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?
Answer: Yes: a careful measurement gives a ratio between 3 and 4, and usually between 3.1 and 3.2 (about 3.14), because L = 20 × circumference. Without measuring, trap the circle between a polygon drawn inside it and one drawn outside it: hexagons give 3 < π < 3.46.
Step-by-step solution
Idea: Wrapping 20 turns makes the thread 20 circumferences long, so L20D = CD, and any measuring error is shared out over 20 turns.
- Why L20D: one turn of thread is one circumference C, so 20 turns give L = 20C. Then L20D = 20C20D = CD.1 mark
- Sample result: reel diameter D = 2.5 cm, thread for 20 turns L = 157.5 cm. Ratio = 157.5 ÷ (20 × 2.5) = 157.5 ÷ 50 = 3.15. This is between 3 and 4 and also between 3.1 and 3.2. (The true value is π ≈ 3.1416.)1 mark
- Pure geometry: draw a regular hexagon inside a circle of radius 1 (perimeter 6) and one around it (perimeter 4√3 ≈ 6.93). The circumference 2π lies between them, so 3 < π < 3.46. Using polygons with more and more sides squeezes π closer. Archimedes went up to 96 sides and found 31071 < π < 317.1 mark
The measurement gives a ratio between 3 and 4, and with care between 3.1 and 3.2 (π ≈ 3.14). Geometry estimates π by trapping the circle between inscribed and circumscribed polygons: hexagons give 3 < π < 2√3 ≈ 3.46.
Answer to write in the exam
20 turns ⇒ L = 20C ⇒ L20D = CD
Sample: D = 2.5 cm, L = 157.5 cm ⇒ 157.550 = 3.15
∴ Ratio lies between 3 and 4, and between 3.1 and 3.2.
Geometry: perimeter of inscribed hexagon < 2π < perimeter of circumscribed hexagon
6 < 2π < 4√3 ⇒ 3 < π < 2√3
Common mistakes that cost marks
- Dividing L by D only, which gives a number about 20 times too big. Divide by 20D.
- Using a thick thread. Each turn then sits on a bigger circle, which makes the ratio too large.
- Measuring the reel’s radius instead of its diameter.
How this can come in the exam
MCQ (1 mark)
A thread wound 10 times around a reel of diameter 3 cm is 94.5 cm long. The estimate of π from this is
- 3.0
- 3.2
- 31.5
- 3.15
Show answer
(D) 3.15
94.5 ÷ (10 × 3) = 94.5 ÷ 30 = 3.15.
Try one yourself
A thread wound 25 times around a reel of diameter 2 cm measures 157 cm. Estimate the C/D ratio.
Show answer
157 ÷ (25 × 2) = 157 ÷ 50 = 3.14.
More questions like this
- 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)?
- The figure depicts a 400 m athletics track. You can see two straight sections of length 84.39 m each, and two curved portions, which are semicircles with a common centre (points A and B); the innermost semicircle on each side has radius 36.5 m. The width of each lane is 1.22 m. Let an athlete make one complete circuit of the track. What is the total distance she runs?