The Mesopotamian Hexagon-to-Circle comparison (see the figure). Can you see why this shows that π > 3?
Step-by-step solution
Idea: A straight line is the shortest path between two points, so each arc of the circle is longer than the side of the hexagon that joins its ends.
- Join O to the six corners. The angles at O are 360° ÷ 6 = 60° each, and two sides of each triangle are radii (= 1). A triangle with two equal sides and a 60° angle between them is equilateral, so every side of the hexagon is 1.1 mark
- Perimeter of the hexagon = 6 × 1 = 6. Circumference of the circle = 2π × 1 = 2π.1 mark
- Each side of the hexagon is a straight segment; the arc above it joins the same two points but is curved, so it is longer. Adding all six: 2π > 6, so π > 3.1 mark
Check: 2π ≈ 6.28, which is indeed a little more than 6 ✓.
Answer to write in the exam
Central angle = 360° ÷ 6 = 60°; OA = OB = 1 ⇒ each triangle is equilateral ⇒ side = 1
Perimeter of hexagon = 6 × 1 = 6
Circumference = 2π × 1 = 2π
Arc > chord (straight line is the shortest path)
⇒ 2π > 6
∴ π > 3
Common mistakes that cost marks
- Taking the hexagon’s side as 2 (the diameter). Each side equals the radius, 1.
- Comparing the hexagon’s perimeter with π instead of 2π.
- Not giving a reason why the arc is longer than the side.
How this can come in the exam
A regular hexagon is drawn inside a circle of radius 5 cm with all its corners on the circle. Its perimeter is
- 15 cm
- 25 cm
- 30 cm
- 31.4 cm
Show answer
(C) 30 cm
Each side equals the radius, 5 cm, so the perimeter is 6 × 5 = 30 cm.
Try one yourself
A square is drawn inside a circle of radius 1 with its corners on the circle. Its side is √2. What lower limit for π does this give?
Show answer
Perimeter of square = 4√2 ≈ 5.66 < 2π, so π > 2√2 ≈ 2.83 (weaker than π > 3).
More questions like this
- 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?
- What is the difference in radius between the first and second lanes? Use the figure to find the stagger needed by the runner in the second lane. Will an equal stagger be needed between the third and second lanes?