Cone

In this mini-lesson, we shall understand the basic properties of a cone.

Most of us associate it with an ice-cream. That delicious thin-waver that can be had plain or with chocolates or sprinkles makes our mouth water.

While the cone may normally give us pleasure, studying it is fascinating, as it is one of the few three-dimensional figures that has a total surface area and a curved surface area. We will also derive the various formulae for a cone, understand the different parts of a cone, and even see how many sides a cone has.

Before we proceed into the theory of the cone, why don't you try your hand at the simulation below?

Lesson Plan

What is a Cone?

Definition:  A cone is a three-dimensional solid geometric figure having a circle at one end and a pointed edge at the other.  

Cone Figure

A cone can be carved out from a cylinder.

The three elements of the cone are its radius, height, and slant height.

Dimensions of a Cone

  • Radius (r) of a cone is the same as the radius of the circle at the end of the cone.  
  • Height (h) is the distance from the center of the circle at one end to the pointed edge at the other end.
  • Slant Height (s) is the distance along the curved surface, drawn from the edge at the top to the circumference of the circle at the base.

Examples of a Cone Shape in Geometry

The following pictures are daily life examples of a cone.

Cone Shaped TentBirthday CapPlastic Cone Road Divider

  1. A Cone Shaped Tent
  2. Birthday Cap
  3. Road Divider

Formulae of a Cone

The relation between slant height, height and rdius of a cone in math.

\[\begin{align}(s~height)^2 &= (height) ^2 + (radius) ^2 \end{align}\]

\[s^2 = r^2 + h^2\]

\[s = \sqrt (r^2 + h^2)\]

The above formula is derived using Pythagoras Rule.

Curved Surface Area

It includes the area of only the curved part of the cone. 

For a cone of radius 'r', height 'h', and slant height 'l', the curved surface area is as follows.

\[Curved~Surface~Area = \pi rl\]

Total Surface Area

Total surface area is the sum of the area of the circle at the base and the area of the curved part of the cone.

Total Surface Area (TSA) = Area of the base (Circle) + Curved Surface Area of the Cone(CSA)

\[\begin{align} TSA = \pi r^2 + \pi rl \end{align}\]

\[TSA = \pi r(r + l)\]

Total surface area is sometimes referred to as only surface area.

Volume of a Cone

 It is the space occupied by the Cone.

\[V =\frac {1}{3} \pi r^2 h\]

Let us now use the below simulation to calculate the volume and area of the cone.

Please enter the dimensions of a cone in the below form, to calculate the area and the volume of the cone.

 
tips and tricks
Tips and Tricks

The constant of \(\pi \) is used in the calculation of area and volume of all two dimensional and three-dimensional figures.

\[\pi = \frac{22}{7} = 3.14 \]

Volume of Cone

The space occupied by the Cone is the Volume of a Cone.

Volume of a cone

(The amount of water in the above cone, represents its volume.)

Let A = Area of Base of the Cone and h = Height of the Cone

\[\begin{align} Volume~of~a~Cone(V) &= \frac{1}{3} \times A \times h \\ V &= \frac{1}{3} \times \pi r^2 \times h \end{align}\]

\[V =\frac{1}{3} \pi r^2h\]

Also, the volume of a cone is one-third of the volume of a cylinder.
\[\begin{align} Volume~of~a~Cone = \frac{1}{3} \times V(Cylinder) \end{align}\]

 
Thinking out of the box
Think Tank

FRUSTUM OF A CONE:  It is partly cylindrical and partly conical in shape. It is slant across the curved surface and the two bases are circles of different radii.  

An example of a frustum of a cone is a tumbler having the base and the top circular edge of different radii.

\(Volume = \frac{1}{3}\pi(r_1^2 + r_2^2 +r_1r_2)\)
\(Curved~Surface~Area =\pi(r_1 + r_2)l\)
\(TSA =\pi(r_1 + r_2)l + \pi r_1^2 + \pi r_2^2 \) 
\(l^2 = h^2 + (r_1 - r_2)^2\)

Here h is the height, \(l \) is the slant height and \(r_1\) and \(r_2 \) are the two radii.

What are the different parts of a Cone?

A cone is made up of two parts.

Curved Surface

The curved surface of the cone covers the area from the circumference of the circle at the base and till the vertex of the cone.

Different Parts of a Cone

The base circle

The base of the cone is a flat surface and is a circle.


How many sides does a Cone have?

The Cone does not have any sides. 

But the Cone edge is its Vertex, which is the endpoint of the Cone.

Vertex of a Cone


Shapes of a Cone

Broadly there are two types of Cones.

(1) Right Circular Cone (2) Oblique Cone

Right Circular Cone

Oblique Cone

This Cone Shape is symmetric if it is cut laterally.

This Cone Shape is not symmetric.

Right Circular Cone Visual

The line representing the height of the cone passes through the center of the base circle and is perpendicular to the radius.

Oblique Cone Visual

The line representing the height of the cone does not pass through the center of the base circle.


Solved Examples

Example 1

 

 

Sam has to find the ratio of the volume of a cone and the volume of a cylinder.  How can you help Sam to find the required ratio?

Solution

To find the ratio, Sam first needs to find the volume of the cone and the cylinder.

Using the cone formula and cylinder formula we have:

Volume of a cone = \(\frac{1}{3} \pi r^2 h\)

Volume of a cylinder = \(\pi r^2h \)

Ratio of the volume of a cone and volume of a cylinder = \(\frac{1}{3} \pi r^2 h : \pi r^2 h \)

\(\begin{align} V~of~a~cone : V~of~a~cylinder &= \frac{1}{3} : 1 \\ V~of~a~cone : V~of~a~cylinder &= 1 : 3\end{align}\)

\(\therefore \) The ratio is 1 : 3
Example 2

 

 

Mary uses a thick sheet of paper and prepares a birthday cap in the shape of a cone. The radius of the cap is 3 inches and the height is 4 inches. How can Mary find the slant height of the birthday cap?

Solution

Given radius (r) = 3 inches and height (h) = 4 inches

Required slant height (s) = ?

\[\begin{align} (Slant~height)^2 &=(radius)^2 + (height)^2 \\ s^2 &= r^2 + h^2 \\  &=3^2 + 4^2 \\ &=9 + 16 \\ s^2 &= 25 \\ s &= \sqrt 25 \\ s &= 5 \end{align}\]

\(\therefore \) The slant height is 5 inches
Example 3

 

 

Jane was camping over the weekend. There she observes a conical tent and approximates that the height of the tent is three times the radius(r) of the tent. You need to help Jane, to find the approximate volume of the tent, in terms of its radius.

Tent in the shape of a Cone

Solution

Given that the height is three times the radius.

h = 3r

The cone volume calculation is as follows:

\[\begin{align} Volume~of~a~Cone(V) &= \frac{1}{3} \pi r^2 h \\ V &= \frac{1}{3} \pi r^2 (3r) \\ V &= \frac{1}{3} \times 3 \times \pi r^3 \\ V &= \pi r^3\end{align}\]

\(\therefore \) The volume of the tent is \(\pi r^3 \) cubic units
Example 4

 

 

The area of the circular base of the cone is 12\(\pi \) square units and its height is 6 units. Find the volume of the cone.

Solution

Area of the circular base of the cone (A) = 12\(\pi \) square units

Height (h) = 6 units

Using the cone formula we have:

\[\begin{align} Volume~of~the~Cone(V) &= \frac{1}{3} \times A \times h \\ V &= \frac{1}{3} \times 12\pi \times 6 \\ V &= 24 \pi~cu~units \end{align}\]

\(\therefore \) The volume of the cone is 24\(\pi \) cubic units 
Example 5

 

 

Two children planned to dig the ground in the shape of a cone and found that the soil of 48 cubic units was dug-out. If the depth of the pit was 9 units, find the radius of the pit.

Solution

Volume of the pit = 48 cubic units

Depth of the pit (h) = 3 units

Applying the cone volume calculation we have:

\[\begin{align} Volume~of~the~pit &= \frac{1}{3} \pi r^2 h \\ 48 \pi &= \frac{1}{3} \times \pi \times r \times r \times 9 \\ 48 \pi &=3 \times \pi \times r \times r \\ \frac{48 \pi}{3 \pi} &= r \times r \\ 16 &= r \times r \\4 \times 4 &= r \times r \\ r &= 4 units \end{align}\]

\(\therefore \) The radius of the pit is 4 units

Interactive Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.  

 

 
 
 
 
 
 

Let's Summarize

The mini-lesson targeted the fascinating concept of cone. The math journey around cone starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

About Cuemath

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Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.


Frequently Asked Questions

1. What is a cylinder?

A cylinder is a three-dimensional solid figure, having two circles at its ends, whose circumference is connected through a curved surface.

The dimensions of a cylinder are its radius "r" and the height "h".

Cylinder

Examples of a cylinder: a tumbler, a water pipe, the barrel of a pen, a doctor's syringe.

2. Does a cone have vertices?

A cone has a vertex at one end and a circle at another end.

The pointed edge of the cone is its vertex.

  
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