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Surface Area of a Cone(Definition, Formulas & Examples)

surface area of a cone

The surface area of a 3D shape (solid object) is a measure of the total area of the object’s surface. The surface area of a cone is the total area covered by its flat and curved surfaces. The total surface area of a cone can be calculated if we calculate the area of its base and the area of the curved face.

Let’s learn how to find the surface area of a cone and its methods and formulas.

Cone – A 3D Solid Shape

A cone is a shape formed by using a set of line segments or lines which connects a common point, called the apex or vertex, to all the points of a circular base(which does not contain the apex). The distance from the cone’s vertex to the base is the height of the cone. The circular base has a measured value of radius. And the length of the cone from apex to any point on the circumference of the base is the slant height. Based on these quantities, there are formulas derived for the surface area of a cone.

Types of Cone

The cones are broadly divided into two categories.

  • Right Circular Cone: A cone that has a circular base and the axis from the vertex of the cone towards the base passes through the center of the circular base. The vertex of the cone lies just above the center of the circular base. The word “right” is used here because the axis forms a right angle with the base of the cone or is perpendicular to the base. These are the most common types of cones which are used in geometry. See the figure below which is an example of a right circular cone.
  • Oblique Cone: A cone that has a circular base but the axis of the cone is not perpendicular to the base, is called an Oblique cone. The vertex of this cone is not located directly above the centre of the circular base. Therefore, this cone looks like a slanted cone or tilted cone.
surface area of a cone
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Right Circular Cone

A right circular cone is one whose axis is perpendicular to the plane of the base. We can generate a right cone by revolving a right triangle about one of its legs.

In the figure, you can see a right circular cone, which has a circular base of radius r and whose axis is perpendicular to the base. The line which connects the vertex of the cone to the centre of the base is the height of the cone. The length at the outer edge of the cone, which connects a vertex to the end of the circular base is the slant height.

surface area of a cone

Relation Between Radius, Height, and Slant Height of a Right Circular Cone

Since, the radius, height, and the slant height of a right circular cone form a right triangle, therefore these three are related to each other by a Pythagoras theorem, where slant height is the hypotenuse and the radius and height are the two legs of the right triangle.

surface area of a cone

Therefore, according to the Pythagoras theorem, $l^{2} = r^{2} + h^{2} => l = \sqrt{r^{2} + h^{2}}$

Similarly, $r = \sqrt{l^{2} – h^{2}}$ and $h = \sqrt{l^{2} – r^{2}}$

where $r$ is the radius, $h$ is the height and $l$ is the slant height of a cone.

What is the Surface Area of a Cone?

The area occupied by the surface/boundary of a cone is known as the surface area of a cone. It is always measured in square units. As it has a flat base, thus it has a total surface area as well as a curved surface area. The vertex in the right circular cone is usually vertically above the center of the base whereas the vertex of the cone in an oblique cone is not vertically above the centre of the base.

The surface area of a cone is measured as the “number of square units” ($cm^{2}$, $m^{2}$, $in^{2}$, $ft^{2}$, etc.). There are two types of surface areas of a cone

  • Curved Surface Area
  • Total Surface Area

Curved Surface Area of a Cone

The curved surface area of a cone refers only to the curved part of the cone which is other than the circular flat base. To find the curved surface area of a cone, we multiply the radius and slant height of the cone by $\pi$, i.e., the curved surface area of a cone is calculated using the formula $\pi r l$.

surface area of a cone

The formula for the curved surface area of a cone is $\pi r l$.

Total Surface Area of a Cone

The total surface area of the cone is obtained by adding the area of the base and the area of the curved surface. Thus, the formula for the total surface area of the cylinder is given as:

The total surface area of a cone = (Area of base) + (Area of the curved surface). Since the base of a cone is circular in shape, its area will be $\pi r^{2}$. 

We already know that the curved surface area of a cone is $\pi rl$.

Therefore, the total surface area of a cone is $\pi rl + \pi r^{2} = \pi r \left(r + l \right)$.

The formula for the total surface area of a cylinder is $\pi r \left(r + l \right)$.

Types of Coordinate Systems

Examples

Ex 1: Find the total surface area and curved surface area of the cone whose radius is $3.5 cm$ inches and slant height is $3 cm$.

The radius of the cone $r = 3.5 cm$

The slant height of the cone $l = 3 cm$

The curved surface area of the cone = $\pi r l = \frac {22}{7} \times 3.5 \times 3 = 33 cm^{2}$

The total surface area of the cone =  $ \pi r \left(r + l \right) = \frac {22}{7} \times 3.5  \times \left(3.5 + 3 \right) = 71.5 cm^{2}$

Ex 2: Find the lateral surface area and total surface area of a right cone if the radius is $4 cm$ and the height is $3 cm$.

The radius of the cone $r = 4 cm$

The height of the cone $h = 3 cm$

The slant height of a cone  $l = \sqrt{r^{2} + h^{2}} => l = \sqrt{4^{2} + 3^{2}} => l = \sqrt{16 + 9} => l = \sqrt{25} => l = 5 cm$

The curved surface area of the cone = $\pi r l = \frac {22}{7} \times 4 \times 5 = 62.86 cm^{2}$.

The total surface area of the cone =  $ \pi r \left(r + l \right) = \frac {22}{7} \times 4  \times \left(4 + 5 \right) = 113.14 cm^{2}$

Ex 3: A cone-shaped roof has a diameter of $12$ ft. and a height of $8$ ft. If roofing material comes in $12$ square-foot rolls, how many rolls will be needed to cover this roof? 

The diameter of the conical roof  $d = 12 ft$.

Therefore, the radius of the cone $r = \frac {d}{2} = \frac {12}{2} = 6 ft$

Height of the conical roof $h = 8 ft$.

The slant height of the conical roof  $l = \sqrt{r^{2} + h^{2}} => l = \sqrt{6^{2} + 8^{2}}  => l = \sqrt{36 + 64} => l = \sqrt{100} => l = 10 ft$

The area to be covered by the roofing material is the curved surface area of the cone = $\pi r l = \frac {22}{7} \times 6 \times 8 = 150.86 ft^{2}$

Area of a roofing material roll = $12 ft^{2}$

Therefore, the number of rolls required = $\frac {150.86}{12} = 12.57$

Hence, the number of rolls that need to be purchased = $13$.

Conclusion

The surface area of a 3D shape (solid object) is a measure of the total area of the object’s surface. There are two types of surface areas in a cone, viz., curved surface area calculated using the formula $\pi r l$ and total surface area calculated using the formula $ \pi r \left(r + l \right)$, where $r$ and $l$ are the radius and slant height of a cone respectively.

Practice Problems

  1. A cone has a circular base of a radius of $10 cm$ and a slant height of $30 cm$. Calculate the curved surface area and the total surface area.
  2. Find the curved surface area, where $r = 9cm$, $h = 12cm$ and $l = 15cm$.
  3. Find the total surface area, where $r = 7cm$, $h = 24cm$. 
  4. A cone has a radius of $3cm$ and a height of $4cm$, find the total surface area of the cone.
  5. The slant height of a cone is $13cm$. the diameter of the base is $10cm$. Find the total surface area of the cone.
  6. Find the total surface area of a cone, if its slant height is $21m$ and the diameter of its base is $24 m$.

Recommended Reading

FAQs

What is the surface area of a cone?

The area occupied by the surface/boundary of a cone is known as the surface area of a cone. It is always measured in square units. As it has a flat base, thus it has a total surface area as well as a curved surface area. 

The surface area of a cone is measured as the “number of square units” ($cm^{2}$, $m^{2}$, $in^{2}$, $ft^{2}$, etc.). There are two types of surface areas of a cone
a) Curved Surface Area: The formula for the curved surface area of a cone is $\pi r l$.
b) Total Surface Area: The formula for the total surface area of a cylinder is $ \pi r \left(r + l \right)$.

What are the TSA and CSA of a cone?

a) Curved Surface Area (CSA) of Cone = $\pi r l$
b) Total Surface Area (TSA) of Cone = $ \pi r \left(r + l \right)$

How do you find the surface area?

The total surface area is calculated by adding all the areas on the surface: the areas of the base, top, and lateral surfaces (sides) of the object. This is done using different area formulas and measured in square units.

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