Arc lengths & sectors · Ages 14–15
Liu Hui's method for π
Perimeter ÷ diameter ≈ π, better with more sides.
How to do it
Method
Perimeter ÷ diameter ≈ π, better with more sides
Hexagon: 6r ÷ 2r = 3
Another way
By area: the 12-gon's 3r²
6 × (1 × 1 ÷ 2) = 3
In full
Liu Hui (3rd century China) put regular polygons inside a circle and kept doubling their sides (6, 12, 24, 48, ...): the polygon hugs the circle ever closer — 'the finer the cut, the less is lost'. A regular hexagon inside a circle of radius 1 has six sides each equal to the radius, perimeter 6; the circumference 2π is longer, so π is more than 3. The regular 12-gon splits into 6 kites, each with diagonals 1 (a radius) and 1 (a hexagon side), area 1 × 1 ÷ 2, 6 × 12 = 3 = 3r² in all. So an n-gon's perimeter ÷ the diameter, or a 2n-gon's area ÷ r², estimates π, better as the sides grow. Doubling the sides, the new side comes from the old by Pythagoras' theorem: radius 1, the 12-gon's side is about 0.5176, perimeter about 6.2117, so π ≈ 6.2117 ÷ 2 ≈ 3.1058. Another way: polygons round the outside give estimates bigger than π, so π lies between the two.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
The circle in the picture has a radius of 16 cm. The regular hexagon inside it has a perimeter of cm, and perimeter ÷ diameter = : the hexagon's estimate of π.
Fill in each blank
Show the answer and working
Answer: 96, 3
Each side equals the radius: 6 × 16 = 96; 96 ÷ 32 = 3.
Example 2
Inside a circle of radius 9 cm is a regular 24-gon, each side about 2.3499 cm long. Liu Hui used it to estimate the circumference: about how long is its perimeter, in cm?
Show the answer and working
Answer: 56.3976 cm
Perimeter = 24 × 2.3499 = 56.3976 cm, a little short of the circumference 2π × 9; more sides, closer still.
Example 3
A regular 6-gon's perimeter ÷ the circle's diameter gives 3.4641 as an estimate of π. π is more than 3.4641.
True or false?
Show the answer and working
Answer: False
π ≈ 3.1416: a polygon inside the circle is shorter, its estimate below π; one outside is longer, above π.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.The circle in the picture has a radius of 25 cm, and inside it is a regular hexagon with all six corners on the circle. What is the hexagon's perimeter, in cm?
Answer:
2.Inside a circle of radius 4 cm is a regular 12-gon, each side about 2.0704 cm long. Liu Hui used it to estimate the circumference: about how long is its perimeter, in cm?
Answer:
3.The circle in the picture has a radius of 2 cm, with a regular 12-gon inside it. It splits into 6 kites, each with both diagonals 2 cm long (one a radius, one a side of the hexagon). What is the 12-gon's area, in cm²?
Answer:
4.A regular 48-gon's perimeter ÷ the circle's diameter gives 3.1461 as an estimate of π. π is more than 3.1461.
True or false?
Answer:
5.A regular 12-gon inside a circle of radius 7 cm has sides of about 3.6232 cm. Liu Hui: the regular 24-gon's area = 12 × the radius × the 12-gon's side ÷ 2. Correct to 1 decimal place, about what is the 24-gon's area, in cm²?
Answer:
All the answers
- 1. 150 cm
- 2. 24.8448 cm
- 3. 12 cm²
- 4. False
- 5. 152.2 cm²
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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