Year 7 Maths • AC9M7M03 • Homework
Circles, Circumference and π
Use the relationships between centre, radius, diameter, circumference and π. Show working for calculations and explain your reasoning where requested.
Part A • 10 marks
Short-response questions
Define radius and diameter, and state their relationship. (2 marks)
A circle has a radius of 6 cm. Calculate its diameter. (2 marks)
Write the formula for circumference using the radius. (2 marks)
A circle has a diameter of 10 cm. Estimate its circumference using π ≈ 3.14. (2 marks)
Explain what π represents in relation to circles. (2 marks)
Part B • 25 marks
Extended-response questions
A circle has a circumference of 44 cm. Calculate its radius and diameter. Show working. Use π ≈ 3.14. (5 marks)
A student draws three circles with radii 3 cm, 5 cm and 8 cm. Calculate and compare their circumferences using π ≈ 3.14. State which has the greatest circumference. (5 marks)
Describe how historical civilisations approximated π. Give at least three examples from Egypt, Babylon, Greece, India or China, and explain why these approximations were useful. (5 marks)
Explain how circles occur in First Nations Australian contexts such as basketry, symbols or architecture. Describe how centre, radius, diameter or circumference can help describe the circular form. (5 marks)
A circular garden has a radius of 12 m. Calculate (a) its diameter, (b) its circumference, and (c) the length of fencing needed to go once around the garden. Use π ≈ 3.14. (5 marks)
Part C • Enrichment
Circular meeting-space design
11. You are designing a circular meeting space. The circumference must be 50 m. Calculate (a) the radius and (b) the diameter. Then explain why π is essential for maintaining the correct relationship between the circumference and diameter of any circular design. Use π ≈ 3.14 and show all working.
Teacher / checking section
Answers
1. Radius and diameter
Radius: distance from the centre to the circumference. Diameter: distance across the circle through the centre. The diameter is twice the radius: d = 2r.
2. Diameter
d = 2 × 6 = 12 cm.
3. Circumference formula
C = 2πr.
4. Circumference from diameter
C = πd = 3.14 × 10 = 31.4 cm.
5. Meaning of π
π is the constant ratio of a circle's circumference to its diameter: π = C/d.
6. Circumference 44 cm
r = C/(2π) = 44/(6.28) ≈ 7.01 cm, so d ≈ 14.01 cm. With π ≈ 22/7 this familiar example gives 7 cm and 14 cm exactly.
7. Three circumferences
r = 3: C = 18.84 cm; r = 5: C = 31.4 cm; r = 8: C = 50.24 cm. The 8 cm radius circle has the greatest circumference.
8. Historical approximations
Accept accurate examples such as Babylon ≈ 3.125; ancient Egypt ≈ 3.1605; Archimedes bounded π between about 3.1408 and 3.1429; Aryabhata ≈ 3.1416; Zu Chongzhi used 355/113 ≈ 3.1415929. Approximations supported increasingly accurate measurement, construction and mathematical calculation.
9. First Nations Australian contexts
Responses should respectfully identify a relevant circular context, such as some forms of basketry, symbols or architecture, and correctly connect geometric features: the centre locates the middle, radius measures centre-to-edge distance, diameter gives full width through the centre, and circumference measures distance around the boundary.
10. Circular garden
(a) d = 24 m. (b) C = 2 × 3.14 × 12 = 75.36 m. (c) One complete boundary therefore requires 75.36 m of fencing (before allowing for gates, joins or wastage).
11. Enrichment
r = 50/(2 × 3.14) ≈ 7.96 m. d = 2r ≈ 15.92 m. π is the constant C/d for every circle, so it connects the required boundary length to the correct diameter and radius.