Practice SL 3.4—The circle, arc and area of sector, degrees only with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.
Paper
Level
Question 1
SLPaper 2
A circular garden has a radius of 7 m . A path runs along a quarter of the perimeter.
1.[3]
Find the length of the path.
Question 2
SLPaper 2
The arc length of a sector is 20 cm and the radius is r .
1.[5]
If the central angle is , find r.
Question 3
SLPaper 2
A circular track has radius 25 m . A runner jogs along a path that covers an angle of .
1.[4]
Calculate the distance the runner travels.
Question 4
SLPaper 2
A circular sign has a shaded sector with angle .
1.[3]
If the radius is , find the length of the arc.
Question 5
SLPaper 2
The arc length of a sector is 18.85 cm and the angle is .
1.[5]
Find the radius of the circle.
Question 6
SLPaper 2
A rotating wheel has a radius of 0.5 m . A mark on the rim travels along an arc of 1.75 m .
1.[6]
Estimate the angle the wheel has rotated through in degrees, and how many full rotations this represents.
Question 7
SLPaper 1
Joey is making a party hat in the form of a cone. The hat is made from a sector, AOB, of a circular piece of paper with a radius of 18 cm and AOB = θ as shown in the diagram. 
To make the hat, sides [OA] and [OB] are joined together. The hat has a base radius of 6.5 cm. 
1.[1]
Write down the perimeter of the base of the hat in terms of π.
2.[2]
Find the value of θ.
3.[2]
Find the surface area of the outside of the hat.
Question 8
SLPaper 1
In a circular playground with a diameter of 10 cm, a rope is stretched across the playground, forming a chord of length 8 cm.
1.[2]
Find the distance from the center of the circle to the midpoint of the chord, representing the height of the segment above the chord.
2.[2]
Calculate the angle subtended by the chord at the center of the circle.
3.[4]
Determine the area of the segment of the circle bounded by the chord and the arc.
Question 9
HLPaper 1
A circle has a radius of 5 cm.
1.[2]
Express an angle of in radians.
2.[4]
Find the area of the sector subtended by the angle .
3.[3]
Determine the length of the arc subtended by the angle .
Question 10
SLPaper 2
In a triangular plot of land , the sides and measure cm and cm, respectively, with an angle between them.
1.[5]
Use the cosine rule to calculate the length of side AC, which represents the boundary distance across the plot.
2.[3]
Calculate the area of the triangle using the formula to determine the total land area available for use.
3.[3]
Using your answer from part (a), determine whether the triangle is acute, right-angled, or obtuse, by checking if the angles meet the conditions for a right-angled or obtuse triangle. Justify your answer using trigonometric properties.