Please give me a clear concept of Surface area and volume...
Answers
We come across a lot of solids which are a combination of one or more basic shapes. Tents, capsules, and Ice-cream cones are the most common examples. You might have also seen trucks with capsule-shaped containers carrying petrol or Liquefied Petroleum Gas. Are they similar to the basic shapes, or a combination of different shapes?
These are certainly a combination of two or more shapes. So in a nutshell, the combination of solids include shapes formed from the fusion of two or solid shapes, which together form a new shape. The shapes formed by the combination of different shapes are called composite shapes.
While calculating the surface area and volume of these shapes we need to first observe the number of solid shapes that form these shapes. As we already know that, solid shapes are three-dimensional structures of a one-dimensional shape, for example, a cube is formed when six square-shaped cards are assembled adjacent to each other.
When we measure the surface area and volume of these solid shapes we consider all the three dimensions: length, breadth, and height. Now, when we combine these solid shapes to form a new shape, we end up calculating these at a different level. Calculating the surface area and volume of the composite shapes takes us to a new level of thinking.
For this, our understanding towards shapes and structures must be accurate. When we calculate the surface area and volume of composite shapes, we break the shape into its constituting shapes. This process of calculation is exactly similar to breaking a bigger problem into a smaller problem, to reach an accurate solution.
Surface Area of Composite Shapes
The surface area of any solid shape is the sum of the areas of all faces in that solid shape. For example, when finding the surface area of a cuboid we add the area of each rectangle constituting the cuboid. Similarly, when finding the surface area of composite shapes, we add the area of all the surfaces of structures constituting that composite structure.
As said earlier we first break the composite structure into its smaller constituents and then add all the solutions to get the major solution to our problem. Here to understand the subject, we first with a simple composite structure of an Ice-cream cone:
An ice-cream filled cone constitutes a cone and a hemisphere-shaped ice-cream. So when we need to find the surface area covered by the whole structure we add the individual surface areas. So, the total surface area of the cone shall be the sum of individual surface areas of the constituting shape. In case of an ice-cream filled cone:
Total surface area of the Ice-cream cone = Curved Surface Area of Hemisphere+ Curved Surface Area of the Cone. Curved Surface area of a hemisphere = 2πr2 and Curved Surface area of the cone = πrl. So,
The Total Surface Area of Ice-cream Cone = 2πr2 + πrl
Let’s take another example of a tent. In a tent we see two structures, one is the cone while the other is a cylinder. To Calculate the Total Surface Area of the tent we calculate the individual surface areas of shapes constituting the tent’s structure. Hence, Total Surface Area of a tent house = Curved Surface Area of the cone + Curved Surface Area of the cylinder = πrl + 2πrh.
Volumes of Composite Shapes
When we want to find the volume of a container we intend to calculate the capacity it can hold. While finding the volume of composite shapes we calculate the capacity of that structure if its hollow. But if its a concrete structure then the calculation for volume is done just to get an idea of the density of that structure.
For finding the volume of combined solids we follow the same level of calculation as we did in the surface area of a composite solid shape. So let’s find the volume of a capsule container on a truck carrying petroleum:
The capsule-shaped container loaded on a truck is a combination of a cylinder with adjoined hemispheres on both sides. Now, when we find the volume of this capsule-shaped container we add the individual volumes of all the constituting shapes. So, in this case, we add the individual volumes of 2 hemispheres and one cylinder.
The volume of a capsule container= Volume of hemisphere + Volume of Cylinder + Volume of Hemisphere = 2/3πr3 + πr2h + 2/3πr3 . This method of finding volumes of combined shapes is used in all types of composite shapes.