 # Surface Area Of Cone

Surface area of a cone can be defined as the total area covered by its surface. The total surface area of cone will include the base area as well as the lateral surface area of the cone. Cone can be defined as a three-dimensional solid structure having a circular base and is in the form of a pyramid-like structure. As discussed earlier, the area of the given object can be classified into three types. In the case of cones, the area will comprise two sections, which are ‘Curved Surface Area’ and ‘Total Surface Area’.

## Definition of Surface Area

The area or region occupied by the surface of the object can be defined as the surface area of the given object. For a given three-dimensional shape, the area of the given object can be classified into three types. They are as follows:

• Curved Surface Area
• Lateral Surface Area
• Total Surface Area

Curved Surface Area: The area of all the curved regions of the solid is called the curved surface area of the object.

Lateral Surface Area: The area of all the regions except the bases of the object, i.e., top and bottom, are called the lateral surface area of the object.

Total Surface Area: The area of all the sides, top and bottom of the solid object is called the total surface area of the object.

### Curved Surface Area of Cone

The surface which excludes the base of the cone is termed as the ‘curved surface’ of a cone. And the area of that surface is called the ‘Curved Surface Area of Cone’. The Curved Surface Area of Cone is calculated as follows:

For finding the Curved Surface Area of the cone use where, r = radius of the circular base of the cone, and l = Slant height of the cone

### Total Surface Area of a Cone

The total surface area of a cone can be defined as the total area which a cone can occupy in a three-dimensional space. It is the summation of the curved surface area of the cone and the area of the base of the cone. The Total Surface Area of Cone is calculated as follows:

Total Surface Area of the cone = CSA + Area of Circular Base  = πrl+πr2

Total Surface Area of cone =πr(l+r)

## Surface Area of Sphere

A sphere is a 3-d object which is perfectly round shaped. The meaning of being a 3-d object is that it is defined in three axes, i.e., x-axis, y-axis, and z-axis. This creates the major difference between a circle and a sphere. Like other 3D shapes, a sphere does not contain any edges or vertices. The area which covers the outer surface of the given sphere is called the surface area of the given sphere. Visualizing a sphere, we can see it has a three-dimensional shape that is formed by rotating a disc that is circular with one of the diagonals.

Let us consider, there’s an unpainted ball which is to be painted. Now to paint the whole ball, the paint quantity required has to be calculated early. Because of this, the area of each and every face has to be identified to calculate the paint quantity to paint the ball.  We call this term(area) the total surface area of the ball.

## Formula for the Surface Area of Sphere

The formula for the surface area of sphere depends on the radius of the sphere. If the radius of the sphere is ‘r’ and the surface area of the sphere is S. Then, the surface area of the sphere will be given by:

Surface area ‘S’ of the sphere is given by, S = 4πr2

In case the diameter is given, then the surface area of the sphere is given as, S = 4π 2.

Where ‘d’ is termed as the diameter of the sphere. You can practice this topic at Cuemath to learn it in a fun way.

## Applications of the Surface Area of Sphere

Surface area of an object can be used for knowing things that are proportional to the surface area.

• If a spherical object is to be painted then we can find how much paint will it take to cover the object.
• To find out the surface tension in case of liquid bubbles.
• If the surface area is known then the costing of the materials applied on the surface of the sphere can be calculated accurately.

As a practical application of surface area of sphere, the calculation of total and curved surface area of the hemisphere can also be included as it itself is a part of the sphere.

## Calculation of Total and Curved Surface Area of Hemisphere

As we can see from the word hemisphere it depicts a sphere that is cut into half. As the sphere is cut into half, therefore the hemisphere will have curved as well as total surface area. So, accordingly, the two surface areas will be calculated.

Since the hemispheres are the sphere cut into two equal parts. So,

Curved Surface Area of Hemisphere = 1/2×4πr2 (Total Surface Area of Sphere)

= 2πrr2

Total Surface of Hemisphere = Curved Surface Area of Hemisphere + Area of Base (circle with radius ‘r’)= 2πrr2+πr2

=3πrr2