Area of a Circle

A circle is a perfectly rounded shape where all points are equidistant from its centre. This is considered one of the most critical concepts in geometry, with primary elements such as radius, diameter, circumference, and area being applied in virtually all mathematical applications.

1.0Basics of Circles 

A circle is a two-dimensional geometric shape in which all points are equidistant from any fixed point called the centre. The distance from the centre to any point on the circle is called the radius (r).

Formulas Related to the circle

1. Area of a circle (A) Formula: $\pi r^2$
2. Area of Circle when the diameter is given: $\pi (D/2{)}^2$
3. Circumference of any circle or Perimeter of a circle is $2\pi r$
4. Diameter of a circle: 2r
5. Area of half of circle or semicircle: $\frac{\pi r^2}{2}$

Here, the value $\pi$ is taken as 3.14 or 22/7.

Applications of the Area of a Circle

• The area of the circle formula is used daily in real life, like calculating the space of circular objects, for example, tables, pizza, and coins.
• It is used in construction, architecture, and engineering to calculate the required material for circular components.
• It is essential in fields such as astronomy, design, and technology for accurate measurements of circular objects or spaces.

2.0Steps to Calculate the Area of a Circle 

• Find the radius of the circle.
• Square the radius (multiply the radius by itself).
• Multiply the squared radius by π, using the value of π as 3.14159 or approximately for simpler calculations.

3.0Perimeter and area of Circle class 7

Solved Examples

Problem 1: A park has two concentric circular paths. The inner circle has a radius of 40 meters, and the outer circle has a radius of 50 meters. The authorities want to build a fence around the outer circle and pave the path between the inner and outer circles. The cost of fencing is Rs 150 per meter, and the cost of paving is Rs 100 per square meter. Find the total cost of fencing and paving the path.

Solution: Let the radius of the inner circle r = 40m 

Let the radius of the outer circle R = 50m 

Area of the path: 

Area of the outer circle(A) – Area of the inner circle(a). 

$\pi R2-\pi r2=\pi (R2-r2)$

$=\frac{22}{7}(50^2-40^2)$

$=\frac{22}{7}(2500-1600)$

$=\frac{22}{7}(900)=2827.43m^2$

Cost of the paving the path: $100\times 2827.43=Rs2,82,743$

The perimeter of the outer circle: 

$2\pi r=2\times \frac{22}{7}\times 50=314.16m$

The cost of fencing on the outer circle = $150\times 314.16=Rs47,124$

The total cost of fencing and paving the path: 2,82,743 + 47,124 = Rs 3,29,867

Problem 2: A truck is moving at a speed of 72 km/h. The radius of the tyre is 0.5 meters. Calculate the area of the tyre, and then find the distance the truck covers in one complete rotation of the tyre.

Solution: Let the radius of tyre r = 0.5m 

Speed of the truck v = 72 km/h 

The area of the tyre = $\pi r^2$

$A=\frac{22}{7}\times 0.52=0.785m^2$

The distance the truck covers in one complete rotation: 

The distance the truck covers in one full rotation of the tyre is equal to the circumference of the tyre. Hence, 

$C=2\pi r=2\times \frac{22}{7}\times 0.5=3.14m$

Problem 3: The circumference of a circular swimming pool is 44 meters. Find the area of the pool.

Solution: The circumference of the swimming pool = 44m 

$2\pi r=44$

$2\times \frac{22}{7}\times r=44$

r=7m

The Area of the Pool: $\pi r^2=\frac{22}{7}\times 7\times 7=154m^2$

4.0Practice Problems 

Problem 1: A farmer wants to build a fence around a circular field with a radius of 60 meters. The cost of fencing is Rs 100 per meter. How much will it cost to build the fence?

Problem 2: A circular garden has a radius of 20 meters. A path with a width of 5 meters is being constructed around the garden. The cost of paving the path is Rs 25 per square meter. Calculate the total cost of paving the path.

Problem 3: Two concentric circles have radii of 12 cm and 20 cm, respectively. Find the difference in the areas of the two circles.