Circle geometry

Explain — 2 marks

A satellite orbits Earth in a circular path. Engineers need to understand the geometry of the orbit to calculate safe distances and predict the satellite's position at any time.

(a) The satellite completes one full orbit. Explain why the angle subtended at the centre of Earth during one complete orbit is 360°. [1 mark]
(b) The satellite travels along an arc that subtends an angle of 90° at Earth's centre. Explain what fraction of the total orbital circumference the satellite has travelled. [1 mark]

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(a) One complete revolution/orbit returns the satellite to its starting position, which corresponds to a complete rotation of 360° (or 2π radians)
(b) The angle 90° is one quarter of 360°, so the satellite has travelled one quarter (or 1/4 or 0.25) of the total circumference

State — 4 marks

A circular running track has a radius of 50 m. Athletes train on this track, and coaches need to understand the geometric properties of the circular path to plan training routes and measure distances accurately.

(a) State the name of the line segment that connects the centre of the circle to any point on the circumference. [1 mark]
(b) State what the diameter of the running track is, given that the radius is 50 m. [1 mark]
(c) State the relationship between the radius and the diameter of a circle. [1 mark]
(d) State the name of the curved line that forms the outer edge of the circular running track. [1 mark]

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(a) Award 1 mark for 'radius' or 'radii' (plural form also acceptable).
(b) Award 1 mark for '100 m' or 'diameter = 2 × radius = 100 m'. Numerical answer must be present.
(c) Award 1 mark for 'diameter is twice the radius' or 'd = 2r' or 'diameter = 2 × radius'. Equivalent statements accepted.
(d) Award 1 mark for 'circumference' or 'perimeter'. Accept 'the circumference of the circle'.

Compare — 3 marks

A designer is creating two circular logos for a company. Logo A has a radius of 5 cm. Logo B has a diameter of 8 cm. Both logos will be printed on promotional materials.

(a) Compare the radii of Logo A and Logo B. [1 mark]
(b) Compare the circumferences of the two logos. Show your working. [1 mark]
(c) Compare the areas of Logo A and Logo B. Which logo has the greater area and by how much? Show your working. [1 mark]

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(a) Logo A has a radius of 5 cm; Logo B has a radius of 4 cm (diameter ÷ 2); Logo A has the larger radius by 1 cm. (Award 1 mark for correct identification that Logo A's radius is larger)
(b) Logo A: C = 2πr = 2π(5) = 10π ≈ 31.4 cm; Logo B: C = 2πr = 2π(4) = 8π ≈ 25.1 cm; Logo A has the larger circumference (or equivalent statement with correct calculations). (Award 1 mark for correct method and conclusion)
(c) Logo A: A = πr² = π(5)² = 25π ≈ 78.5 cm²; Logo B: A = π(4)² = 16π ≈ 50.3 cm²; Logo A has the greater area by 9π cm² or approximately 28.2 cm² (or 28.3 cm²). (Award 1 mark for both areas calculated correctly and difference identified)

Calculate — 2 marks

A gardener is designing a circular pond for a park. The pond has a diameter of 4 metres.

(a) Calculate the circumference of the pond. Give your answer in terms of π. [1 mark]
(b) The gardener wants to place decorative stones around the edge of the pond, leaving a 1 metre wide border. Calculate the area of the border. [1 mark]

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(a) 4π (m) or equivalent
(b) π × 3² − π × 2² or 9π − 4π or equivalent method
(b) 5π (m²) or 15.7 (m²) or 15.71 (m²)

Calculate — 4 marks

A circular garden has a diameter of 12 metres. A landscaper needs to install a circular border around the edge and calculate the area to be seeded with grass.

(a) Calculate the circumference of the circular garden. Give your answer to 1 decimal place. [2 marks]
(b) Calculate the area of the circular garden that needs to be seeded with grass. Give your answer to 1 decimal place. [2 marks]

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(a) Correctly identifies radius as 6 metres (from diameter 12) or uses diameter directly in formula
(a) Correct application of circumference formula C = πd or C = 2πr and correct calculation to 37.7 metres
(b) Correct application of area formula A = πr² using radius of 6 metres
(b) Correct calculation to 113.1 square metres

Explain — 2 marks

State — 4 marks

Compare — 3 marks

Calculate — 2 marks

Calculate — 4 marks

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