Cone

Introduction

A cone is a 3-dimensional figure with a circular base and a sharp top known as the vertex. The term cone defines it as a solid with one flat base and one curved face. The form of the cone is found frequently in everyday life such as in ice-cream cones and funnels. Key characteristics of cone are its height, radius, and slant height. We apply the formula for surface area of cone to calculate the area covered by it, i.e., curved surface area of cone and total surface area of cone. The volume of cone informs us about the amount of space it will cover.

 

Table Of Contents

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

 

Cone

A cone is a three-dimensional geometric object with a flat circular base and a single curved surface that tapers smoothly from the base to a point known as the apex or vertex. The conic shape is that of an ice-cream cone or party hat.

 

Definition of Cone

The definition of cone is that it is a solid whose base is circular and has one curved surface that terminates at a pointed vertex. The cone is created when a right triangle is rotated about one of its perpendiculars.

 

Shape of a Cone

The cone's shape is distinct in the way it presents a flat surface (the base, which is circular) 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 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.

 

Surface Area of Cone

Surface area of cone consists of the curved surface area of cone and the base. Surface area of cone is the sum of area occupied by its curved surface and base.

Surface area of cone formula is:

 

Total Surface Area of Cone

Total Surface Area of Cone = π r l + π r²

Where:

• r = radius of base

• l = slant height

• π ≈ 3.1416

This cone surface area formula consists of:

Curved Surface Area of Cone = π r l

Base Area = π r²

Therefore, the total surface area of cone = curved surface area + base area

 

Curved Surface Area of Cone

Curved surface area of cone is the area of the cone minus its base. It is the curved side area of the cone.

Formula:

Curved Surface Area of Cone = π r l

The curved surface area of cone comes to be highly useful in day-to-day life while calculating the wrapping material of a conical gift.

 

Volume of Cone

The volume of cone is the amount of space contained by the cone.

Formula:

Volume of Cone = (1/3) π r² h

Where:

• r = radius of the base

• h = height of the cone

• π ≈ 3.1416

Volume of cone determines the capacity of objects that are cone-shaped in nature such as funnels or ice cream cones.

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

= 3.14 × 7 × 25

= 549.5 cm²

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 = π × r × l + π × r²

= 3.14 × 6 × 10 + 3.14 × 6²

= 188.4 + 113.04

= 301.44 cm²

Answer: 301.44 cm²

 

Example 3: 

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

Solution:

 Volume of Cone = (1/3) × π × r² × h

= (1/3) × 3.14 × 25 × 12

= (1/3) × 3.14 × 300

= 314 cm³

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 = π × r × l + π × r²

= 3.14 × 4 × 5 + 3.14 × 4²

= 62.8 + 50.24

= 113.04 cm²

Answer: 113.04 cm²

 

Practice Problems

1. Determine the curved surface area of 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 cone
   (Use π = 3.14)
   

 

Conclusion

The cone is a fundamental 3D shape in geometry with real-world applications. Understanding the definition of cone, properties of cone, shape of a cone, surface area of cone, curved surface area of 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.

 

Related Links

Volume of Cone - Learn how to calculate the volume of a cone using the standard formula, with clear diagrams and practical examples.

Cylinder - Understand the properties of a cylinder, including surface area and volume, with real-life examples and visual explanations.

Geometric Shapes - Explore various geometric shapes, both 2D and 3D, their characteristics, and how they’re used in mathematics and everyday life.

 

Frequently Asked Questions on Cone

1. What is TSA and CSA of cone?

 Ans:

 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?

Ans: 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?

Ans: 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?

 Ans: 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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