Circumference of Circle
The circumference of a circle is the distance around the circle — its perimeter. Just as a rectangle has a perimeter measured by adding its four sides, a circle has a boundary length called its circumference.
The concept of circumference is closely tied to the constant π (pi), which is the ratio of any circle's circumference to its diameter. This ratio is the same for every circle, no matter how large or small.
In Class 7 Mathematics (NCERT), circumference of a circle is studied in the chapter Perimeter and Area. You will learn the formula, understand the value of π, and solve problems involving circular shapes.
Knowing how to calculate circumference is essential for real-life applications such as finding the length of a circular track, the amount of fencing for a round garden, or the distance covered by a wheel in one rotation.
What is Circumference of Circle?
Definition: The circumference of a circle is the total length of the boundary (perimeter) of the circle.
Key terms:
- Circle: A closed curve where every point is at the same distance from a fixed point (centre).
- Radius (r): The distance from the centre to any point on the circle.
- Diameter (d): The distance across the circle through the centre. d = 2r.
- Circumference (C): The perimeter (boundary length) of the circle.
- π (pi): The ratio of circumference to diameter. π = C/d ≈ 3.14159... or 22/7.
Important:
- π is an irrational number — its decimal expansion never terminates or repeats.
- For calculation purposes, we use π ≈ 22/7 or π ≈ 3.14.
- The circumference is always π times the diameter, regardless of the circle's size.
Circumference of Circle Formula
Circumference of Circle:
C = 2πr
Where:
- C = circumference
- π = 22/7 or 3.14
- r = radius of the circle
Using diameter:
C = πd
Where:
- d = diameter = 2r
Related formulas:
- To find radius from circumference: r = C / (2π)
- To find diameter from circumference: d = C / π
- Distance covered in n rotations of a wheel: Total distance = n × C = n × πd
Derivation and Proof
Understanding π and the circumference formula:
Activity to discover π:
- Take several circular objects (coins, plates, lids, bangles).
- Wrap a string around each object and measure the length. This is the circumference (C).
- Measure the diameter (d) of each object using a ruler.
- Calculate the ratio C/d for each object.
- You will find that C/d is approximately 3.14 for every object.
This constant ratio C/d is called π.
- Since C/d = π, we get C = πd.
- Since d = 2r, we can also write C = π × 2r = 2πr.
Historical note:
- Ancient mathematicians (Archimedes, Aryabhata) calculated π by inscribing and circumscribing polygons around circles.
- Archimedes showed that π is between 3 + 10/71 and 3 + 1/7 (approximately 3.1408 to 3.1429).
- The Indian mathematician Aryabhata gave π ≈ 3.1416 in the 5th century.
Why C/d is the same for all circles:
- All circles are similar figures — they have the same shape.
- When you scale a circle by a factor k, both the diameter and circumference scale by k.
- So the ratio C/d stays constant.
Types and Properties
Types of circumference problems:
1. Finding circumference given radius:
- Directly apply C = 2πr.
- Example: r = 7 cm → C = 2 × 22/7 × 7 = 44 cm.
2. Finding circumference given diameter:
- Apply C = πd.
- Example: d = 14 cm → C = 22/7 × 14 = 44 cm.
3. Finding radius or diameter from circumference:
- r = C/(2π) or d = C/π.
- Example: C = 88 cm → d = 88 × 7/22 = 28 cm, r = 14 cm.
4. Wheel rotation problems:
- Distance in one rotation = circumference.
- Number of rotations = Total distance / circumference.
- Perimeter of semicircle = πr + 2r = r(π + 2).
- This includes the curved part (half of circumference = πr) and the diameter (2r).
6. Combined shapes:
- Perimeter of shapes with circular arcs (e.g., quarter circles, semicircles attached to rectangles).
- Add the arc lengths and straight edges separately.
Solved Examples
Example 1: Example 1: Circumference from radius
Problem: Find the circumference of a circle with radius 14 cm. (Use π = 22/7)
Solution:
Given:
- r = 14 cm
Using the formula:
- C = 2πr
- C = 2 × (22/7) × 14
- C = 2 × 22 × 2
- C = 88 cm
Answer: The circumference is 88 cm.
Example 2: Example 2: Circumference from diameter
Problem: Find the circumference of a circle with diameter 21 cm.
Solution:
Given:
- d = 21 cm
Using the formula:
- C = πd
- C = (22/7) × 21
- C = 22 × 3
- C = 66 cm
Answer: The circumference is 66 cm.
Example 3: Example 3: Finding radius from circumference
Problem: The circumference of a circle is 132 cm. Find the radius.
Solution:
Given:
- C = 132 cm
Using the formula:
- C = 2πr
- 132 = 2 × (22/7) × r
- 132 = (44/7) × r
- r = 132 × 7/44
- r = 924/44
- r = 21 cm
Answer: The radius is 21 cm.
Example 4: Example 4: Wheel rotations
Problem: A wheel has diameter 70 cm. How many rotations does it make to cover a distance of 1.1 km?
Solution:
Given:
- d = 70 cm, Distance = 1.1 km = 1,10,000 cm
Step 1: Circumference of wheel:
- C = πd = (22/7) × 70 = 220 cm
Step 2: Number of rotations:
- Rotations = Distance / C = 1,10,000 / 220 = 500
Answer: The wheel makes 500 rotations.
Example 5: Example 5: Fencing a circular garden
Problem: A circular garden has radius 35 m. Find the cost of fencing it at Rs 50 per metre.
Solution:
Given:
- r = 35 m, Rate = Rs 50/m
Step 1: Find circumference:
- C = 2πr = 2 × (22/7) × 35 = 2 × 22 × 5 = 220 m
Step 2: Cost of fencing:
- Cost = 220 × 50 = Rs 11,000
Answer: The cost of fencing is Rs 11,000.
Example 6: Example 6: Perimeter of semicircle
Problem: Find the perimeter of a semicircle with diameter 28 cm.
Solution:
Given:
- d = 28 cm, so r = 14 cm
Perimeter of semicircle = curved part + diameter:
- Curved part = half of circumference = πr = (22/7) × 14 = 44 cm
- Straight part = diameter = 28 cm
- Total perimeter = 44 + 28 = 72 cm
Answer: The perimeter of the semicircle is 72 cm.
Example 7: Example 7: Comparing circumferences
Problem: Circle A has radius 7 cm and Circle B has radius 21 cm. How many times is the circumference of B greater than that of A?
Solution:
Circumference of A:
- C₁ = 2π × 7 = 14π
Circumference of B:
- C₂ = 2π × 21 = 42π
Ratio:
- C₂/C₁ = 42π/14π = 3
Answer: The circumference of B is 3 times that of A. (When the radius triples, the circumference also triples.)
Example 8: Example 8: Distance on circular track
Problem: A circular track has radius 56 m. Rahul runs 5 complete rounds. Find the total distance he covers.
Solution:
Given:
- r = 56 m, rounds = 5
Step 1: Circumference of track:
- C = 2πr = 2 × (22/7) × 56 = 2 × 22 × 8 = 352 m
Step 2: Total distance:
- Distance = 5 × 352 = 1,760 m = 1.76 km
Answer: Rahul covers 1,760 m (1.76 km).
Example 9: Example 9: Finding diameter from given circumference
Problem: The circumference of a circular pond is 176 m. Find its diameter and radius.
Solution:
Given:
- C = 176 m
Finding diameter:
- C = πd
- 176 = (22/7) × d
- d = 176 × 7/22
- d = 1232/22
- d = 56 m
Finding radius:
- r = d/2 = 56/2 = 28 m
Answer: Diameter = 56 m, Radius = 28 m.
Example 10: Example 10: Wire bent into circle
Problem: A wire of length 88 cm is bent into a circle. Find the radius of the circle.
Solution:
Given:
- Length of wire = 88 cm = circumference of the circle
Using the formula:
- C = 2πr
- 88 = 2 × (22/7) × r
- 88 = (44/7) × r
- r = 88 × 7/44
- r = 616/44
- r = 14 cm
Answer: The radius of the circle is 14 cm.
Real-World Applications
Real-world applications of circumference:
- Wheels and tyres: The circumference of a wheel determines the distance it covers in one full rotation.
- Circular tracks: Athletic tracks, cycling tracks, and race circuits use circumference to measure lap distance.
- Fencing: The circumference determines how much fencing material is needed for a circular garden or park.
- Bangles and rings: The circumference of a bangle or ring determines whether it will fit — this is the "ring size".
- Pipes and cables: The circumference of a pipe determines how much tape or insulation is needed to wrap around it once.
- Clock hands: The tip of a clock hand traces a circle; the distance it covers in one revolution is the circumference.
- Planetary orbits: The circumference of a planet's (approximately circular) orbit determines the distance it travels in one year.
Key Points to Remember
- The circumference is the perimeter (boundary length) of a circle.
- The formula is C = 2πr or C = πd.
- π = 22/7 ≈ 3.14 — it is the ratio of circumference to diameter for every circle.
- π is an irrational number (its decimal never terminates or repeats).
- Diameter = 2 × radius. Always check which is given in the problem.
- The circumference is directly proportional to the radius — double the radius → double the circumference.
- Distance covered by a wheel in one rotation = circumference.
- Perimeter of a semicircle = πr + 2r (half the circumference + diameter).
- Use π = 22/7 when r or d is a multiple of 7; otherwise use π = 3.14.
- All circles are similar, which is why C/d is the same constant (π) for every circle.
Practice Problems
- Find the circumference of a circle with radius 21 cm.
- The diameter of a circle is 35 cm. Find its circumference.
- The circumference of a circle is 154 cm. Find its radius.
- A wheel has radius 28 cm. How many rotations does it make to cover 352 m?
- Find the perimeter of a semicircle with radius 10.5 cm.
- A circular park has diameter 140 m. Find the cost of fencing at Rs 75 per metre.
- A wire of length 132 cm is bent into a circle. Find the radius and diameter of the circle.
- The radii of two circles are 3.5 cm and 10.5 cm. Find the ratio of their circumferences.
Frequently Asked Questions
Q1. What is the circumference of a circle?
The circumference is the total length of the boundary (perimeter) of a circle. It is calculated using the formula C = 2πr or C = πd.
Q2. What is π (pi)?
π (pi) is the ratio of the circumference of any circle to its diameter. Its value is approximately 22/7 or 3.14159. It is the same for every circle.
Q3. What is the difference between circumference and area?
Circumference is the length of the boundary (measured in cm, m) — it is one-dimensional. Area is the space enclosed inside the circle (measured in cm², m²) — it is two-dimensional.
Q4. Is circumference the same as perimeter?
Yes, circumference is the perimeter of a circle. The word 'circumference' is specifically used for circles, while 'perimeter' is a general term for any shape.
Q5. How do you find the radius if the circumference is given?
Use r = C/(2π). Divide the circumference by 2π to get the radius.
Q6. What is the perimeter of a semicircle?
Perimeter of a semicircle = πr + 2r (or πr + d). It includes the curved part (half the circumference) and the straight diameter.
Q7. How far does a wheel travel in one rotation?
A wheel covers a distance equal to its circumference in one complete rotation. Distance = circumference = πd or 2πr.
Q8. When should you use π = 22/7 vs π = 3.14?
Use π = 22/7 when the radius or diameter is a multiple of 7 (7, 14, 21, 28, 35...) for cleaner calculations. Use π = 3.14 otherwise. Follow the instruction in the problem.
Q9. If the radius doubles, what happens to the circumference?
The circumference also doubles. Circumference is directly proportional to radius: C = 2πr. If r becomes 2r, C becomes 2π(2r) = 4πr = 2 × original C.
Q10. Is π exactly equal to 22/7?
No. π is irrational, meaning it cannot be expressed as a simple fraction. 22/7 = 3.142857... is only an approximation. The true value of π = 3.14159265... with infinitely many non-repeating decimals.
