Cone

A cone is more than just a 3D shape—it's a perfect blend of geometry and everyday life. From ice cream cones to traffic markers, this sleek structure pops up everywhere. In this lesson, you’ll be uncovering some of the most important properties, formulas, and real-world uses of these basic yet captivating three-dimensional figures.

1.0Introduction to Cone

A cone is a three-dimensional, solid geometric figure with a circular base and an apex or vertex, which is a pointed tip. It is formed by rotating a right triangle around one of the sides that is perpendicular to the other. The most widely used is the right circular cone, where the apex is directly above the centre of the base.

A cone is one of the members of the family of geometric solids and is crucial to theoretical mathematics and practical uses. Some common real-life examples of cones include an Ice Cream cone, a traffic cone, a party hat, etc. 

Interesting Fact: The word “cone” is derived from a Greek word, konos, which simply means “pine cone,” reflecting its characteristic tapering shape. 

2.0Cone Shape Properties

To understand what a cone actually holds, let’s get an insight into its various important properties: 

• Base: Unlike a cylinder, a cone only has one circular base at one end, and tapers at the other end.
• Apex: It is the highest point where the curved surface tapers off, also known as the vertex of the cone.
• Height (h): It is the perpendicular or the shortest distance from the base of the cone to its apex. 
• Radius (r): The distance from the centre of the base to its circular end’s edge is referred to as the radius of a cone.
• Slant Height (l): It is the total length of the line connecting the apex to a point on the circular edge.
• Three-dimensional: Cones are three-dimensional figures, meaning cones, unlike 2D shapes, have volume and take up space.
• Curved Surface: In contrast to a pyramid, the side surface of a cone is smooth and continuous.
• Single Edge: The circular base is the only edge a cone possesses.
• One Vertex: Similar to the edge, the apex is also the sole or only vertex in the conical figure.

3.0Cone Geometry Formulas

Now that we have the basic understanding of the cone shape, let’s get into some of its important geometrical formulas for practical implications of the figure: 

Surface Area of Cone

The surface area of a cone is the sum of all the areas covering the outer surface of the cone; that is, it consists of:

• The lateral (curved) surface area, which encircles the sides of the cone, and
• The area of the circular base

The surface area of a cone can be classified into two categories, that are: 

Curved Surface Area (CSA) of a Cone: 

Also referred to as Lateral Surface Area, it is the curved side area of the cone, meaning the side that encircles the cone from base to apex and does not include the base. The formula for the curved surface area of a cone is expressed as: 

Total Surface Area (TSA) of a Cone: 

The Total Surface Area is simply the addition of the curved surface area and the area of the circular base of the cone. The formula for the total surface area of a cone can be expressed as: 

Here, in both the formulas:  

• r = radius of the base
• l = slant height

Volume of Cone

The capacity of a cone is the volume of space inside it. Surprisingly, the volume of a cone is precisely one–third the capacity of a cylinder with the same base and height. The formula for the volume of a cone is written as:  

Here, 

• r = radius of the base
• h = height (perpendicular from base to apex)

Slant Height of Cone

The slant height (l) is the length from the apex to any point on the base's circumference. It is the hypotenuse of a right triangle, with the radius and height being the other two sides. The slant height of the cone is calculated as: 

4.0Example Problems on Cones

Problem 1: An ice cream cone has a circular base with a radius of 3.5 cm and a height of 12 cm. How much ice cream can it hold?

Solution: It is given that, 

Radius of the ice cream cone = 3.5

Height of the ice cream cone = 12 cm

The volume of ice cream the cone holds = $\frac{1}{3}\pi r^{2}h$

The volume of ice cream the cone holds=$\frac{1}{3}\times \frac{22}{7}\times 3.5^2\times 12$

The volume of ice cream the cone holds=$154c{m}^3$

Problem 2: A conical water tank has a slant height of 30 cm and a base radius of 24 cm. If the tank is filled to 80% of its total capacity, find the volume of water in it.

Solution: Given that, the slant height of the tank is 30 cm and the base radius is 24cm.

Slant Height of water tank(l)= $\sqrt{r^2+h^2}$

$30=\sqrt{24^2+h^2}$

$30^2=\sqrt{24^2+h^2}$

$h^2=900-576=324$

$h=\sqrt{324}=18cm$

Total volume of the water tank = $\frac{1}{3}\pi r^{2}h$

Total volume of the water tank = $\frac{1}{3}\times \frac{22}{7}\times 24^2\times 18$

Volume of water = $\frac{80}{100}\times \frac{1}{3}\times \frac{22}{7}\times 24^2\times 18$

Volume of water in the tank = $\frac{4}{5}\times \frac{1}{3}\times \frac{22}{7}\times 576\times 18=8689c{m}^3$

Problem 3: A conical tent is to be made of canvas cloth. The diameter of the base is 14 meters, and the vertical height is 24 meters.

• Find the amount of cloth required (curved surface area).
• If canvas costs ₹40 per m², find the total cost.

Solution: Given that the vertical height of the conical tent is 24 meters and the diameter of the base is 14 meters. Hence, the radius of the cone is 7m. 

1. Find the amount of cloth required (curved surface area)

Slant height of tent = $\sqrt{r^2+h^2}$

Slant height of tent = $\sqrt{7^2+24^2}=\sqrt{49+576}=\sqrt{625}=25m$

The amount of cloth required = curved surface area of the conical tent 

The amount of cloth required = $\pi rl=\frac{22}{7}\times 25\times 7=550m^2$

2. The total cost if canvas costs ₹40 per m²

It is given that the cost of the canvas is ₹40 per m²

Hence, the total cost of canvas = 40550=Rs 22000