Cone: Definition, Properties, Surface Area & Volume

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Cone is a three-dimensional shape with a smooth transition from a flat base, usually a circular base, to the point at the top, also known as the apex or vertex. A cone is made up of line segments that connect the apex (vertex), the common point, to every point of a circular base (which does not contain the apex). Cone can also be defined as a pyramid which has a circular cross-section, unlike a pyramid which has a triangular cross-section.

Table of Content

Definition of Cone
Properties of Cone
Types of Cone
- Right Circular Cone
- Oblique Cone
Surface Area of Cone
Volume of Cone
Things to Remember
Sample Questions

Key Terms: Cone, Total Surface Area, Curved Surface Area, Volume, Right Circular Cone, Oblique Cone, Vertex, Slant Height

Definition of Cone

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A cone can be defined as a three-dimensional shape formed using a set of line segments or the lines that connects the apex or vertex (common point) to all the points of the circular base, which does not contain the apex.

Height of Cone: The distance of the apex or vertex of the cone to the base is referred to as the height of the cone.
Radius of Cone: The value of the radius is measured from the circular base.
Slant Height of Cone: The length of the cone from apex to any point on the circumference of the base is the slant height. The formula for the slant height of the cone is given as l = √(r2+h2), where l denotes slant height, r denotes radius and h denotes the height of the cone.

Cone

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Properties of Cone

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A cone has one face only, which is the circular base of the cone.
A cone has no edges.
A cone has only one apex or vertex point.

Types of Cones

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There are essentially two types of cones. The right circular cone is one, and the oblique cone is another.

Right Circular Cone

The vertex of a right circular cone is located on the other side of the base. The radius-perpendicular line that represents the cone's height passes through the centre of the base circle.

Right Circular Cone

Oblique Cone

The vertex of an oblique cone is not located precisely across from the circular base. The base circle's centre is not intersected by the line that represents the cone's height.

Oblique Cone

Surface Area of Cone

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The amount of space that a cone's surface takes up is known as its surface area. A cone is a three-dimensional shape with a circular base. This indicates that a radius or diameter makes up the base. The height of the cone is measured from the centre of the base to the topmost piece of the cone. There are two approaches to calculating the cone's surface area:

Total Surface Area
Curved Surface Area

While the curved surface area just covers the area of the curved surface, the total surface area also includes the flat circular region. The total and curved surface area of a cone is given as follows if the radius of the base of the cone is "r" and the height of the cone's slant is "l":

TSA of Cone = π r(r + l)

CSA of Cone= πrl

Volume of Cone

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The amount of room or capacity a cone takes up is known as its volume. Cones' volume is expressed in cubic units such as cm3, m3, in3, etc. The volume of a cone, whose radius is 'r' and height is 'h' will be calculated as Volume of Cone = (1/3) πr2h cubic units Apart from that, the volume of a cone is one-third of the volume of a cylinder.

Volume of Cone = (1/3) × Volume of Cylinder Applications of Cone Here are some everyday items where we can see the conical shape:

Traffic Cones
Prism
Birthday Caps
Ice-Cream Cones
Funnel
Megaphones

Things To Remember

A cone is a three-dimensional shape that has a curved surface. It is formed using a set of line segments which connects the apex or vertex to all the points of a circular base.
A cone can be categorized as either an oblique cone or a right circular cone. In contrast to an oblique cone, which has a vertex that is not vertically above the base centre, the right circular cone has a vertex that is vertically above the centre of the base.
The perpendicular height is the distance from the centre of the base to the vertex. The circular base of the cone has a measured value of radius.
Slant Height of the cone is the distance between the vertex and the edge of the circular base of the cone is the slant height of the cone. The slant height of a cone is calculated using the formula l = √(h2+r2).
The total surface area (TSA) of the cone is πr(l + r), whereas, the curved surface area (CSA) of the cone is πrl.
The volume of the cone is calculated as ⅓ πr2h cubic units.

Read More: Surface Areas and Volumes

Sample Questions

Ques. Find the volume of a cone, if the radius is 4 cm and the height is 9 cm. (3 Marks)

Ans. Radius r = 4 cm

Height h = 9 cm

Using the volume of a cone formula,

Volume of cone =1/3 (π r2h) Volume of cone = 1/3 x 3.14 x 4² x 9 Volume of cone = 150.72 cm³

Ques. The circumference of the base of a 12 m high wooden solid cone is 44 m. find the volume. (3 Marks)

Ans. Circumference of cone = 44 m Height (h) = 12 m

2πr = 44 2 x (22/7) x r = 44 r = 44 x (1/2) x (7/22) r = 7 cm

Volume of cone = (1/3)πr²h = (1/3) x (22/7) x 72 x 12 = (1/3) x (22/7) x 7 x 7 x 12 = 616 cm3

The volume of the cone = 616 cm3

Ques. A vessel is in the form of a frustum of a cone. Its radius at one end is 8 cm and the height is 14 cm. If its volume is 5676/3 cm3, then find the radius at the other end. (3 Marks)

Ans. Volume of the frustum cone = (5676/3) cm3

Let r be the required radius Radius (R) = 8 cm

Height (h) = 14 cm

(1/3)πh (R2+r2+Rr) = (5676/3) (1/3) ⋅ (22/7) ⋅ (14) (82+ r2+8r) = 5676/3 r2+8r+64 = 129 r2+ 8r+64-29 = 0 r2+8r-65 = 0 (r+13) (r-5) = 0 r = -13, r = 5 cm (r= -13 is rejected as radius can never be negative)

So, the required radius = 5 cm

Ques. The perimeter of the ends of a frustum of a cone is 44 cm and 8.4π cm. If the depth is 14 cm, then find its volume. (3 Marks)

Ans. Perimeter of upper end = 44 cm

Perimeter of lower end = 8.4π cm

Height of frustum cone = 14 cm 2πR = 44 2 x (22/7) x R = 44 R = 44 x (1/2) x (7/22) R = 2 x (1/2) x 7 R = 7 2πr = 8.4π r = 8.4π x (1/2Π) r = 4.2

Volume of the frustum cone = (1/3)πh (R2+r2+Rr) = (1/3) x (22/7) x 14 (72+4.22+7(4.2)) = (44/3) (49+29.4+17.64) = (44/3) (96.04) = (44) (32.013) = 1408.57 cm3

The volume of the frustum cone = 1408.57 cm3

Ques. Calculate the curved surface area, total surface area and volume of the given cone. (3 Marks)

Ans. Here,

Height of cone (h) = 8 cm

Slant height (l) = 10 cm

Now, Curved surface area (CSA) = πrl = 22/7 × 6 × 10 = 188.57 cm2

Total surface area (TSA) = πr(r + l) = 22/7 × 6 (6 + 10) = 301.71 cm2

Volume of cone (V) = πr2h/3 = 1/3 × 22/7 × 62 × 8 = 22/21 × 36 × 8 = 301.71 cm3

Ques. If the total surface area of a cone is 704 cm2 and the radius of its base is 7 cm, find the volume of the cone. (3 Marks)

Ans. Here, Radius of cone (r) = 7 cm

Total surface area of cone = 704 cm2 Πr (r + l) = 704 22/7 × 7 (7 + l) = 704 22 (7 + l) = 704 154 + 22l = 704 22l = 704 – 154 l = 550/22 l = 25 cm

Now, Volume of cone (V) = πr2h/3 = 1/3 × 22/7 × 72 × 24 = 1232 cm3

Ques. The radius of a cone is 3 cm and the vertical height is 4cm. Find the slant height of the cone. (3 Marks)

Ans. We have r = 3cm, h = 4 cm Let l cm be the slant height of the cone.

Then Slant height = √r2+h2 = √32+42 = √9 + 16 = √25 l = 5 cm

Thus, the slant height of the cone calculated as 5 cm.

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Cone: Definition, Properties, Surface Area & Volume

Definition of Cone

Properties of Cone

Types of Cones

Right Circular Cone

Oblique Cone

Surface Area of Cone

Volume of Cone

Things To Remember

Sample Questions

CBSE X Related Questions

1.
The value of p for which roots of the quadratic equation $x^2 - px + 6 = 0$ are rational, is

2.
Prove that: $\frac{\tan \theta}{1 - \cot \theta} + \frac{\cot \theta}{1 - \tan \theta} = 1 + \tan \theta + \cot \theta$

3.
Two dice are rolled together. The probability of getting an outcome $(x, y)$ where $x \gt y$, is

4.
The dimensions of a window are $156\text{ cm} \times 216\text{ cm}$. Arjun wants to put grill on the window creating complete squares of maximum size. Determine the side length of the square and hence find the number of squares formed.

5.
A chord of a circle, of radius 14 cm, subtends an angle of $60^\circ$ at the centre. Find the area of the smaller sector and perimeter of the smaller segment.

6.
A bag contains 25 balls. Some of them are yellow and others are green. One ball is drawn at random. If probability of getting a green ball is $3/5$, then find the number of yellow balls.

Similar Mathematics Concepts

Comments

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Cone: Definition, Properties, Surface Area & Volume

Definition of Cone

Properties of Cone

Types of Cones

Right Circular Cone

Oblique Cone

Surface Area of Cone

Volume of Cone

Things To Remember

Sample Questions

CBSE X Related Questions

1.The value of p for which roots of the quadratic equation $x^2 - px + 6 = 0$ are rational, is

2.Prove that: $\frac{\tan \theta}{1 - \cot \theta} + \frac{\cot \theta}{1 - \tan \theta} = 1 + \tan \theta + \cot \theta$

3.Two dice are rolled together. The probability of getting an outcome $(x, y)$ where $x \gt y$, is

4.The dimensions of a window are $156\text{ cm} \times 216\text{ cm}$. Arjun wants to put grill on the window creating complete squares of maximum size. Determine the side length of the square and hence find the number of squares formed.

5.A chord of a circle, of radius 14 cm, subtends an angle of $60^\circ$ at the centre. Find the area of the smaller sector and perimeter of the smaller segment.

6.A bag contains 25 balls. Some of them are yellow and others are green. One ball is drawn at random. If probability of getting a green ball is $3/5$, then find the number of yellow balls.

Similar Mathematics Concepts

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
The value of p for which roots of the quadratic equation $x^2 - px + 6 = 0$ are rational, is

2.
Prove that: $\frac{\tan \theta}{1 - \cot \theta} + \frac{\cot \theta}{1 - \tan \theta} = 1 + \tan \theta + \cot \theta$

3.
Two dice are rolled together. The probability of getting an outcome $(x, y)$ where $x \gt y$, is

4.
The dimensions of a window are $156\text{ cm} \times 216\text{ cm}$. Arjun wants to put grill on the window creating complete squares of maximum size. Determine the side length of the square and hence find the number of squares formed.

5.
A chord of a circle, of radius 14 cm, subtends an angle of $60^\circ$ at the centre. Find the area of the smaller sector and perimeter of the smaller segment.

6.
A bag contains 25 balls. Some of them are yellow and others are green. One ball is drawn at random. If probability of getting a green ball is $3/5$, then find the number of yellow balls.