's try the former.
The measures of the cone formed by Metal B are:
r=2 cm\text{r} = 2\text{ cm}
h=124=8 cm\text{h} = 12 - 4 = 8\text{ cm}
Applying the formula we get:
VB=13πr2h\text{V}_B = \frac{1}{3}\pi\text{r}^2\cdot\text{h}
VB=13π228\text{V}_B = \frac{1}{3}\pi\text{2}^2\cdot\text{8}
VB=33.51 cm3\text{V}_B = 33.51\text{ cm}^3
So, the company will need 217.82cm3\text{cm}^3 of Metal A and 33.51cm3\text{cm}^3 of Metal B to make each machine part.

Alternative formulas for the volume of a right circular cone

You may not necessarily always have access to the desired measures of a cone. For example, if you are given the slant height instead of the height, we can’t directly apply the formula of the volume of a cone discussed previously.
Cone with labeled Radius, Height, Angle and Slant Height
Here is a list of some of the commonly used alternatives:
Given ParametersAlternate Formulae for Volume of a right circular cone
Slant Height(c) and Radius(r)
13πr2c2r2\frac{1}{3}\pi\text{r}^2\cdot\sqrt{\text{c}^2 - \text{r}^2}
Slant Height(c) and Height(h)
13π(c2h2)h\frac{1}{3}\pi\left(\text{c}^2 - \text{h}^2\right)\cdot\text{h}
Slant Height(c) and Semi Vertical Angle θ\theta
13πc3cosθsin2(θ)\frac{1}{3}\pi\text{c}^3\cdot\cos{\theta}\cdot\sin^2({\theta})
Semi Vertical Angle(θ\theta) and Radius(r)
13tanθπr3\frac{1}{3\tan{\theta}}\cdot\pi\text{r}^3
Semi Vertical Angle(θ\theta) and Height(h)
13πh3tan2(θ)\frac{1}{3}\cdot\pi\text{h}^3\cdot\tan^2({\theta})
Although these formulas seem to be quite different from one another, they can be derived through the original formula by applying basic trigonometric ratios.
Question
Quiz on the cone
A pencil manufacturer has worked out that for every 10 minutes of writing, their pencil loses about 0.04mm^3 of graphite. Assuming the tip of the pencil is cone-shaped with an cone angle of 15°15\degree, work out the decrease in height of the tip after writing for 1 hour in mm.
Pencil with cone-shaped tip zoomed in
Solution:
Since in this question we are trying to determine the height of the cone, let's make h the subject of the formula for the right circular cone.
Step 1. Rearrange the formula of the volume of a cone to make h(height) the subject
V=13πr2h\text{V} = \frac{1}{3}\pi\text{r}^2\cdot\text{h}
Since the radius (r) is not available, we have to substitute r in terms of h and the semi vertical angle of the cone.
By applying the tan ratio to the right angle triangle:
r=htanθ\text{r} = \text{h}\tan{\theta}
Replacing r in the formula for V:
V=13πh3tan2(θ)\text{V} = \frac{1}{3}\pi\text{h}^3\tan^2({\theta})
Now let's make h the subject:
h=3Vπtan2(θ)3\text{h} = \sqrt[3]{\frac{3V}{\pi\tan^2({\theta})}}
Step 2. Determine the volume of graphite lost altogether.
The rate of graphite loss is :
Rate of graphite loss=0.0410 mm3/min\text{Rate of graphite loss} = \frac{0.04}{10} \text{ mm}^3\text{/min}
Rate of graphite loss=0.004 mm3/min\text{Rate of graphite loss} = 0.004 \text{ mm}^3\text{/min}
Hence, if the pencil is used for 1 hour (60 minutes):
Volume of grpahite lost=0.00460\text{Volume of grpahite lost} = 0.004*60
Volume of graphite lost=0.24 mm3\text{Volume of graphite lost} = 0.24 \text{ mm}^3
Step 3. Substitute the measures of the cone into the formula obtained in Step 1
h=3Vπtan2(θ)3\text{h} = \sqrt[3]{\frac{3V}{\pi\tan^2({\theta})}}
h=3×0.24πtan2(15°)3\text{h} = \sqrt[3]{\frac{3\times 0.24}{\pi\tan^2({15\degree})}}
h1.47 mm\text{h} \approx 1.47\text{ mm}