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Cone: Definition, Formula, Types, Examples & Properties

A Cone is a three-dimensional Geometric Figure that has a flat and curved surface called a base along with a pointed tip known as a vertex or apex. It is one of the most common figures studied in geometry.

In this article, we will learn about what is a cone, the parts of a cone, the number of faces, edges, and Vertices in a cone, cone formulas, and many more concepts related to Cones in detail.

Table of Content

What is a Cone?

The cone is a 3-D figure whose one of the portions is circular and flat and the other portion is pointed. Cone in math is one of the most commonly used figures. It appears to be a combined version of two 2-D figures namely a triangle and a circle. We can also say that Cone is a stack of many rings with a decreasing radius. The most common examples of cones are an ice cream cone, a sharpened pencil lead, a birthday cap, and the list goes on. Even our eyes have 6–7 million cone cells which help them adjust to color sensitivity.

Cone Definition

Cone is a three-dimensional figure whose base is flat and circular and the apex is pointed.

Till now we have got basic information about cones. Now let’s discuss its different parts.

Cone

Parts of a Cone: Vertex, Base, and Axis

A cone has three parts: a vertex, a base, and an axis.

  • Base: It is the flat surface which is generally circular and is also the largest cross-section of the cone.
  • Vertex: A cone converges to a point that is directly above the center of its base, this point is known as its vertex.
  • Axis: The line joining the vertex and the center of the base is known as the axis of the cone. A cone has a circular symmetry around its axis.
  • Radius: The radius of a cone is the radius of the circular base of the cone. It is basically the distance from the center of the base to its circumference.

Slant Height of a Cone

A slant height of the cone is measured along its curved surface which is also the longest height of the cone. It is the line segment that connects the tip of the cone (vertex) to a point on the boundary of its base. It is generally measured in units such as centimeters (cm) and meters (m).

Formula for Slant Height of a Cone

Formula for calculating the slant height of the cone can be derived by using the Pythagorean theorem. It can be defined as:

l = √h2 + r2

where,

  • l is Slant Height of Cone
  • h is Altitude(height) of tCone
  • r is Radius of Base

Slant Height vs Height of a Cone

  • Slant Height: It is defined as the distance from the vertex or apex of the cone to a point on the circumference of its base.
  • Height: The height of a cone is defined as the distance from the vertex or apex of the cone to the center of its base.

Key differences between both slant height and height of a cone is given as follows:

Slant Height Height
Slant Height is the distance along the curved surface of the cone. Height is the vertical distance between the vertex and the center of the base.
It is the longest distance between the base and the vertex. It is the shortest distance between the base and the vertex.
Slant Height is also called hypotenuse. Height is also called altitude.
It is denoted by ‘l’. It is denoted by ‘h’.

Faces, Edges, and Vertices of a Cone

A cone has one flat face, one curved face, an edge, and a vertex.

Face 2 (1 flat face, 1 curved face)
Vertex 1
Edge 1 (curved)

Cone Shape in Everyday Objects

In our day-to-day life, we can see various objects which are conical in shape.

A few examples are:

  • Birthday cap
  • Traffic cone
  • Christmas tree
  • Funnel
  • Waffle cone
  • Radish
  • Megaphone
  • Tent
  • Sharpened pencil

Real Life Examples of Cone

Surface Area of a Cone

In solid shapes, a surface area can be defined as the total area covered by all its faces(i.e. flat as well as curved). In a cone, a surface area is the sum total of the area of its flat surface and the curved surface. It is the area that covers the outer surface of the cone. It is measured in square units like m2,cm2 etc.

Curved Surface Area of Cone Formula

A curved surface area is the area covered by the lateral or curved part of the cone. It can be calculated by the given formula:

Curved Surface Area (CSA) = πrl = πr√h2 + r2

where

  • r is Radius of Base
  • l is Slant height of Cone
  • h is Altitude of Cone

Total Surface Area of Cone

Total surface area is the sum of the curved surface area and the flat surface area. Above we have already discussed the curved surface area of the cone. Now let’s find out about its flat surface area.

A flat surface area also known as base area is the area covered by the base of the cone which is circular in general, so it can be calculated by the given formula:

Base area = πr2, where r is the radius of the base.

Now, the total surface area of the cone can be given by:

Total Surface Area (TSA) = Base area + CSA

(TSA) = πr2 + πrl

(TSA) = πr(r + l)

where

  • r is Radius of Base
  • l is Slant Height of Cone

How to Find Surface Area of a Cone?

To find the Surface area of a Cone the above-mentioned formula can be used.

Let’s take a few examples to understand that better.

Example 1: Find the surface area of a cone whose slant height is 20 cm and the radius of the base is 14 cm. (Use π=22/7)

Given,

  • Slant height (l) = 20 cm
  • Radius (r) = 14 cm

We know that,

Total surface area of cone (TSA) = πr(r + l)

TSA = 22/7 × 14 × (14+20)

⇒ TSA = 22/7 × 14 × (34)

⇒ TSA= 22/7 × 476

⇒ TSA= 1496 cm2

Hence, total surface area of the cone is 1496 cm2

Example 2: Measure the total surface area of a cone whose height is 4 cm and the radius of the base is 9 cm.(Use π=22/7)

Given,

  • Height (h) = 4 cm
  • Radius (r) = 9 cm

We need to calculate slant height first by formula

l = √h2 + r2

⇒ l = √42 + 32

⇒ l = √16 + 9

⇒ l = √25

⇒ l = 5 cm

Slant height (l) = 5 cm

We know that,

Total surface area of cone (TSA) = πr(r + l)

TSA = 22/7 × 3 × (3+5)

⇒ TSA = 22/7 × 3 × 8

⇒ TSA = 75.43 cm2

Hence, the total surface area of cone is 75.43 cm2

Volume of a Cone

Cone being a 3-D shape occupies space and thus has a volume which can be described as the amount of space it occupies or in simple words it can be said to be the capacity of the cone. It is measured in cubic units like m3, cm3, in3 etc.

Cone Volume Formula

Volume of a Cone can be determined by multiplying one-third of its base area(πr2) with its height(h). Thus the formula is:

V = (πr2h)/3

Volume being product of three units (r × r × h) has a cubic unit.

How to Find Volume of a Cone?

To find the volume of the cone we will use its above-mentioned formula i.e.

V = (πr2h)/3

Let’s take an example to understand that better.

Example: Find the volume of a cone whose height is 10 cm and the radius of the base is 3.5 cm (Use π=22/7).

Given,

  • Height (h) = 10cm
  • Radius (r) = 3.5cm

We know that,

Volume of Cone = (πr2h)/3

⇒ V = 1/3 ×22/7 × (3.5) × (3.5) × 10

⇒ V = 1/3 × 22/7 × 122.5

⇒ V = 1/3 × 385

⇒ V = 128.33 cm3

Hence, volume of cone is 128.33 cm3

Cone Formulas

We have already discussed various formulas related to a solid shape CONE. Let’s take a quick recap of the same.

Name Formula Measured In
Slant Height (l) √(h2 + r2) Unit (m, cm)
Curved Surface Area (CSA) πrl = πr√(h2 + r2) Square unit (m2, cm2)
Base Area πr2 Square unit (m2, cm2)
Total Surface Area (TSA) πr(r + l) Square unit (m2, cm2)
Volume (v) (πr2h)/3 Square unit (m3, cm3)

Types Of Cones

Based on the alignment of the vertex with its circular base, Cones are broadly classified into two types, namely:

  • Right Circular Cone
  • Oblique Cone

Right Circular Cone and Oblique Cone

Right Circular Cone

Right Circular Cone is a cone whose altitude makes a right angle (90°) with the center of its base and the base of the cone is circular. In the right circular cone, the axis and vertical height (altitude) coincide with each other. If we rotate a right-angled triangle along its legs, a right circular cone can be generated.

Oblique Cone

Oblique cone is a cone whose vertex is not perpendicularly aligned to the center of its circular base. In an oblique cone, the vertex is not directly above the center of the base. It is always ’tilted’ towards one side.

Right Circular Cone vs Oblique Cone

Basic difference between Right Circular Cone and Oblique Cone are added in table below,

Right Circular Cone Oblique Cone
Its vertex is directly above the center of the base. Its vertex is not directly above the center of the base.
Its altitude and axis coincide with each other. Its altitude and axis domakethe  not coincide with each other.
Axis of right circular cone always makes a right angle with the base. Axis of an oblique cone does not makes a right angle with the base.

Double Napped Cone

A double-napped cone is made of two cones joined at their vertex. An hourglass is a perfect example of a double-napped cone. A double-napped cone consists of the following parts:

  • A generator and a generator angle: A generator is an oblique line that is rotated to produce a double-napped cone and the angle it makes with the axis is known as the generator angle.
  • A vertex and a vertex angle: A vertex is a point where both the cones meet and the angle made at the vertex is called vertex angle.
  • A lower nappe and an upper nappe: The lower cone and the upper cone of the double-napped cone are called the lower nappe and the upper nape respectively.
  • Axis of symmetry: The line joining both the axis of the individual cone is known as the axis of symmetry of the double-napped cone.

Double Napped Cone

Frustum of a Cone

The term “frustum” is a Latin word meaning ‘a piece’. If we take a cone and slice it into two parts (cut parallel to the base). The upper part of the cone will maintain its shape (i.e. a cone) and the lower part will be the frustum. In other words, the frustum can be said to be the flat-top cone (i.e. a cone whose upper part is flattened). Some common facts about the Frustum of a Cone:

  • It is also known as a truncated cone.
  • It has no vertex.
  • It has three faces(2 flat and 1 curved) and 2 edges.
  • It has two bases (a top and a bottom) so it has two radii for the same.
  • Flat part of frustum is known as floor of frustum.

Frustum-of-cone

Volume of Frustum of a Cone

Frustum of a cone is a three-dimensional figure and thus has a volume. The volume of frustum of a cone is the total amount of space it can occupy or we can say it is the total capacity of the frustum of the cone. It is measured in cubic units like m3, cm3, etc.

Volume of Frustum of Cone (V) = 1/3 πh(R2 + r2 + Rr)

where

  • r is Radius of Lower Base of Frustum of Cone
  • R is Radius of Upper Base of Frustum of Cone
  • h is Height of Frustum of Cone

Surface Area of Frustum of a Cone

Surface area of a Frustum of a cone is determined by adding the area of all its faces. Since the Frustum of a cone has 3 faces (1 curved and 2 flat), we need to sum up the area of a curved surface along with the area of the two bases.

Surface Area of Frustum of Cone = CSA + UBA + LBA

Hence, Surface Area of Frustum of a Cone = πl(R + r) + πR2 + πr2

Surface Area of Frustum of Cone = πl(R + r) + π(R2 + r2)

where,

  • r is Radius of Lower Base of Frustum of Cone
  • R is Radius of Upper Base of Frustum of Cone
  • l is Slant Height of Frustum of Cone

Also Check,

Solved Examples on Cone

Example 1: Find the slant height of a cone whose Curved Surface Area is 330m2 and whose diameter of base is 10 m.

Solution:

Given,

Curved Surface Area (CSA) = 330 m2

Diameter = 10 m

radius (r) = diameter/2

r = 10/2 = 5 m

Also, CSA = πrl

Putting given values we get

330 = 22/7 × 5 × l

⇒ 330 = 22/7 × 5 × l

⇒ 330 × 7 = 110 × l

⇒ 2310/110 = l

⇒ l = 21 m

Hence, slant height (l) is 21 m

Example 2: Calculate the height of a frustum of a cone whose volume is 616 cm3 and the radii of the two bases are 3 cm and 5 cm respectively.

Solution:

Radius of Upper Base (r) = 3 cm

Radius of Lower Base (R) = 5 cm

Volume (V) = 616 cm3

We know that

Volume of Frustum of Cone (V) = 1/3 × h π(R2 + r2 +Rr)

Putting given values we get

616 = 1/3 × h × 22/7(52 + 32 + 5×3)

⇒ 616 = 1/3 × h × 22/7(25 + 9 + 15)

⇒ 616 = 1/3 × h × 22/7 × 49

⇒ 616 × 3 × 7 = h × 22 × 49

⇒ 12936 = 1078 × h

⇒ h = 12936/1078

⇒ h = 12 cm

Hence the height of the frustum of the cone is 12 cm.

Example 3: In a cone, the volume and its height are in the ratio 66: 7. Find the diameter of the base of the cone.

Solution:

Let, Volume of cone = 66a and Height of Cone = 7a

We know that,

Volume of Cone (V) = (πr2h)/3

Putting given values we get,

66a = 22/7 × r2 × 7a × 1/3

⇒ 66a × 3 × 7 = 22 × 7a × r2

⇒ 1386a = 154a × r2

⇒ 1386a/154a = r2

⇒ r2 = 9

⇒ r = √9 = 3

Also, we know Diameter(d) = 2r

d = 2 × 3 = 6 units

Hence diameter of base of cone is 6 units.

Example 4: If the Total Surface Area of the cone is 3300 cm2 and the radius of the base is 21 cm. Calculate the slant height and volume of the cone.

Given,

Total Surface Area (TSA) = 3300 cm2

Radius of Base (r) = 21 cm

We know that,

Total Surface Area (TSA) = πr(r + l)

Putting given values we get,

3300 = 22/7 × 21 × (21 + l)

⇒ 3300 × 7 = 22 × 21 × (21 + l)

⇒ 23100 = 462 × (21 + l)

⇒ 23100/462 = (21 + l)

⇒ 21 + l = 50

⇒ l = 50 – 21

⇒ l = 29 cm

Hence, Slant Height of cone is 29 cm.

Now to find volume we first need to find height of cone by formula: l2 = h2 + r2 

Putting values we get,

292 = h2 + 212

⇒ 841 = h2 + 441

⇒ 841 – 441 = h2

⇒ h = √400

⇒ h = 20 cm

Now, Volume of cone (V) = (πr2h)/3

By putting values we get,

V = 22/7 × 21 × 21 × 20 × 1/3

⇒ V = (22 × 21 × 21 × 20) / (7 × 3)

⇒ V = 194040 / 21

⇒ V = 9240 cm3

Hence, Volume of a Cone is 9240 cm3.

Example 5: Find the Total Surface Area of a cone whose Curved Surface Area is 264 m2 and the radius of the base is 6 m.

Given,

Curved Surface Area (CSA) = 264 m2

Radius of Base (r) = 6 m

We know that,

Total Surface Area (TSA) = Base Area + CSA

TSA = πr2 + πrl

Here CSA = πrl = 264 m2 (given)

Putting values in above formula we get,

TSA = (22/7 × 62 ) + 264

⇒ TSA = (22 × 36)/7 + 264

⇒ TSA = 792/7 + 264

⇒ TSA = 113.14 + 264

⇒ TSA = 377.14 m2

Hence, Total Surface Area of Cone is 377.14 m2.

Example 6: Calculate the Volume and the Total Surface Area of a cone whose diameter of base is 10 cm and height is 12 cm.

Given,

Height of cone (h) = 12 cm

Diameter of base (d) = 10 cm

Also Radius of base (r) = d/2 = 10/2

r = 5 cm

We know that, Volume of Cone (V) = (πr2h)/3

By putting values we get,

V = (22/7 × 52 × 12)/3

⇒ V = (22 × 25 × 12)/ (7 × 3)

⇒ V = 6600/21

⇒ V = 314.29 cm3

Hence, volume of a cone is 314.29 cm3

Now, Total Surface Area (TSA) = πr(r + √h2 + r2 )

By putting values we get,

TSA = 22/7 × 5 × (5 + √122 + 52 )

⇒ TSA = 22/7 × 5 × (5 + √144 + 25 )

⇒ TSA = (22 × 5 × (5 + √169 ))/7

⇒ TSA = (110 × (5 + 13 ))/7

⇒ TSA = (110 × 18)/7

⇒ TSA = 1980/7

⇒ TSA = 282.86 cm2

Hence, Total Surface Area of Cone is 282.86 cm2.

Practice Questions on Cone

Some practuice question on Cone Formulas are,

Q1: Find the volume of a cone whose radius of the base is 12.2 cm and the height is 13.3 cm.

Q2: If the volume of a cone is 1540 m3 and the height is 10 m. Find the radius of the base.

Q3: The total surface area of the cone is 418 m2 and the radius of the base is 7 m. Find the slant height of the cone.

Q4: A bucket has a height of 14 cm and the radii of two bases are 6cm and 9 cm respectively. Find the volume of the bucket.

Q5: If the slant height of the cone is 29 units and the radius of the base is 20 units. Calculate the altitude (height) of the cone.

Q6: Calculate the volume of a cone whose slant height is 14 m and the diameter of the base is 12 m.

Q7: If the Base Area of a cone is 38.5 cm2 and the slant height of the cone is 5 cm. Find the Total Surface Area of the cone.

Q8: Find the Total Surface Area and volume of the cone whose radius is 8 cm and height is 15 cm.

FAQs on What is a Cone

What is a Cone Shape?

A cone is a 3D figure whose one of the portion is flat and circular while other end is curved and pointed.

How Many Faces Does a Cone Have?

A cone has 2 faces (one curved and one flat).

How to Find Height of a Cone?

Height of a cone can be calculated by using the formula,

h = √l2 – r2 

How to Calculate Volume of a Truncated Cone?

Volume of a truncated cone be calculated by using the formula

v = 1/3 × h π(R2 + r2 +Rr)

How to Calculate Slant Height of a Cone?

Formula to calculate the slant height of the cone is:

l = √h2 + r2

How many Vertex does a Cone have?

A cone has 1 vertex only.

How many Edges do a Cone have?

A cone has only 1 edge.

How to Find Surface Area of a Cone?

Surface area of a cone is calculated by the formula,

TSA = πr(r + l)

How to Find Radius of a Cone?

Radius of the cone can be calculated by the formula

r = √l2 – h2

What is Frustum of a Cone?

Frustum of a Cone is the cut out portion of a cone whose aapex portion has been cut as a result it has circular and flat surface on both the ends

How many Faces does a Frustum have?

A frustum has three faces (2 flat and 1 curved).


 


Reffered: https://www.geeksforgeeks.org

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