What about a circle? What is its perimeter (usually called the circumference) in terms of its diameter? Is the ratio of circumference (C) to diameter (D) the same for circles of all sizes (see the figure)? What do you think?
Answer: Yes, C ÷ D is the same for every circle. This constant is π ≈ 3.14, so the circumference is C = πD.
Step-by-step solution
Idea: All circles have the same shape. Enlarging a circle k times makes both its diameter and its circumference k times longer, so their ratio does not change.
- Enlarge a circle so that its diameter becomes k times longer. Every length in the picture, including the curved edge, also becomes k times longer. So C ÷ D = (kC) ÷ (kD) stays the same. The ratio is the same for circles of all sizes.1 mark
- This fixed ratio is called π (about 3.14). So C ÷ D = π, which gives C = πD.1 mark
Yes, C/D is the same for all circles. It is the constant π (about 3.14), so the circumference is C = πD.
Check: A coin of diameter 2.5 cm has a circumference of about 3.14 × 2.5 ≈ 7.9 cm, and a plate of diameter 25 cm about 78.5 cm. The ratio is 3.14 both times ✓.
Answer to write in the exam
All circles are enlargements of one another ⇒ C : D is constant.
C ÷ D = π ≈ 3.14
∴ C = πD
Common mistakes that cost marks
- Thinking a bigger circle has a bigger C/D ratio. Both C and D grow by the same factor.
- Writing C = πr. With the radius it is C = 2πr; with the diameter it is πD.
How this can come in the exam
MCQ (1 mark)
Circle P has diameter 7 cm and circle Q has diameter 14 cm. Then (circumference ÷ diameter) for Q is
- half that of P
- equal to that of P
- twice that of P
- four times that of P
Show answer
(B) equal to that of P
C ÷ D = π for every circle.
Try one yourself
The circumference of a circular table top is 88 cm. Find its diameter. (Use π = 227.)
Show answer
D = C ÷ π = 88 × 722 = 28 cm.
More questions like this
- What is the value of the C/D ratio? How would you estimate this ratio?
- You can do a simple measurement at home to estimate the C/D ratio. Take a cotton reel with thin thread around it. Measure the diameter D of the reel as accurately as possible. Unwrap and then tightly wrap the thread around the reel 20 times. Unwrap it again; measure its length L, and calculate L20D. This is the ratio we want. For accuracy, the thread should be very thin. Please do the experiment! Do you get a ratio between 3 and 4? Between 3.1 and 3.2? It is also possible to estimate the C/D ratio using pure geometry, i.e., without any measurements at all! Can you imagine how?
- The Mesopotamian Hexagon-to-Circle comparison (see the figure). Can you see why this shows that π > 3?
- Archimedes’ method utilising inscribed and circumscribed polygons (see the figure). Can you see why this diagram of an inscribed and circumscribed hexagon tells us that π is between 3 and 2√3? (Hint: Use the Baudhāyana–Pythagoras Theorem.)
- What will be the length of a semicircle with the same radius r (see the figure)?