What is a Circle?
A circle is a closed 2D shape. The distance from a fixed point inside a circle to any point on its boundary is constant. This fixed point is known as the center of the circle and the fixed distance is known as the radius of the circle.
The radius is usually represented by ‘r’ and is fixed for any given circle with a specified boundary.
Formula for the Diameter of a Circle
The diameter of a circle is the distance between two points on the boundary of a circle, and a line connecting these two points always passes through its center. The diameter is represented by ‘d’ and is twice the length of the radius of a circle, that is,
Diameter, d = 2r.
Diameter is measured in units of length such as inches, yards, centimeters, feet, meters and even in miles.
Formula for the Circumference of a Circle
The length of the boundary of a circle is called its circumference.
This boundary or circumference is equal to twice the product of its radius and the mathematical constant \(\pi\).
Circumference of a circle, C = 2\(\pi\)r
Since d = 2r or r = \(\frac{d}{2}\),
Circumference of circle, C = \(\pi\)d
The circumference is also measured in terms of the units of length.
Formula for the Area of a Circle
The area of a circle is the measure of total space enclosed within its boundary. It is equal to the the product of and the square of its radius.
Area of circle, A = \(\pi r^2\)
Since d = 2r or r = \(\frac{d}{2}\) ,
Area of circle, A = \(\pi \frac{d^2}{4}\) .
Area is measured in square units.
Solved Examples
Example 1: Find the circumference of a circle with a radius of 12 cm. (Use \(\pi~=~\frac{22}{7}\))
Solution:
The radius of the given circle is 12 cm.
Circumference of the same circle is calculated with the formula,
C = 2\(\pi r\) [Formula for circumference of circle]
= 2 x \(\frac{22}{7}\) x 12 [Substitute 12 for r]
= 75.43 cm
So, the circumference of the circle is 75.43 cm.
Example 2: Find the area of a circle with a diameter of 18 cm. (Use \(\pi~=~\frac{22}{7}\))
Solution:
The diameter of the given circle is 18 cm.
Area of circle will be given by,
A = \(\pi \frac{d^2}{4}\) [Formula for the area of a circle]
= \(\frac{22}{7}\) x \(\left(\frac{18}{2}\right)^2\) [Substitute 18 for d]
= 254.57 sq. cm
So, the area of the circle is 254.57 square centimeters.
Example 3: Tom is riding a bicycle and covers 1000 m in 400 rounds of the bicycle wheel. Find the diameter of the bicycle wheel. (Use \(\pi~=~\frac{22}{7}\))
Solution:
Total distance covered by Tom is 1000 m in 400 rounds of the wheel.
Distance covered by the wheel in 1 round = \(\frac{1000}{400}\) = 2.5 m
Distance covered in 1 round is equal to the circumference of the circular wheel.
So, the diameter of the wheel can be calculated by using the formula for circumference of a circle:
C = d [Formula for circumference of circle]
2.5 = \(\frac{22}{7}\) x d [Substitute the value of C and \(\pi\).]
d = \(\frac{2.5~\times~7}{22}\) [Solve for d]
d = 0.80 m
Therefore the diameter of the wheel is 0.80 m or 80 cm.
Example 4: Annie is eating pizza of area 154 cm\(^2\). Now she wants to calculate the length of the boundary of the pizza. What is the length of the boundary of pizza? (Use \(\pi~=~\frac{22}{7}\))
Solution:
To calculate the length of the boundary of the circular pizza we need to calculate its circumference. For this let us first determine its radius.
The area of pizza is about 154 cm\(^2\).
A = \(\pi r^2\) [Formula for area of circle]
154 = \(\frac{22}{7}~~\times~~r^2\) [Substitute the value of A and \(\pi\).]
r\(^2\) = \(\frac{154~~\times~~7}{22}\) [Simplify]
r\(^2\) = 49
r = 7. [Take positive square root]
The radius of the pizza is 7 cm.
So, the circumference of the pizza is,
C = 2\(\pi r\) [Formula for the circumference of a circle]
= 2 x \(\frac{22}{7}\) x 7 [Substitute 7 for r]
= 44 cm
Therefore, the length of the boundary of the pizza or its circumference is 44 cm.
Circles that have a common center but different radii are called concentric circles.
The area of a circle is the measure of space enclosed by its curved boundary.
We can not compare the circumference and the area of a circle because the measurement unit of the circumference is the unit of length and the measurement of the unit of area is in square units.
The radius of a circle is half of its diameter.
If an object is in the shape of a circle, then we say that the object has a circular shape.