Surface Area of a Cone Formula: Check Solved Examples

Surface Area of a Cone Formula: A cone is a three-dimensional shape with a circular base. This indicates that the base has a radius and a diameter. The height of the cone is the distance between the centre of the base and the apex. The area occupied by the surface of a cone is known as its surface area. Students must understand the surface area of a cone formula to score well in the exam.

We will learn from this article what a cone is and how to calculate the total surface area of a cone formula, the curved surface area of a cone formula, the lateral surface area of the cone and more. Along with this, the student will learn how to calculate the surface area of any three-dimensional object. Stay tuned to discover more about the area of cone formula through illustrations and examples.

Surface Area of a Cone Formula: What is a Cone?

A \(3 – D\) shape that tapers smoothly from a flat circular base to a point called the vertex or the apex is the cone. It is formed by a set of line segments, connecting an apex to all the points on a circular plane base. A cone is made up of one curved face, one flat face, one curved edge and one vertex.

The radius, height, and slant height are the three components of a cone. The radius \(r\) is defined as the distance between the circular base’s centre and any point on its circumference. The distance between the apex and the centre of the circular base is the height \(h\) of the cone.

The distance between the apex of the cone and any point on its circumference is defined as the slant height \(h\). The radius, height, and slant height of a cone are shown in the diagram below. A party hat, a tent, an ice cream cone, and a road barrier are all examples of cones in the real world.

What is the Surface Area of Cone?

The surface area of a cone is the area occupied by the surface of a cone. The shape of a cone is formed by arranging several triangles and spinning them around an axis. It has a total surface area and a curved surface area because it has a flat base.

A cone can be classified as a right circular or an oblique cone. In a right circular cone, the vertex is vertically above the centre of the base, but in an oblique cone, the vertex is not vertically above the centre of the base.

Total Surface Area of Cone Formula

There are two types of surface area in a cone:

1. Total Surface Area: A cone’s total surface area is defined as the cone’s total area occupied in a three-dimensional space. It is the same as the sum of the curved surface and the cone’s base.
2. Curved Surface: The curved surface of a cone is the area of the cone, excluding the base.

Learn About Volume of Right Circular Cones

If the radius of the base of the cone is \(r\) and the slant height of the cone is \(l\), then the total surface area of cone \( = \,\pi r\,(r\, + l)\).

And the curved surface area of cone \( = \,\pi rl\)

We can find the relationship between the cone’s surface area and its height by applying the Pythagoras theorem.

We know that \({h^2} + {r^2} = {l^2},\) where \(r\) is the cone’s height, \(l\) is the base’s radius, and is the cone’s slant height.

\(l\, = \,\sqrt {({h^2} + {r^2})} \)

The total surface area of the cone in terms of height \(\, = \pi r(r + \,\sqrt {({h^2} + {r^2})} )\)

And the curved surface area of the cone in terms of height \( = \pi r\,\sqrt {({h^2} + {r^2})} \)

Derivation of the Surface Area of Cone

Consider a situation in which we must paint the faces of a conical flask. Before we start painting, we need to figure out how much paint we’ll need to cover all of the walls. To estimate the amount of paint needed, we need to know the area of each face of the flask, which is known as the total surface area. The sum of a cone’s face area equals its total surface area.

To see the figure created by the surface of a paper cone, cut it along its slant height. Assign the letters \(A\) and \(B\) to the two endpoints and the letter \(O\) to the point of intersection of the two lines.

If you cut this figure into several parts, such as \(O{b_1},\,O{b_2},\,O{b_3},……..O{b_n},\) each measuring the same length as the original cone’s slant height, you’ll see that \(n\) triangles are produced.

You can now calculate the overall area of this figure by adding the areas of these individual triangles. Hence,

Area of figure \( = \frac{1}{2} \times l \times {b_1} + \frac{1}{2} \times l \times {b_2} + \frac{1}{2} \times l \times {b_3} + …\,\,…\,\,…\,…\, + \frac{1}{2} \times l \times {b_n})\)

\( = \frac{1}{2} \times l \times ({b_1} + {b_2} + {b_3} + …\,\,…\,\,…\,\,…\, + {b_n})\)
\( = \left( {\frac{1}{2}} \right) \times l \times ({\rm{length}}\,{\rm{of}}\,{\rm{an}}\,{\rm{entire}}\,{\rm{curved}}\,{\rm{boundary}})\)

Length of entire curved boundary \( = \) Circumference of base \( = \,2\pi \, \times r\)
(where \(r\) is the radius of the base)

Thus, area of figure \( = \,\frac{1}{2} \times 2\pi  \times r \times l\, = \,\pi rl\)

Hence, the curved surface area of a cone \( = \,\pi rl\)

Total Surface Area of a Cone Formula

\({\rm{Total}}\,{\rm{Surface}}\,{\rm{Area}}\,{\rm{(TSA)}}\,{\rm{ = }}\,{\rm{CSA}}\,{\rm{ + }}\,{\rm{Area}}\,{\rm{of}}\,{\rm{Circular}}\,{\rm{Base}}\)

Base Area \( = \pi {r^2}\)

Curved Surface Area \({\rm{CSA}}= \pi rl\)

Thus, the total surface area is given by

\({\rm{TSA}} = \pi {r^2} + \pi rl = \pi r(r + l)\)

Surface Area of Right Circular of a Cone Formula

A circular cone has a right circular section. A right circular cone has an axis that is perpendicular to the base.

The surface area of cone \( = \,\,\pi r(r + \,\sqrt {{h^2} + {r^2}} )\)
where \(r\) is the radius of the circular base
\(h\) is the height of the cone

The slant height of the cone, \(l\, = \,\sqrt {{h^2} + \,{r^2}} \)

Therefore, surface area \( = \pi r(r + l)\)

Solved Examples on Surface Area of a Cone Formula

Q.1. Calculate the total surface area of the cone and curved surface area of the cone whose radius is \(14\,{\rm{cm}}\) and slant height is \(4\,{\rm{cm}}\) \((Use\,\pi \, = \,22/7)\).
Ans: Given that: \(r\, = \,14\,{\rm{cm}}\,{\rm{,}}\,{\rm{l}}\,{\rm{ = }}\,{\rm{4}}\,{\rm{cm}}\,{\rm{,}}\,{\rm{and}}\,\pi \,{\rm{ = }}\,\frac{{22}}{7}\,\,\)
We know, the total surface area of the cone \( = \,\pi r\,(r\, + \,l),\,\)
\( = \,\frac{{22}}{7}\, \times \,14\, \times \,(14\, + 4)\, = \,\frac{{22}}{7}\, \times \,14\, \times \,18\, = \,792\,{\rm{c}}{{\rm{m}}^2}\)
And the curved surface area of a cone \( = \,\pi rl\)
\( = \,\left( {\frac{{22}}{7}} \right)\, \times \,14\, \times \,4\, = \,\,176\,\,{\rm{c}}{{\rm{m}}^2}\)
Hence, the total surface area of the cone is \(792\,\,{\rm{c}}{{\rm{m}}^2}\) and the curved surface area of the \(176\,\,{\rm{c}}{{\rm{m}}^2}\).

Q.2. What is the slant height of the cone if the total surface area of the cone is \(308\,i{n^2}\) and radius \(7\) inches?
Ans: Given that the total surface area of cone \( = \,308\,\,in{\,^2}\) and the radius of the cone \( = \) \( = \,7\,{\rm{inches}}\).
We know that the total surface area of the cone \( = \,\pi r(r + l)\)
\( \Rightarrow \left( {\frac{{22}}{7}} \right) \times 7 \times (7 + l) = 308\)
\( \Rightarrow 22 \times (7 + l) = 308\)
\( \Rightarrow 7 + l\, = \,14\)
\( \Rightarrow l\, = 7\,{\rm{inches}}\)
Hence, the slant height of the cone is \(7\,{\rm{inches}}\).

Q.3. What is the height of the cone whose radius is \({\rm{7 cm}}\) and curved surface area is \({\rm{440}}\,\,{\rm{c}}{{\rm{m}}^2}\).\((Use\,\pi  = \,22/7)\).
Ans: Given, the curved surface area of cone \( = \,440\,{\rm{c}}{{\rm{m}}^2}\) and the radius of the cone \( = \,7{\rm{cm}}\)
We know, the curved surface area of the cone \( = \,7{\rm{cm}}\)
\( \Rightarrow \,\left( {\frac{{22}}{7}} \right) \times 7 \times l = 440\)
\( \Rightarrow \,22 \times l = 440\)
\( \Rightarrow l = \frac{{440}}{2}\)
\( \Rightarrow l = 20\,{\rm{cm}}\)
\(h\, = \,\sqrt {{l^2} – {r^2}\,}  = \,\sqrt {{{20}^2} – {7^2}}  = \sqrt {400 – 49}  = 18.73\,{\rm{cm}}\)
Hence, the height of the cone is \(18.73\,{\rm{cm}}\).

Q.4. Calculate the total surface area of a cone whose radius is \({\rm{7 cm}}\) \({\rm{7 cm}}\) is \(6\,{\rm{cm}}\).
Ans: Given that: \(r\, = \,4\,{\rm{cm}}\,,\,h\, = 6\,{\rm{cm}}\)
We know, \(l\, = \,\sqrt {{h^2} + {r^2}}  = \sqrt {{6^2} + {4^2}}  = \sqrt {36 + 16}  = 7.2\,{\rm{cm}}\)
And the total surface area of the cone \( = \,\pi r\,(r + l)\)
\( = \,\frac{{22}}{7}\, \times 4 \times (4 + 7.2) = \frac{{22}}{7} \times 4 \times 11.2 = \frac{{22}}{7} \times 4 \times 11.2 = \frac{{985.6}}{7} = 140.8\,{\rm{c}}{{\rm{m}}^2}\)
Hence the total surface area of the cone is \(140.8\,{\rm{c}}{{\rm{m}}^2}\).

Q.5. The height of a cone is \(8\,{\rm{cm}}\) and its base radius is \(6\,{\rm{cm}}\) Find the curved surface area of the cone (Use \(\pi \, = 3.14\)).
Ans: Given: \(h\, = \,8\,{\rm{cm}}\,\,{\rm{and}}\,{\rm{r}}\,{\rm{ = }}\,{\rm{6}}\,{\rm{cm}}\,\,\)
We know that the curved surface area of a cone \( = \,\pi rl\,\,\)
First, we need to find the slant height \(l\)
We know that slant height \(l\, = \,\sqrt {{h^2} + {r^2}} \)
\( \Rightarrow \,l\, = \,\sqrt {{6^2} + \,{8^2}}  = \sqrt {36 + 64}  = \sqrt {100}  = 10\,{\rm{cm}}\)
Now, the curved surface area of cone \( = 3.14\, \times \,6\, \times \,10\,\,{\rm{c}}{{\rm{m}}^2} = 188.4\,{\rm{c}}{{\rm{m}}^2}\).

Summary

In this article, we have covered what a cone is, what are real-life examples of cones, the curved surface area of a cone formula, the lateral surface area of a cone, the total surface area of the cone formula, and the right circular cone formula, etc. We saw the surface area of a cone, the surface area of a cone formula, and its derivation is. Along with the solved examples, we also examined the surface area of a right circular cone, curved surface area, and total surface area of a cone.

FAQs on Surface Area of a Cone Formula

Q.1. What is the formula for the curved surface area of a right circular cone?
Ans: Curved surface area \( = \,\pi rl\)
Where \(r\) is the radius of the circular base.
\(h\) is the height of the cone.
\(l\) is the slant height of the cone.

Q.2. What is the formula for the total surface area of a right circular cone?
Ans: Total surface area of a right circular cone
where \(r\) is the radius of the circular base.
\(h\) is the height of the cone.
\(l\) is the slant height of the cone.

Q.3. How do you find the surface area of a cone?
Ans: If we know the radius of the base of the cone and the height of the cone, then we can find the surface area of the cone by using the formula The surface area of cone \( = \,\pi r(r\, + \,\sqrt {{h^2} + {r^2}} )\)
Where \(r\) is the radius of the circular base, \(h\) is the height of the cone.

Q.4. How to calculate the slant height of a cone?
Ans: The slant height of a cone is calculated using the formula \(l = \,\sqrt {{h^2} + {r^2}} \,{\rm{units}}\) where \(r\) is the radius of the circular base, \(h\) is the height of the cone.

Q.5. What is a real-life example of a cone?
Ans: Real-life examples of the cone are ice cream cone, funnel, Christmas tree, birthday cap, conical tent etc.