\( \DeclareMathOperator{cosec}{cosec} \)

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International Baccalaureate Mathematics

Geometry and Trigonometry

Syllabus Content

The circle: radian measure of angles; length of an arc; area of a sector

Here are some exam-style questions on this statement:

"The following diagram shows a circle with centre O and radius 9 cm." ... more

"The perimeter of the sector is 8 cm and the area of the sector is 3 cm²." ... more

"The diagram shows a circle with a minor sector shaded blue and a major sector shaded yellow." ... more

"The following diagram (not to scale) shows a circle with centre O and radius 7 cm." ... more

"The following diagram shows part of a circle with centre O and radius 10cm." ... more

"The following diagram shows a semicircle with centre O. Points A. B and C lie on the circumference of the circle and AC is a diameter. Angle BOC = \(\theta\), where \(0 < \theta < \frac{\pi}{2}\)." ... more

"An Asian water buffalo is tethered in a rectangular field by a rope of length \(r\) metres. One end of the rope is securely tied to point \(P\) as shown in this diagram (not drawn to scale)." ... more

See all these questions

Here is an Advanced Starter on this statement:

Pizza Slice
A problem which can be solved by considering the areas of a triangle and a sector of a circle. more

Click on a topic below for suggested lesson Starters, resources and activities from Transum.

Angles Pupils should understand that angles represent an amount of turning and be able to estimate the size of angle. When constructing models and drawing pupils should be able to measure and draw angles to the nearest degree and use appropriate language associated with angles. Pupils should know the angle sums of polygons and that of angles at a point and on a straight line. They will learn about angles made in circles by chords, radii and tangents and recognise the relationships between them. Pupils will work with angles using trigonometry, transformations and bearings. In exams pupils are often instructed that while non-exact answers should be given to three significant figures, angle answers should be given to one decimal place.

Furthermore

Official Guidance, clarification and syllabus links:

Radian measure may be expressed as exact multiples of \( \pi \), or decimals.

Formula Booklet:

Length of an arc is \( l = r\theta \), where \( r \) is the radius, \( \theta \) is the angle measured in radians

Area of a sector is \( A = \frac{1}{2}r^2\theta \), where \( r \) is the radius, \( \theta \) is the angle measured in radians

In the study of circles, the radian measure is a way of expressing angles that is directly related to the radius of the circle. It provides a simple relationship between the length of an arc and the angle subtended at the centre of the circle by that arc. Similarly, the area of a sector can be found by knowing the angle in radians and the radius of the circle. This approach to measurement simplifies many mathematical calculations and is fundamental in trigonometry and calculus.

Key Formulae:

The length of an arc \( l \) in a circle of radius \( r \) with a central angle \( \theta \) measured in radians is given by:

\[ l = r\theta \]

The area \( A \) of a sector of a circle with radius \( r \) and central angle \( \theta \) in radians is given by:

\[ A = \frac{1}{2}r^2\theta \]

Example:

Consider a circle with a radius of 4 cm. Calculate the length of the arc and the area of the sector formed by a central angle of 1.5 radians.

The length of the arc is:

\[ l = r\theta = 4 \times 1.5 = 6 \text{ cm} \]

The area of the sector is:

\[ A = \frac{1}{2}r^2\theta = \frac{1}{2} \times 4^2 \times 1.5 = 12 \text{ cm}^2 \]

A nice video explanation of what a radian is (from SuperScript)

This video on Radians, Lengths of Arcs and Areas of Sectors is from Revision Village and is aimed at students taking the IB Mathematics Analysis and Approaches SL and HL courses.

This video on Degrees v Radians is from Revision Village and is aimed at students taking the IB Maths AA SL/HL level course

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