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Cone Definition

A cone can be defined as a three-dimensional shape of geometry that is narrow from the top and from a flat base, the flat base is generally circular to a point that forms an axis to the center of the base is called the vertex.


A cone is a set of lines that connect a common point called the apex or vertex to the points on the circumference of the circular base. The distance from the vertex of the cone to the base is the height of the cone. The cone has a circular base whose radius is known as the radius of the cone and the length of the apex to any point on the circumference of the base is the slant height.


In the following figure of the cone, you can see the height (h) and the radius (r) of its base and the slant height (l) of the cone.

Cone Formulas

Now we can derive the cone formulas for the slant height, the surface area, and volume of the cone with the help of its height, and the radius of the circular base.

  • Slant Height

It is defined as the distance from the top of the vertex to the point on the outer line or the circumference of the base of the cone. With the help of the Pythagoras Theorem in the right circular cone, the formula for slant height is obtained. Three dimensions form a right triangle, where the slant height acts as the hypotenuse of the right triangle.


Hence according to the Pythagorean Theorem, the formula for the slant height is given by as slant height, l2 = r2+h2

Where the radius of the base of the cone is r, h is the height of the cone and l is the slant height.


  • The volume of the Cone

If the volume of the cone is V, the radius of its circular base is r, height from the vertex to the base is h then the formula for measuring the volume of the cone is given as follows:

The Volume of Cone (V) = $\frac{1}{3} $πr2h cubic units.


  • Surface Area of the Cone

We can calculate the value of the surface area of the cone with slant height and radius of the circular base. In a right circular cone, the surface area is equal to the summation of the lateral surface area of the cone that is given by πrl and the surface area of the circular base that is given by πr2. So πrl + πr2 is the total surface area.


So, Surface Area of the cone = πr(l + r) square units.


Types of Cone

After knowing the definition and formula of cones, we can move forward with the types of cones. There are mainly two types of cones in geometry. These are as follows:

  • Right Circular Cone

  • Oblique Cone

Let’s discuss each cone in detail.

  • Right Circular Cone

A cone has a circular base and the axis from the vertex of the cone towards the base passes through the center of the circular base. The vertex of the cone lies just above the center of the circular base.


The term right is used here because the axis is perpendicular to the base or forms a right angle with the base of the cone. In geometry, right circular cones are the most common types of cones.


The following figure is an example of a right circular cone.

  • Oblique Cone

A cone has a circular base but the main difference that differentiates it from a right circular cone is that the axis of the cone is not perpendicular to the base. This type of cone is called an Oblique cone. Therefore, the Oblique cones look like slanted cones.


The following figure is an example of the Oblique cone.

Properties of Cone

Some of the cone properties are given as follows.

  • A cone possesses a single face of the circular base without any edge.

  • A cone has only one vertex point at the top of the cone.

  • “V = $\frac{1}{3} $πr2h” is the formula to measure the volume of the cone, where r is the radius of the base and ‘h’ is the height of the cone.

  • The TSA of the cone is obtained by πr(l + r).


Frustum of Right Circular Cone

The frustum of a cone is a piece of the given circular or right circular cone. When from the top of a right circular cone a small conical portion is removed, the solid we get is called a Frustum of a cone.


In other words, we can say that when a plane intersects the cone parallel to the base of the cone then the part with the base of the cone is called the frustum of a cone.

(Image will be uploaded soon)


Did you know

We can create a cone by revolving a right-angled triangle around any of the short sides of the right-angled triangle.


Solved Examples

Example 1: Calculate the volume of the cone if radius, r = 8 cm and height, h = 14 cm.

Solution: According to the formula of volume of the cone, we will get,

V = $\frac{1}{3} $πr2h

V = $\frac{1}{3} $ × $\frac{22}{7}$ × 64 × 14

V = 938.28 Cubic cm

Hence the volume of the cone is 938.28 Cubic cm.


Example 2: Calculate the total surface area of the cone having a radius of 6 cm and height = 10 cm?

Solution: According to the formula of the surface area of the cone, we will get,

Total Surface Area = πr(l + r)

Slant height l2 = r2+h2= (36)2+(100)2= 11.66 cm

Therefore,

Total Surface Area, A = π × 6(11.66 + 6) = π × 105.96= 333.017 $cm^{2} $.

So the TSA of the cone is 333.017 $cm^{2} $.


Practice Questions

  • Calculate the radius of the cone if height, h = 21 cm and the volume V = 198 $cm^{2} $.

  • Calculate the height of the cone if radius, r = 3 cm and the volume V = 66 $cm^{2} $.


Conclusion

Hence cones are an important structure of geometry. We see several examples of cones in daily life such as the cap of a clown, Ice cream cone and funnel etc. With the help of formulas of volume, surface area and slant height we can calculate different parameters of the cone.

FAQs on Cone

1. How will you differentiate between height and slant height?

The height (or altitude) of the cone is usually denoted by h, it is the vertical height of the cone which is the perpendicular distance from the top down to the base. On the other side, the slant height is the distance from the top to a point on the base circumference down the side. The Pythagoras theorem is used to measure the slant height of the cone with the help of the value of the radius of the base and height and it is usually denoted by l.


The relation between slant height and height is: l2 = r2+h2

2. How is the Volume of the Cone calculated?

With the help of the values of the height and radius of a cone, the volume of a cone can be calculated. Hence with the help of the formula of the volume of the cone that is given as one-third the product of the area of the circular base and the height of the cone. If the r is the radius of the base and h is the height of the cone, then the volume of the cone can be calculated as V=$\frac{1}{3} $πr2h.

3. What is the relation between the volume of the cone and volume of the cylinder?

Volume of the cylinder is given as the product of the area of the base and the height of the cylinder. If the radius of base of cylinder is r and height of cylinder is h then the formula of volume of cylinder is V = πr2h. 


If the radius of base of cone is r and height of cone is h then the formula of volume of cone is V = $\frac{1}{3} $πr2h.


So, the volume of a cone is one third of volume of the cylinder if the radius of base and height is same in both.


Therefore, Vcone=$\frac{1}{3} $VCylinder

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