A cone is a three-dimensional geometric object that has a flat circular base, and a single apex (or vertex) which is not in the plane of the base. The line segment joining the apex to any point on the edge of the base is termed the slant height.

Formula

Curved surface area of a cone =

Total surface area of a cone =

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

Derivation:

For calculating the curved surface area and the total surface area of a cone, we cut one into a circular base and top slanted part. The area of the slanted part is the curved surface area. The total surface area is the sum of this circular base and curved surface areas.

Area of the circular base:

The base is just a circle and we know that area of circle is given as:

Area of a circle=πr²

Where r is the base radius of the cone

Area of the curved surface:

Now open the curved top and cut into small pieces, so that each cut portion is a small triangle, whose height is the slant height l of the cone.

Now the area of each triangle =1/2× base of each triangle × l.

∴Curved surface area = sum of the areas of all the triangles

From the figure, we know that, the curved surface is equivalent to the perimeter of the base of the cone.

Circumference of the base of the cone =

∴ Area of the curved surface =

Area of the curved surface= πrl

Total Surface Area of a Cone = Area of the circular base + Area of the curved surface

Total Surface Area of a Cone =

Total surface area of a cone = πr (l + r)

Solved Problems

Problem 1 : Find the total surface area of a cone, if radius is 8.2 cm and height is 16 cm.

Solution:

Given,

Radius r = 8.2 cm

Height h= 16 cm

Slant height,

Total surface area of a cone =

Total surface area of a cone = 3.14 x 8.2(17.98 + 8.2) = 674.4cm²

Problem 2: Find the curved surface area of a cone with base radius 5 cm and slant height 20 cm. (Take )

Solution:

Given: Slant height l= 20 cm

Radius r= 8 cm

Now, the formula for the curved surface area of a cone is the following:

Curved surface area = πrl

Curved surface area =

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Cone Formula

Definition of a Cone

Formula

Curved surface area of a cone =

Total surface area of a cone =

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

Derivation:

Area of the circular base:

The base is just a circle and we know that area of circle is given as:

Area of a circle=πr²

Where r is the base radius of the cone

Area of the curved surface:

Now open the curved top and cut into small pieces, so that each cut portion is a small triangle, whose height is the slant height l of the cone.

Now the area of each triangle =1/2× base of each triangle × l.

∴Curved surface area = sum of the areas of all the triangles

From the figure, we know that, the curved surface is equivalent to the perimeter of the base of the cone.

Circumference of the base of the cone =

∴ Area of the curved surface =

Area of the curved surface= πrl

Total Surface Area of a Cone = Area of the circular base + Area of the curved surface

Total Surface Area of a Cone =

Total surface area of a cone = πr (l + r)

Solved Problems

Problem 1 : Find the total surface area of a cone, if radius is 8.2 cm and height is 16 cm.

Solution:

Given,

Radius r = 8.2 cm

Height h= 16 cm

Slant height,

Total surface area of a cone =

Total surface area of a cone = 3.14 x 8.2(17.98 + 8.2) = 674.4cm²

Problem 2: Find the curved surface area of a cone with base radius 5 cm and slant height 20 cm. (Take )

Solution:

Given: Slant height l= 20 cm

Radius r= 8 cm

Now, the formula for the curved surface area of a cone is the following:

Curved surface area = πrl

Curved surface area =

Other Related Sections

List of Physics Formulas

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