A cone is a three-dimensional geometric figure with a flat circular base and a pointed apex. It is formed by stacking a series of circles with decreasing radius, resulting in a tapered shape. Common examples of cones include ice cream cones, birthday hats, and traffic cones. Even our eyes have 6–7 million cone cells, which help them adjust to color sensitivity.
Below is the figure of the cone with radius r, height h, and length L:
Parts of a Cone:
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 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).
The 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 the height of a cone are 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’. |
Note: A cone has one flat face, one curved face, an edge, and a vertex.
Examples of Conical Shapes in Everyday Life
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
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 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.
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
The 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
Read More: Surface area of a Cone
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.
The 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
The volume being product of three units (r × r × h) has a cubic unit.
Read More: Volume of a Cone
Let's quickly recap the formulas related to a coneTypes 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
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
An 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
The basic difference between the Right Circular Cone and Oblique Cone are added in the 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 do not coincide with each other. |
| The Axis of the right circular cone always makes a right angle with the base. | The 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 a 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.
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.
- The flat part of the frustum is known as the floor of frustum.
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
The 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.
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