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.
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(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]
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(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]
Show mark scheme
- (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.
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(a) State the name of the line segment that connects the centre of the circle to any point on the circumference.
[1 mark]
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(b) State what the diameter of the running track is, given that the radius is 50 m.
[1 mark]
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(c) State the relationship between the radius and the diameter of a circle.
[1 mark]
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(d) State the name of the curved line that forms the outer edge of the circular running track.
[1 mark]
Show mark scheme
- (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.
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(a) Compare the radii of Logo A and Logo B.
[1 mark]
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(b) Compare the circumferences of the two logos. Show your working.
[1 mark]
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(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]
Show mark scheme
- (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.
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(a) Calculate the circumference of the pond. Give your answer in terms of π.
[1 mark]
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(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]
Show mark scheme
- (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)
Describe — 3 marks
The diagram shows a circle with centre O. Points A, B and C lie on the circumference of the circle. Angle AOC = 124° where AOC is the angle at the centre subtended by the minor arc AC. Point B is on the major arc AC.
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(a) Describe the relationship between the angle at the centre and the angle at the circumference when both angles are subtended by the same arc.
[1 mark]
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(b) Given that angle AOC = 124°, describe how to calculate the size of angle ABC, and state the value of angle ABC.
[2 marks]
Show mark scheme
- (a) Angle at centre is twice angle at circumference (or equivalent)
- (b) Divide 124 by 2 (or half the angle at centre)
- (b) 62°
GCSE Perfect