Measures and solids · Ages 14–15
Liu Hui's method for π worksheet
Part of: Arc lengths & sectors
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Liu Hui's method for π worksheet
Measures and solids · Ages 14–15
Numsprite
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Name:Date:Score:／48
Write each answer in the space. For word problems, show your working and write a sentence answer.
Part A: Working out and filling in (2 marks each)
1.The circle in the picture has a radius of 3 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.The circle in the picture has a radius of 19 cm. The regular hexagon inside it has a perimeter of cm, and perimeter ÷ diameter = : the hexagon's estimate of π.
Fill in each blank
Answer:
3.The circle in the picture has a radius of 12 cm, and inside it is a regular hexagon with all six corners on the circle. What is the hexagon's perimeter, in cm?
Answer:
4.The circle in the picture has a radius of 10 cm. The regular hexagon inside it has a perimeter of cm, and perimeter ÷ diameter = : the hexagon's estimate of π.
Fill in each blank
Answer:
5.Inside a circle of radius 8 cm is a regular 24-gon, each side about 2.0888 cm long. Liu Hui used it to estimate the circumference: about how long is its perimeter, in cm?
Answer:
6.The circle in the picture has a radius of 12 cm, with a regular 12-gon inside it. It splits into 6 kites, each with both diagonals 12 cm long (one a radius, one a side of the hexagon). What is the 12-gon's area, in cm²?
Answer:
7.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?
Answer:
8.A regular hexagon inside a circle of radius 28 cm has its sides doubled into a regular 12-gon. By Pythagoras' theorem, how long is a new side, in cm, correct to 2 decimal places?
Answer:
9.A regular 12-gon inside a circle of radius 3 cm has sides of about 1.5528 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:
Part B: Choose, or true or false (2 marks each)
10.The circle in the picture has a radius of 3 cm; the regular hexagon inside it has a perimeter of 18 cm, shorter than the circumference. What does that say about π?
- A. π is less than 3
- B. π is exactly 3
- C. π is more than 3
- D. π is more than 6
Choose the right answer
Answer:
11.A regular 24-gon's perimeter ÷ the circle's diameter gives 3.1597 as an estimate of π. π is more than 3.1597.
True or false?
Answer:
12.A regular 12-gon's perimeter ÷ the circle's diameter gives 3.1058 as an estimate of π. π is more than 3.1058.
True or false?
Answer:
Part C: Word problems (4 marks each)
13.Ming draws a regular 24-gon inside a circle of radius 3 cm and measures each side as about 0.7833 cm. Its perimeter is about cm, and perimeter ÷ diameter about : his estimate of π.
Fill in each blank
Working:
Answer:
14.A round pond has a radius of 8 m, and inside it is a platform shaped as a regular 12-gon, its twelve corners on the pond's edge. What is the platform's area, in m²?
Working:
Answer:
15.A round table top has a radius of 9 dm, and a regular 12-gon of glass lies on it, its twelve corners just touching the edge. Taking π as 3.14, how many dm² of the top are not under the glass?
Working:
Answer:
16.A hexagonal gazebo stands inside a circular path of radius 16 m, its six corners just on the path. The gazebo's perimeter is m; taking π as 3.14, the path is m round, m more than the gazebo.
Fill in each blank
Working:
Answer:
17.A round pond has a radius of 17 m, and inside it is a platform shaped as a regular 12-gon, its twelve corners on the pond's edge. What is the platform's area, in m²?
Working:
Answer:
18.A hexagonal gazebo stands inside a circular path of radius 6 m, its six corners just on the path. The gazebo's perimeter is m; taking π as 3.14, the path is m round, m more than the gazebo.
Fill in each blank
Working:
Answer:
Show the answers and working
Liu Hui's method for π worksheet: Answers and working
Part A: Working out and filling in
1. 18 cm
Joining the centre to the corners makes 6 equilateral triangles, so each side equals the radius 3: 6 × 3 = 18 cm.
2. 114, 3
Each side equals the radius: 6 × 19 = 114; 114 ÷ 38 = 3.
3. 72 cm
Joining the centre to the corners makes 6 equilateral triangles, so each side equals the radius 12: 6 × 12 = 72 cm.
4. 60, 3
Each side equals the radius: 6 × 10 = 60; 60 ÷ 20 = 3.
5. 50.1312 cm
Perimeter = 24 × 2.0888 = 50.1312 cm, a little short of the circumference 2π × 8; more sides, closer still.
6. 432 cm²
A kite is its diagonals multiplied ÷ 2 = 12 × 12 ÷ 2 = 72; six make 432 cm², that is 3r²: π is about 3.
7. 56.3976 cm
Perimeter = 24 × 2.3499 = 56.3976 cm, a little short of the circumference 2π × 9; more sides, closer still.
8. 14.49 cm
The hexagon's side is 28, half of it 282; the centre is 282 − (282)2 from that side, the rest of the radius 28 less that; the new side = (282)2 + that2 ≈ 14.49 cm.
9. 28 cm²
The 24-gon splits into 12 kites with diagonals the radius 3 and a 12-gon side 1.5528: 12 × 3 × 1.5528 ÷ 2 ≈ 28 cm², about 3.1056 × 32.
Part B: Choose, or true or false
10. C. π is more than 3
The circumference 2π × 3 is longer than 18: π × 6 > 18, so π > 18 ÷ 6 = 3.
11. False
π ≈ 3.1416: a polygon inside the circle is shorter, its estimate below π; one outside is longer, above π.
12. True
π ≈ 3.1416: a polygon inside the circle is shorter, its estimate below π; one outside is longer, above π.
Part C: Word problems
Marking: 2 for the working, 1 for the answer, 1 for the sentence.
13. 18.7992, 3.1332
24 × 0.7833 = 18.7992; 18.7992 ÷ (2 × 3) = 3.1332.
14. 192 m²
Liu Hui: the 12-gon is 6 kites with both diagonals 8: 6 × 8 × 8 ÷ 2 = 3 × 82 = 192 m².
15. 11.34 dm²
The circle 3.14 × 92 = 254.34, the 12-gon 3 × 92 = 243: 254.34 − 243 = 11.34 dm².
16. 96, 100.48, 4.48
Each side equals the radius: 6 × 16 = 96; 2 × 3.14 × 16 = 100.48; 100.48 − 96 = 4.48.
17. 867 m²
Liu Hui: the 12-gon is 6 kites with both diagonals 17: 6 × 17 × 17 ÷ 2 = 3 × 172 = 867 m².
18. 36, 37.68, 1.68
Each side equals the radius: 6 × 6 = 36; 2 × 3.14 × 6 = 37.68; 37.68 − 36 = 1.68.
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