Cone

Introduction

A cone is a solid shape in geometry that has special features like a circular base, a curved surface, and a vertex (point). It is studied as one of the basic 3D shapes along with cylinders, spheres, and cubes. To understand a cone, we look at important parts such as its radius, height, and slant height. These measurements are used to calculate values like the surface area and volume of the cone. Learning about cones helps us practice important geometry concepts and apply formulas correctly in problems.

In this guide, we will learn the definition of a cone, its shape and properties, and the formulas for finding its surface area and volume. We will also work through solved examples, practice problems.

 

Table Of Contents

• Definition of Cone
• Shape of a Cone
• Properties of a Cone
• Curved Surface Area of Cone
• Surface Area of Cone
• Volume of Cone
• Solved Examples on Cone
• Practice Problems on Cone
• Conclusion
• FAQs On Cone

 

Definition of Cone

A cone is a three-dimensional (3D) solid shape. It has one flat circular base at the bottom and one curved surface that rises from the base and narrows smoothly to meet at a single point called the vertex (or apex). You can think of a cone as a triangle that is rotated around one of its sides. When a right-angled triangle is spun around its height, it forms a cone.

 

Shape of a Cone

The cone's shape is distinct in the way it presents a flat surface (the circular base) and a curved surface that progresses to a single point. The predominant features of the shape of a cone are:

• Base: Flat and circular

• Vertex: The point

• Height (h): The distance from the vertex to the center of the base perpendicularly

• Slant height (l): The distance from the vertex to the edge of the base diagonally

• Radius (r): Distance from the center to the edge of the circular base

 

Properties of a Cone

The properties of cone are:

• A cone has a single circular base.

• It has one curved surface.

• It has one vertex.

• The cone is a three-dimensional solid

• A cone may be a right circular cone or an oblique cone.

• It has slant height, height, and radius.

These cone properties are used in calculating different measurements such as volume and surface areas.

 

Curved Surface Area of Cone

Like other three-dimensional solids, a cone is made up of both a flat surface and a curved surface. The flat surface is the circular base, and the curved surface is the slanted, round part that connects the base to the vertex. The curved surface area of a cone refers only to this curved portion, excluding the flat base.

When calculating the curved surface area (CSA), we focus only on how much area the slanted part of the cone covers. This is very useful in real-life applications such as finding the paper needed to make a party hat or the material required to cover a cone-shaped tent.

Formula:

Curved Surface Area of Cone =πrl

where:

• r = radius of the base
• l = slant height of the cone
• π≈3.1416

The slant height (l) is different from the vertical height of the cone. It is the diagonal length from the vertex of the cone to any point on the circular boundary of the base.

The curved surface area is also known as the lateral surface area of the cone. This means we are only measuring the outer "side wall" of the cone, without including the base.

 

Surface Area of Cone

The portion occupied by the outer surface of a cone is referred to as the surface area of a cone. It is always in square units. A cone is obtained when a right-angled triangle is rotated around one of its sides and, in doing so, produces both a curved surface and a circular base.

As a cone also contains a curved surface and a flat base, its surface area can be separated into two portions:

Curved Surface Area (CSA) - the curved side only.
Total Surface Area (TSA) - curved surface area + base area.

We can also define cones as right circular cones or oblique cones. The vertex is directly above the center of the base in a right circular cone. The vertex is not vertically above the center of the base in an oblique cone.

Surface area of a cone formula is:

Total Surface Area of Cone =πrl+πr2

Where:

• r = radius of base

• l = slant height

• π≈3.1416

 

Derivation of Surface Area of a Cone

To derive the surface area of a cone, let us take a right circular cone with:

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

 

Step 1: Break the cone into parts

A cone has two surface parts:

• Curved Surface Area (CSA) - the slanted curved portion.
• Base Area - the flat circular base.

So,

TSA=CSA+Base Area

Step 2: Formula for Curved Surface Area

Think of the curved surface of the cone as a sector of a circle (when you cut and open the cone).

• The radius of the sector = slant height (l).
• The arc length of the sector = circumference of the base = 2\pi r.

Now, the area of a sector of a circle is:

Sector Area=πl2arc length​2πl

Substitute arc length = 2πr

CSA=πl22πr​2πrl

CSA=πrl

Step 3: Add Base Area

The base is a circle of radius r.

Base Area=πr2

Step 4: Total Surface Area

Now add both parts:

TSA=CSA+Base Area

TSA=πrl+πr2

Factorize:

TSA=πr(l+r)

Therefore, the Total Surface Area of a Cone is:

TSA=πr(l+r)

and the Curved Surface Area is:

CSA=πrl

 

 

Volume of Cone

The volume of a cone is the amount of space contained inside the cone. It tells us how much the cone can hold and is always measured in cubic units.

Formula:

Volume of Cone=13πr2h

Where:

• r = radius of the base

• h = height of the cone

• π ≈ 3.1416

The formula shows that the volume of a cone is one-third of the volume of a cylinder having the same base and height.

Solved Example On Cone

Example 1: 

A cone has radius 7 cm and slant height 25 cm. Find its curved surface area.

Solution:

Curved Surface Area of Cone=π×r×l

Curved Surface Area of Cone=3.14×7×25

Curved Surface Area of Cone=549.5cm2

Answer: 549.5 cm²

 

Example 2: 

A cone has a slant height of 10 cm and a radius of 6 cm. Find total surface area of cone.

Solution:

Total Surface Area of Cone=πrl+πr2

Total Surface Area of Cone=3.14×6×10+3.14×62

Total Surface Area of Cone=188.4+113.04

Total Surface Area of Cone=301.44cm2

Answer: 301.44 cm²

 

Example 3: 

Find the volume of the cone whose radius is 5 cm and height is 12 cm.

Solution:

²Volume of Cone=13×π×r²×h

Volume of Cone=13×3.14×(5)×12

Volume of Cone=13×3.14×25×12

Volume of Cone=13×3.14×300

Volume of Cone=314cm2

Answer: 314 cm³

 

Example 4: 

A cone with a radius = 4 cm and a slant height = 5 cm. Calculate the total surface area using the formula of the surface area of a cone.

Solution:

Total Surface Area=πrl+πr2

 

 

Total Surface Area=3.14×4×5+3.14×42

Total Surface Area=62.8+50.24

Total Surface Area=113.04cm2

Answer: 113.04 cm²

 

Practice Problems

1. Determine the curved surface area of a cone with a radius of 10 cm and a slant height of 15 cm.
    (Use π = 3.14)
2.  A cone has a radius of 8 cm and height of 12 cm. Determine the volume of cone.
    (Use π = 3.14)
3. Apply the formula for surface area of cone to determine the total surface area of cone when radius = 7 cm and slant height = 13 cm.
   (Use π = 3.14)
4. The volume of the cone is 376.8 cm³ and the radius is 6 cm. Determine the height of the cone.
    (Use π = 3.14)
5. A cone has radius = 9 cm and slant height = 10 cm. Determine both:
    a) Curved surface area of cone
    b) Total surface area of the cone
   (Use π = 3.14)
   

 

Conclusion

The cone is a fundamental 3D shape in geometry with real-world applications. Understanding the definition of cone, the properties of a cone, shape of a cone, surface area of cone, curved surface area of a cone, total surface area of cone, and the volume of cone helps in solving practical and mathematical problems. Mastering the surface area of cone formula and volume of cone formula makes it easier to work with cones in real life and academics.

 

Frequently Asked Questions on Cone

1. What is TSA and CSA of cone?

 Answer:

 TSA (Total Surface Area) of Cone = πr(l + r)

 CSA (Curved Surface Area) of Cone = πrl

 Where r is the radius and l is the slant height.

 

2. What is a cone in mathematics?

Answer: A cone is a 3D shape with a circular base and smooth curved surface that tapers from the base to a point known as the vertex. It possesses one face, one curved surface, one edge, and one vertex.

 

3. How many sides does a cone have?

Answer: A cone does not have flat sides like polygons. It has one circular face, one curved surface, and one edge (the circle boundary). So, technically, it has no sides, but it has 1 face and 1 curved surface.

 

4. What are the three types of cones?

 Answer: Types of Cones :

1. Right Circular Cone - Vertex is directly above the center of the base.
   
2. Oblique Cone - Vertex is not at the center of the base.
3. Elliptical Cone - Base is an ellipse rather than a circle.

 

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