2. Circle Formulas
A circle is a closed two-dimensional figure in which every point on the boundary is at the same distance from its centre. That distance is called the radius.
Circle-based questions are important in competitive examinations such as SSC CGL, KPSC, Banking, Railway and other government exams. The most important formulas are for radius, diameter, circumference and area.
What Does a Circle Look Like?
AB = Diameter | OP = Radius | O = Centre
1. Radius of a Circle
The radius is the distance from the centre of a circle to any point on its circumference.
Radius is generally represented by the letter r.
Centre → Circumference = Radius (r)
Example
The radius of a circular garden is 7 m. What is its radius?
The radius is already given.
Answer: 7 m
2. Diameter of a Circle
The diameter is the straight-line distance across a circle passing through its centre.
The diameter is exactly twice the radius.
Where:
d = Diameter
r = Radius
Example 1: Find Diameter
The radius of a circle is 7 cm. Find its diameter.
d = 2 × 7
d = 14 cm
Answer: 14 cm
Example 2: Find Radius from Diameter
The diameter of a circular field is 28 m. Find its radius.
We know:
Therefore:
r = 28 ÷ 2
r = 14 m
Answer: 14 m
If the question gives diameter but the formula requires radius, first divide the diameter by 2.
3. Circumference of a Circle
The circumference is the total distance around the boundary of a circle. It is also called the perimeter of a circle.
Formula Using Diameter
Since d = 2r, circumference can also be written as:
Where:
| Symbol | Meaning |
|---|---|
| r | Radius of the circle |
| d | Diameter of the circle |
| π | Pi; commonly taken as 22/7 or 3.14 |
Example 1: Circumference Using Radius
Find the circumference of a circle whose radius is 7 cm.
Take:
Formula:
Substitute r = 7:
C = 2 × 22/7 × 7
C = 44 cm
Answer: 44 cm
Example 2: Circumference Using Diameter
Find the circumference of a circle whose diameter is 14 cm.
C = 22/7 × 14
C = 44 cm
Answer: 44 cm
Use C = 2πr when radius is given and C = πd when diameter is directly given.
4. Area of a Circle
The area of a circle is the region enclosed by the circumference of the circle.
The standard formula is:
Why is the Radius Squared?
The area depends on the radius in both dimensions of the circular region. Therefore, the radius appears as r² in the formula.
The answer for area must always be expressed in square units.
Example 1: Find Area Using Radius
Find the area of a circle with radius 7 cm.
Take:
Formula:
Substitute r = 7:
A = 22/7 × 7²
A = 22/7 × 49
A = 22 × 7
A = 154 cm²
Answer: 154 cm²
Example 2: Radius Given as 14 cm
Find the area of a circle whose radius is 14 cm.
A = 22/7 × 14 × 14
A = 22 × 2 × 14
A = 616 cm²
Answer: 616 cm²
5. Area of a Circle When Diameter is Given
Sometimes a question gives the diameter instead of the radius. In that situation, first find the radius.
Then use:
Example
The diameter of a circular park is 28 m. Find its area.
Step 1: Find radius
r = d/2
r = 28/2
r = 14 m
Step 2: Find area
A = 22/7 × 14 × 14
A = 616 m²
Answer: 616 m²
6. Value of π (Pi)
The symbol π (Pi) is used in circle formulas. According to the source formula sheet, π may be taken as:
If the question specifies a value of π, always use the value given in the question.
If no value is specified, competitive-exam questions commonly use 22/7 or 3.14, depending on the numbers and instructions.
7. Important Circle Formulas at a Glance
| Quantity | Formula |
|---|---|
| Diameter | d = 2r |
| Radius | r = d/2 |
| Circumference | C = 2πr |
| Circumference using diameter | C = πd |
| Area | A = πr² |
8. Common Mistakes in Circle Questions
1. Confusing radius and diameter
Remember: d = 2r.
The diameter is twice the radius.
2. Forgetting to square the radius
The area formula is: A = πr², not πr.
3. Using the wrong unit
Circumference is measured in units such as cm or m, while area is measured in cm² or m².
4. Using circumference when area is required
If the question asks for the region covered by a circular field, garden or plate, it usually requires area.
9. Real-Life Example of Circle Mensuration
Imagine a circular garden with a radius of 7 m.
If you want to know how much land the garden covers, calculate its area.
If you want to put a fence around the garden, calculate its circumference.
10. Practice Questions
Question 1: Find the diameter of a circle whose radius is 21 cm.
Question 2: Find the circumference of a circle whose radius is 14 cm.
Question 3: Find the area of a circle whose radius is 21 cm.
Question 4: A circular park has a diameter of 28 m. Find its area.
Question 5: A circular track has a radius of 35 m. Find the distance covered in one complete round.
11. Quick Revision
Diameter: d = 2r
Radius: r = d/2
Circumference: C = 2πr
Circumference using diameter: C = πd
Area: A = πr²
π: 22/7 or 3.14
12. Mensuration Exam Tip
For SSC CGL, KPSC and other government examinations, remember these three circle formulas first:
These formulas form the foundation for many circle-based questions.
In Part 3 — Semicircle Formulas, we will cover the perimeter and area of a semicircle with diagrams, formula explanations, solved examples and competitive-exam tips.