Surface Area and Volume- Formulas
Surface Area and Volume
The surface area of a 3D object is the area covered by the surface(s) of the object. On the other hand Volume of an object measures the amount of space available within the object. We can calculate surface area and volume for any three-dimensional geometrical shape or object.
In geometry, we have objects in different shapes and sizes such as cubes, cuboids, spheres, cylinders, cones, etc. For each object, we can calculate the surface area and volume. So let us learn about the formulas for calculating the Surface Area and Volume of the various 3d shapes.
What is meant by the Surface Area of a 3D object?
The total area covered by the outer surfaces of 3D objects is called its Surface Area. It is measured in square units. Following are the types of Surface Area.
- Total Surface Area.
- Curved/ Lateral Surface Area.
Total Surface Area
TSA is the area occupied by the base(s) and the remaining surfaces of the object. It is the total area covered by all the surfaces of the object. TSA is the sum of the area of the side surfaces and the area of the base(s).
Curved/ Lateral Surface Area
Curved/Lateral Surface Area is the area occupied by all the remaining surfaces of the object other than its base(s). The surface area of the base(s) is excluded since only one needs to find only the surface area of the faces of the object.
Volume
The amount of space occupied by the object measured in cubic meters is called its volume. For example, if we have a square shape of paper, it is the 2D figure so, it only has the area. On the other hand, a square box is a 3D object so it has both total surface area and volume.
Now we will discuss the formulas for finding out the total surface area and volume of the different three-dimensional figures in a simple way.
Surface Area and Volume Formulas
The below table contains the Surface Area and Volume Formula for the basic geometrical figures:
Examples
Now let’s learn to calculate the surface area and volume using the formula given in the above table for better understanding.
Q1. Find the curved surface area of the cone whose radius and height are 4 cm and 3 cm.
Solution:
Let l be the slant height of the cone and the formula for calculating the slant height is:
l=\sqrt{r^{2}+h^{2}} \therefore l=\sqrt{4^{2}+3^{2}}=\sqrt{16+9}
=\sqrt{25}=5 \mathrm{~cm}
Thus, the CSA of the cone
=\pi rl
=\frac{22}{7}\times 4\times 5
=62.85 \mathrm{~cm}^{2}
Q2. Two cubes each of volume 27 \mathrm{~cm}^{3} are joined end to end. Find the surface area of the cuboid?
Solution:
Given that
The volume of the cubes = 27\mathrm{~cm}^{3}
If the cubes are joined end to end, the dimension of the cuboid formed will be:
Length = 3 cm,
Breadth = 3 cm,
Height = 6 cm.
Thus,
The surface area of the cuboid:
TSA of cuboid = 2(lb + bh + hl)
= 2((3 × 3) + (3 × 6) + (6 × 3))
= 2(9 + 18 + 18)
= 2 × 45 = 90 \mathrm{~cm}^{2}
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Frequently Asked Questions
1. What is the formula for calculating the volume of the cylinder?
Ans: We can use the following formula to find the volume of the cylinder:
= Area of base × height
=\pi r^{2}\times h
2. How to calculate the volume of the cylinder with a conical cap on the top?
Ans: The volume of the cylinder with a conical cap on top:
The sum of the volume of the cylinder and the volume of the conical cap.
=\pi r^{2}h + \frac{1}{3} \pi r^{2}h
Here,
r = radius of the circular base of the cylinder,
h = height of the cylinder.
3. What are the formulas for the surface area and volume of a sphere?
Ans: The formula to find the surface area of the sphere is 4\pi r^{2}.
The formula to find the volume of the sphere is \frac{4}{3}\pi r^{3}.
Here, r = radius of the circular base of the cylinder.