Surface Area of a Cone Formula: Check Solved Examples (2023)

  • Written By Priya Wadhwa
  • Last Modified 19-07-2022

Surface Area of a Cone Formula: Check Solved Examples (1)

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.

Surface Area of a Cone Formula: Check Solved Examples (2)
(Video) Volume and Surface Area of a Cone & Lateral Area Formula- Basic Geometry
Surface Area of a Cone Formula: Check Solved Examples (3)

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.
Surface Area of a Cone Formula: Check Solved Examples (4)

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.

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

(Video) Surface Area of a Cone | Math with Mr. J

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.

Surface Area of a Cone Formula: Check Solved Examples (5)

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 \( = \,\frac{1}{2} \times 2\pi \times r \times l\, = \,\pi rl\)

(Video) Volume of a Cone | MathHelp.com

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

Surface Area of a Cone Formula: Check Solved Examples (6)

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 \( = \pi r(r + l)\)

(Video) Surface Area of a Cone with Lateral Area - geometry

Solved Examples on Surface Area of a Cone Formula

Q.1. Calculate the total surface area of thecone 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 thetotal surface area of thecone is \(308\,i{n^2}\) and radius \(7\) inches?
Ans: Given that the total surface area of cone \( = \,308\,\,in{\,^2}\) andthe radius ofthe 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 areais \({\rm{440}}\,\,{\rm{c}}{{\rm{m}}^2}\).\((Use\,\pi = \,22/7)\).
Ans: Given, the curved surface area of cone \( = \,440\,{\rm{c}}{{\rm{m}}^2}\) andthe radius ofthe 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 creamcone, funnel, Christmas tree, birthday cap, conical tent etc.

(Video) How to find the surface area of a cone

FAQs

How do you calculate the surface area of a cone? ›

The total surface area of a cone is the combination of the curved surface as well as the base area of a cone. The formula to calculate the total surface area of the cone is: TSA of cone = πr2 + πrl = πr(l+r) square units.

What is the total surface area of cone Class 9? ›

The base shape of the cone is a circle. Cone has a single face and has a base similar to the cylinder. The volume of a cone can be given by V=13πr2h , where r is the radius of the base of the cone and h is the slant height of the cone. The total surface area of the cone is given by A=πr(r+l) .

What is the total surface area of cone Class 10? ›

We know, the total surface area of the cone is πr (r + l), and the lateral surface area of a cone is πrl. Given that: r = 7 inches, l = 3 inches, and π = 22/7. Thus, total surface area of cone, T = πr (r + l) = (22/7) × 7 × (7 + 3) = (22/7) × 7 × 10 = 22 × 10 = 220 in2. ∴ The total surface area of the cone is 220 in2.

How do you find the CSA and TSA of a cone? ›

So here we get curved surface area of a cone as Phi L square units or IR into square root of H

What is a formula of a cone? ›

The formula for the volume of a cone is V=1/3hπr². Learn how to use this formula to solve an example problem.

What is the surface area of this cone to the right? ›

The formula for finding the surface area of a cone is πr2+πrl. In this formula, r represents the radius of the circular base, h represents the height of the cone, l represents the slant height, and π can be approximated as 3.14. If the slant height is not given, it can be found using the formula l=√r2+h2.

What is a cone Class 9? ›

A cone is a three-dimensional shape in geometry that narrows smoothly from a flat base (usually circular base) to a point(which forms an axis to the centre of base) called the apex or vertex.

How do you find the L of a cone in Class 10? ›

l = h 2 + r 2

Where, r is the base radius, h is the height and l is the slant height of the cone.

How many surfaces does a cone have? ›

<br> 2. A cone is a three-dimensional shape. <br> 3. The surface area of a cone is the space occupied by the curved surface and the base surface of the cone.

How do you solve the volume of a cone? ›

A cone is a solid that has a circular base and a single vertex. To calculate its volume, you need to multiply the base area (area of a circle: π * r²) by height and by 1/3: volume = (1/3) * π * r² * h.

How do you find the surface area of half a cone? ›

Volume and Surface Area of a Cone & Lateral Area Formula - YouTube

What is TSA in math? ›

The sum of the areas of all external surfaces of a three-dimensional object is called its total surface area (TSA).

What is the curved surface area of a cone calculator? ›

For a cone with a base radius of 3 inches and a height of 4 inches, we can calculate the surface area like so: A = πr(r + √(h² + r²)) A = π * 3 * (3 + √(4² + 3²)) = 75.4 in²

What is the formula of CSA of cylinder? ›

Curved Surface Area (CSA) of Cylinder

It is also called Lateral surface area (LSA). The CSA of a cylinder having its base radius 'r' and height 'h' is given by: Curved surface area (CSA) of cylinder = 2πrh sq. units.

What is the area and volume of a cone? ›

The volume V of a cone with radius r is one-third the area of the base B times the height h . Note : The formula for the volume of an oblique cone is the same as that of a right one. The volumes of a cone and a cylinder are related in the same way as the volumes of a pyramid and a prism are related.

What is a cone in math? ›

cone, in mathematics, the surface traced by a moving straight line (the generatrix) that always passes through a fixed point (the vertex). The path, to be definite, is directed by some closed plane curve (the directrix), along which the line always glides.

What is a cone 3d shape? ›

In maths, a cone is defined as a distinctive three-dimensional geometric figure with a flat and curved surface pointed towards the top. The term “cone” is derived from the Greek word “konos”, which means a wedge or a peak. The pointed end is the apex, whereas the flat surface is called the base.

How do you find the surface area of a cone without the slant height? ›

Surface Area Of A Cone - Slant Height Not Given - YouTube

How do you find the base of a cone? ›

Base Area of a Cone Formula

For a given cone, with the known base radius, base area (or flat surface area) is π times square of the radius and can also be represented as, A = πr2 where r is the base radius.

What is the total surface area of a right circular cone of height 48 cm and base diameter 28cm? ›

Hence, the height of the cone is 48 cm. Hence, the slant height of the cone is 50 cm. Hence, the curved surface area of the cone is 2200 cm2.

Is the volume of cone? ›

The formula for the volume of a cone is ⅓ 𝜋r2h cubic units, where r is the radius of the circular base and h is the height of the cone.

How do you draw a cone? ›

How to Draw and Shade a Cone - YouTube

What is pi RL? ›

The area of the circular base = \pi \times r^2. Area of the curved surface= \pi r l. Total Surface Area of the Cone = Area of its circular base + Area of the total curved surface.

What is formula of surface area and volume? ›

V o l u m e = 4 3 π r 3. Cone. S A = π r l + π r 2.

How do you solve for surface area and volume? ›

Surface Area and Volume Review (Geometry) - YouTube

What is example of cone? ›

Answer: 5 examples of a cone in real life are Christmas tree, carrot, party hat, ice-cream cone, and traffic cones (used as road-dividers). Let's see some examples of cones in real life. Explanation: Cone is a 3-dimensional solid figure with one pointed edge as a vertex and circular base at the other end.

How many flat surfaces a cone has? ›

A cone has one flat surface and a curved surface.

How do you find the surface area of a cone without radius? ›

Finding The Surface Area Of Cones With And Without The Slant Height

How do you find the surface area of a cone without the slant height? ›

Surface Area Of A Cone - Slant Height Not Given - YouTube

How do you find the surface area of a cone Wikihow? ›

Surface area of a cone = π x radius x side + π x radius2 = π x r x s + πr. Surface area of a sphere = 4 x π x radius2 = 4πr. Surface area of a cylinder = 2 x π x radius2 + 2 x π x radius x height = 2πr2 + 2πrh.

What is the curved surface area of a cone calculator? ›

For a cone with a base radius of 3 inches and a height of 4 inches, we can calculate the surface area like so: A = πr(r + √(h² + r²)) A = π * 3 * (3 + √(4² + 3²)) = 75.4 in²

How do you find the surface area of half a cone? ›

Surface Area of a Cone with Lateral Area - geometry - YouTube

What is the total surface area of a right circular cone of height 48 cm and base diameter 28cm? ›

Hence, the height of the cone is 48 cm. Hence, the slant height of the cone is 50 cm. Hence, the curved surface area of the cone is 2200 cm2.

What is the formula for finding the height of the cone? ›

Input them in the height of a cone formula: h = √(l² - r²) where: l is the slant height; r is the radius; and. h is the resulting height.

How do you find the slant height of a Class 10? ›

The slant height of the cone of height h and radius r is given by l=h+r.

How do you find the base of a cone? ›

Base Area of a Cone Formula

For a given cone, with the known base radius, base area (or flat surface area) is π times square of the radius and can also be represented as, A = πr2 where r is the base radius.

How many surfaces does a cone have? ›

<br> 2. A cone is a three-dimensional shape. <br> 3. The surface area of a cone is the space occupied by the curved surface and the base surface of the cone.

How do you make a cone with specific measurements? ›

To make the cone a precise size, make the length of your semi-circle twice as long as the height you want your cone to have. For example, if you want a 12-inch cone, draw a semi-circle that's 24 inches long. Alternatively, use a compass to draw a circle with a radius equal to the desired height of your cone.

How do I calculate surface area? ›

Surface area is the sum of the areas of all faces (or surfaces) on a 3D shape. A cuboid has 6 rectangular faces. To find the surface area of a cuboid, add the areas of all 6 faces. We can also label the length (l), width (w), and height (h) of the prism and use the formula, SA=2lw+2lh+2hw, to find the surface area.

What is surface area formula? ›

Surface Area Formulas

Find the area of each face. Add up all areas. SA=2B+2πrh. Find the area of the base, times 2, then add the areas to the areas of the rectangle, which is the circumference times the height. SA=4πr2.

How do you solve for surface area and volume? ›

How to Find Volume and Surface Area of a Cube - YouTube

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