What is the value of the C/D ratio? How would you estimate this ratio?
Answer: C/D = π = 3.14159… (about 227 or 3.14). Estimate it by measuring round objects (thread around the edge ÷ diameter), or by geometry: compare the circle with polygons drawn inside and outside it.
Step-by-step solution
Idea: π is the same for every circle, so measuring any circle carefully gives an estimate. Polygons drawn inside and outside the circle give lower and upper limits without measuring.
- Value: C/D = π = 3.14159265… Its digits never end or repeat (π is irrational). Useful approximations: 227 ≈ 3.143 and 3.14.1 mark
- Estimating: (1) Wrap a thin thread around a bangle, tin or cotton reel, measure the thread (C) and the diameter (D), and divide. You should get about 3.1. (2) Without measuring: a regular hexagon inside a circle of radius 1 has perimeter 6, and one outside has perimeter 4√3 ≈ 6.93. The circumference 2π lies between them, so 3 < π < 3.47. Polygons with more sides give closer limits.1 mark
The C/D ratio is π ≈ 3.14159… (often taken as 22/7 or 3.14). It can be estimated by measuring round objects, or by trapping the circle between inscribed and circumscribed polygons.
Answer to write in the exam
C/D = π = 3.14159… ≈ 227 ≈ 3.14
Estimate 1: measure C with a thread and D with a ruler; C ÷ D ≈ 3.1
Estimate 2: inscribed hexagon (perimeter 6) < 2π < circumscribed hexagon (perimeter 4√3)
∴ 3 < π < 2√3 ≈ 3.46
Common mistakes that cost marks
- Writing π = 227 exactly. 227 is only an approximation; π is irrational.
- Measuring the radius instead of the diameter and getting a ratio near 6.3.
How this can come in the exam
Assertion–Reason (1 mark)
Assertion (A): π = 227 exactly.
Reason (R): π is an irrational number.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
Show answer
(D) A is false but R is true.
227 is a fraction, so it is rational; π is irrational, so π ≠ 227. A is false, R is true.
Try one yourself
A thread wrapped once around a bangle of diameter 6.4 cm measures 20.1 cm. Estimate the C/D ratio.
Show answer
20.1 ÷ 6.4 ≈ 3.14.
More questions like this
- 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)?
- What will be the length of a quarter circle with the same radius (see the figure)?