[Maths Class Notes] on Surface Area and Volume Pdf for Exam

In our day-to-day life, we came across various solids having different shapes and size, in which we can calculate both surface area and volume. We need to calculate the surface area and volume of the objects around us. But what if these basic shapes combine and form some different shape other than the original one. Now, the question is how you will calculate the volume, surface, and areas of new objects. While calculating the surface area and volumes of these new shapes, we need to observe the new form. A deep discussion is given below that will create a more clear image of these objects and their calculation. 

 

Solid made up of common geometric solids is known as composite solid.  They are made up of pyramids, prisms, cylinders, pairs, and cones to find out the volume and surface area of a solid. We need to find out the volume and surface area of prisms,  cons, spheres, pyramids, and cylinders. Sum total of the surface areas of the individual solids that make up the combined solid excluding the overlapping parts from each figure is the total area of a combined solid whereas the sum of the volumes of individual solids that make up the combined solid is the volume of a combined solid.

Combination of Solids – Surface Area

We should only calculate the areas that are visible to our eyes while calculating the surface area of the combined solid. For example, we need to just find out the C.S.A. of the hemisphere and of the cone separately and sum them up together if a cone is surmounted by a hemisphere.

The surface area of an object is given by the total area of the surface that an object occupies, or we can say the total area of all the surfaces of any three-dimensional figure. The surface area of figures other than cubes or cuboids can be calculated as the lateral area of the figure plus its every base, in case if prism and cylinder are the same then we can take it as twice the area of the base. The surface area of any given figure can be calculated with the help of the example of a gift as a three-dimensional figure and let the surface area be the wrapping paper, so the amount of wrapping paper used to cover the gift is the surface area of the given three-dimensional figure. Surface area can be given by the following formula:

 

Surface area = Lateral area + (n * base)

 

where, n = no. of bases present (n = 2 for prism/ cylinder, n = 1 for pyramids/ cones, and n = 0 for spheres/ circles)

 

Surface area can be further divided into 2 types such as:

1. Total surface area – The area including the base and the curved part is called a total surface area.

2. Curved surface area – The area of the curved part excluding the base is called the curved surface area.
   

Volumes of any given object can be said as the amount of liquid it can contain in it. Basically, the quantity enclosed by the given three-dimensional objects is called the volume of that object. The volume of the one dimensional (e.g., lines), as well as the two-dimensional object (e.g., squares), are considered zero as the volume is considered as quantity. The basic properties to find the volume of any given object are as follows:

1. Any given object has the volume of length * breadth * height (V = lwh).

2. The total volume of any given object is the sum of all non-overlapping regions.

3. Exactly the same when superimposing figures have the same volume.

4. Depending on the unit cube, every polyhedral region has a unique volume.
   

Volume can be calculated with the help of the following formulas for different figures:

Volume of the sphere (V) = (4/3) π x (radius)³

Volume of prism or cylinder (V) = base area x height

Volume of pyramid or cone (V) = [(frac {1}{3})] x base area x height

Combination of solids

We came across different shapes which are the combination of different solids having one or more basic shapes. Some of the most common examples of the combination of solids include ice-cream cones, capsules, tents, and capsule-shaped trucks carrying petrol or liquefied petroleum gas. Basically, the combination of solids can be explained as those solids which can break into two or more different solids having sides. The combination of solids is also known as composite shapes as the combination of solids formed by the fusion of two or more different shapes to form a new shape.

 

To calculate the surface area or the volume of these types of solids, first, we have to see the number of solid shapes that form these shapes, as these three-dimensional structures contain various one-dimensional shapes having an example of a cube formed with the help of six squares which is a one-dimensional shape. The surface area of a given composite shape is the sum of the area of all the faces in that solid. To understand the combination of solids, we can take one example of ice cream-filled cone, which is the fusion of a cone and the hemisphere-shaped ice cream. So, the total surface area of the ice cream-filled cone is equal to the sum of the curved surface area of the hemisphere and the curved surface area of the cone.

 

The curved surface area of cone = πrl,

And the curved surface area of hemisphere = [pi r^2]

So, the total surface area of ice cream filled cone = [pi r^2 + pi rl]

 

To calculate the volume of the combinational shapes, we have to first figure out the different shapes involved in it to form the composite shape. The volume of the combinational shapes can be calculated by calculating the volume of the specific shapes through which a new combinational shape is formed and adding them to form the total volume of the composite shape.

 

Similarly, in the case of volume, the volume of the ice cream-filled cone can be found out by individually calculating the volume of the hemisphere and the volume of the cone and then adding up to form the volume of the ice cream-filled cone.

 

Volume of cone = [frac {1}{3}] πr²h

 

Volume of hemisphere = [frac {2}{3}] πr³

 

Important formulas of surface area and volume are as below: