Surface Areas and Volume


Mathematics has a branch called Geometry which includes shapes and their properties. These shapes can be two-dimensional or three-dimensional. Two-dimensional shapes have two dimensions: Length and Breadth whereas three-dimensional shapes have three dimensions: Length, Breadth, and Height. 


Area

The space occupied by any two-dimensional shape is known as Area. And, the space occupied by any three-dimensional shape is known as Surface Area. 

For example, sphere is a three-dimensional shape but circle is a two-dimensional shape. Then, space occupied by a sphere, cuboid, cube, cylinder, cone, hemisphere etc. is known as Surface area but the space occupied by the circle, rectangle, square, triangle etc, is known as Area.

Area is measured in Square units.


Total Surface Area

Total surface area is the area of all the surfaces of an object including its base(s).


Curved Surface Area (Lateral Surface Area)

Curved Surface Area is the area of all the curved surfaces of an object excluding its base(s).

Surface areas for some of the three-dimensional geometric shapes are given as:

Cuboid: A Cuboid is the three-dimensional representation of a rectangle. 


It has six surfaces (rectangles). The surface area of a cuboid can be found by summing up all the areas of six rectangles. Thus, the surface area of cuboid is .

The curved (lateral) surface area of a cuboid is the area covered by 4 rectangles. Thus, the lateral surface area of a cuboid is .

Cube: A Cube is the three-dimensional representation of a square.


It has six surfaces (squares). The surface area of a cube can be found by summing up all the area of six squares. Thus, the surface area of cube is .

The curved (lateral) surface area of a cube is the area covered by 4 squares. Thus, the lateral surface area of a cube is .


Cylinder: A cylinder is a three-dimensional rectangle with circular bases. 

Curved (lateral) surface area of the cylinder is .

The total surface area of the cylinder will be the sum of areas of two bases and the curved surface area. Thus, the total surface area of the cylinder is .

Cone: A Cone is a pyramid with the circular base. 

 

Curved (lateral) surface area of the cone is .

The total surface area of the cone will be the sum of area of the circular base and the curved surface area. Thus, the total surface area of the cone is .


Sphere: A Sphere is a solid shape. It is a three-dimensional circle.

The surface area of the sphere is equal to the area of four circles. Thus, surface area of sphere is .

Curved surface area of sphere will be the same, that is, .

Hemisphere: If we cut the sphere into two halves, then the half shape is known as a Hemisphere. Thus, it is the half part of the sphere.


The curved surface area of the hemisphere will be the half of the curved surface area of the sphere. Thus, the curved surface area of the hemisphere is .

The total surface area of the hemisphere is the sum of the curved surface area and the area of the circle (base). Thus, the total surface area of the hemisphere is .


Volume

The volume is the amount of space a three-dimensional object occupies. In other terms, volume is the capacity of an object or a container.An object can be solid or hollow. In order to determine the volume, the three dimensions: length, breadth and height must be known.The two-dimensional shapes do not have volume. The shapes rectangle, square, triangle, circle etc. do not have volume but cuboid, cube, cone, sphere etc. have volume as these are three-dimensional solid objects.

Volume is measured in cube units. 

Volumes for some of the three-dimensional geometric shapes are given as:


Cuboid: A Cuboid is the three-dimensional representation of a rectangle. 

A cuboid is made up of rectangular planes till its height.

Volume of cuboid can be found out using the area of the rectangle as the rectangular planes are stacked up to the height of the cuboid.

Since the area of the rectangle is , the volume of the cuboid will be the multiplication of the area of the rectangle and its height. Thus, the volume of the cuboid is .


Cube: A Cube is the three-dimensional representation of a square.

A cube is made up of square planes till its height.

Volume of cube can be found out using the area of the square as the square planes are stacked up to the height (side) of the cuboid.

Since all the sides of the square are equal, the volume of the cube will be the multiplication of the area of the square and its side. Thus, the volume of cube is .


Cylinder: A cylinder is a three-dimensional rectangle with circular bases. 

Volume of the cylinder will be the multiplication of the area of the circular base and its height. Thus, the volume of cylinder is .

Cone: Cone is a pyramid with the circular base. 


Volume of the cone is one third of the volume of the cylinder. Since the volume of the cylinder is , the volume of cone will be .

Sphere: Sphere is a solid shape. It is a three-dimensional circle.


From the following figure, we can see that the sphere is been inserted within a cylinder. We did so to find out the volume of the sphere.


Here, the height of the cylinder is equal to the diameter of the sphere as the sphere touches the upper and lower bases of the cylinder. Thus,

Volume of the sphere is two third of the volume of the cylinder.Since the volume of the cylinder is , the volume of sphere will be .

Since , then the volume of the sphere will be .


Hemisphere: Hemisphere is the half part of the sphere.

Volume of a hemisphere will be half of the volume of the sphere. Thus, the volume of the hemisphere is .

Below table shows the Surface Area and Volume of the basic geometric objects:

Object Name

Surface Area

Curved Surface Area

Volume

Cuboid

Cube

Cylinder

Cone

Sphere

Hemisphere