Surface Area of A Circle
The surface area of a particular object simply refers to the measurement of the total area which is occupied by the surface of that object. If we talk about a circle, then it is called a shape in which all the points on the plane are at equal distances from the centre which is a fixed point. Any point which is joining the centre of the circle is called the radius and all the radius has the same length. In this article, we will talk about the surface area of a circle. We will learn how to calculate the area with the help of the surface area of the circle formula and a few examples.
With the following area of a circle equation, we can calculate the area of the circle:
The surface area of a circle formula = \[ \pi r^{2}\]
Where,
\[ \pi = \frac{22}{7} \] or 3.14
r = radius of the circle
The above-mentioned formula is the basic and most common one. The total surface area of a circle and its formula to calculate can also be written in the following ways where you don't have radius but has other values:
Surface area = \[ (\pi /4) d^{2}\] or \[\frac{\pi d^{2}}{4}\]
Here, d = diameter
Surface area = \[ C^{2}/4 \pi \] or \[\frac{C^{2}}{4\pi}\]
Here, C = circumference of the circle
Solved Examples
With the help of solved examples, we can better understand how to use the formula of the total surface area of a circle; which are given below:
Problem: The area of a circular field is 13.86 hectares. Calculate the cost of fencing the field at a rate of Rs. 4.40 per metre. ( Source: A.A.O. Exam, 2010 )
Solution:
Area of the circular field = 13.86 hectares ( given )
( 1 hectare = 10000 Square meters )
Therefore,
Area = \[ 13.86 \times 10000 = 138600 m^{2}\]
\[ \pi r^{2} = 138600 \]
\[\frac{22}{7} \times r^{2} = 138600\]
r = 210 m
Circumference = \[ 2 \pi r\]
\[ = 2 \times \frac{22}{7} \times 210 \]
= 1320 m
Cost of fencing circular field = \[1320 \times 4.40\]
= Rs. 5,808
Problem: Calculate the area of the largest circle that can be drawn inside a rectangle whose sides are 9m and 8m.
Solution:
Radius of the required circle = \[ \frac{1}{2} \times 8 = 4m \]
Area of a circle = \[ \pi r^{2}\]
\[ = \frac{22}{7} \times 4 \times 4 \]
\[ = 50.28 m^{2}\]
Problem: If the radius of a circle is 14cm, what is the total surface area of circle in sq. cm. ?
Solution:
Radius of a circle = 14cm ( Given )
Area of a circle = \[ \pi r^{2}\]
\[ = \frac{22}{7} \times 14 \times 14 \]
= 616 sq. cm
Problem: Calculate the diameter of a circle if the surface area of a circle is 1188 sq. cm.
Solution:
Area of a circle = \[ \pi r^{2}\]
\[ 1188 = \frac{22}{7} \times r^{2}\]
\[ 1188 \times \frac{22}{7} = r^{2}\]
\[ 378 = r^{2}\]
\[\text{ Diameter} = r^{2}\]
Therefore, the diameter of a circle is \[378 cm^{2}\].
Problem: A circular grassy plot of land, 42 m in diameter, has a path 3.5 m wide running round it outside, Find the cost of gravelling the path at Rs. 4 per sq. metre.
( Source: M.A.T. 2005 )
Solution:
(Image Will be Updated Soon)
Diameter of a plot = 42 m ( given )
Radius of the plot ( r ) =\[ \frac{\text{diameter}}{2}\]
=\[ \frac{42}{2} = 21m\]
Radius of path = 3.5 m ( given )
Radius of the ( plot + path ) ( R ) = 21 + 3.5 = 24.5 m
Area of the path = \[ \pi ( R^{2} - r^{2} )\]
= \[ \pi ( (24.5)^{2} - (21)^{2} )\]
= \[ \frac{22}{7} \times ( (24.5 + 21) (24.5 - 21)\]
= \[ \frac{22}{7} \times ( (45.5 \times 3.5\]
=\[ 500.5 m^{2}\]
Cost of gravelling the path = 500.5 × 4 = Rs. 2,002.
Problem: The radii of three concentric circles are in the ratio 1:2:3. Calculate the ratio of the area between the two inner circles to that between the two outer circles.
( Source: S.S.C. 2008 )
Solution:
Let us assume the radii of three concentric circles which are in ratio 1:2.3 = r, 2r and 3r respectively.
(Image Will be Updated Soon)
\[ \text{Ratio} = \pi ((2r)^{2} - (r)^{2}) : \pi ((3r)^{2} - (2r)^{2}) \]
= \[3\pi r^{2} : 5\pi r^{2}\]
= 3:5
Therefore, the ratio is 3:5.
Conclusion
To sum up the discussion about the total surface area of a circle, we can say that it is the total area of the surface that is occupied by the circular object which can be calculated by πr². If you don't have a radius, you can use other formulas as well if you are having a diameter or circumference. Suppose if you are asked about the surface area of the disc, then remember a disc is circular and thus the same formula will apply to that. In this article, we have talked about the surface area of the circle, its formula and solved examples.
FAQs on Surface Area of A Circle Formula
1. What do you mean by the surface area of a circle?
A circle is a shape in which all the points on the plane join the fixed center and are at an equal distance from it. When these points join this center, that is called the radius and two radii are said to be having an equal length. The total area occupied by the circular object is called the surface area.
2. What is the formula for the total surface area of a circle?
It can be calculated with the help of the following formula:
Surface area = \[\pi r^{2}\]
Where,
\[\pi = \frac{22}{7}\] or 3.14
r = radius of the circle
Surface area = \[(\frac{\pi}{4})d^{2}\]
Here, d = diameter
Surface area = \[\frac{C^{2}}{4\pi}\]
Here, C = circumference of the circle
3. What is the area of a circular table whose diameter is 70 cm?
Diameter of the table ( d ) = 70 cm
Radius of the table = \[\frac{d}{2} = \frac{70}{2} = 35 cm\]
Area of a circular table = \[\pi r^{2}\]
\[ = \frac{22}{7} \times 35 \times 35\]
\[ = 3850 cm^{2}\]
