Circular measure in A Level Math Paper 1: common mistakes and examiner advice

Circular measure is a short topic with a lot of ways to lose marks. The formulas are few, but the questions usually combine sectors, triangles and segments, and the mistakes come from choosing the wrong angle, the wrong radius or the wrong combination of areas. This guide collects what Cambridge examiners say students get wrong, based on recent reports.

Summarised in our own words from a reading of the examiner reports.

Mistakes that cost marks

• The wrong radius. In composite figures, the sector's radius may be different from the one in the diagram's label. Reports mention students who used r when the sector had radius 2r, who assumed a radius was simply 6, or who forgot to add the straight edges to an arc length when finding a perimeter.
• The wrong angle. Half the angle was used where the whole angle was needed, or the angle in the sector was written as π − θ when it should have been 2π − θ. Weaker students did not make clear which angles they were finding.
• Degrees in radian formulae. Using 180° or 45° in rθ or ½r²θ is a repeated error. In arc length and sector area formulae the angle must be in radians. Some students also used 2π where π was required (or the reverse) when converting.
• Perimeter and area mixed up. Students found the area when the perimeter was required, or the reverse. Others forgot to add the two radii for each sector.
• Missing parts of the figure. A shaded region is usually a combination such as "triangle plus three segments" or "sector plus triangle". Students forgot to add the triangle, left out a segment, or assumed the region was a semicircle.
• Assuming the figure is special. Reports mention the assumption that a triangle is equilateral or that a shape is a semicircle without any justification. Diagrams are often not to scale.
• Premature rounding. Finding an angle in degrees, converting to radians, and rounding at every step loses accuracy. Reports advise keeping more than three significant figures in intermediate steps.
• Forgetting π in a circumference or arc. For example, the circumference of a semicircle without π.
• A different answer form from the one asked. Accuracy marks cannot be awarded if the final answer is not given to the required degree of accuracy, and exact questions need an exact form.

What examiners advise

• Know the formula for the area of a segment. It appears often. The reports describe it as needed in many questions on this topic.
• Use the marks as a guide. A question worth one mark should not need a lot of work.
• Show all the necessary working in "show that" questions, including the cosine rule, sine rule or right-angled triangle step.
• Split an isosceles triangle in half to find its height and base. Remember to double the result afterwards.
• Work in radians from the start where possible.

Habits that help

• Label every angle and every length on a sketch before you write any formula.
• Write the region as a sum or difference of simple shapes, then add or subtract.
• Read the question again at the end and check you have a perimeter, an area or whichever quantity it asked for.

Questions students ask

Can I use degrees in the arc length and sector area formulae?

No. In rθ and ½r²θ the angle must be in radians. Using degrees such as 180° or 45° in these formulae is a repeated error in the reports, and so is using 2π where π was needed when converting.

How do I find the perimeter of a sector or a shaded region?

Add up every edge: the arc and the straight edges. Reports mention students who forgot to add the straight edges to an arc length, or the two radii of each sector, and others who found the area when the perimeter was asked for.

How do I find the area of a shaded region?

Write it as a sum or difference of simple shapes, such as a triangle plus three segments or a sector plus a triangle, then add or subtract. Students lose marks by forgetting the triangle, leaving out a segment, or assuming the region is a semicircle.

Why do I lose accuracy marks in circular measure?

Rounding too early. Finding an angle in degrees, converting and rounding at every step loses accuracy, so keep more than three significant figures in intermediate steps and give the final answer to the accuracy asked for.

How do I find the height of an isosceles triangle in a circular measure question?

Split it in half to make a right-angled triangle, find the height and base, and remember to double the result afterwards. In "show that" questions, show the cosine rule, sine rule or right-angled triangle step.