Circular measure in Additional Math 0606: common mistakes and examiner advice

Circular measure questions ask for arc lengths, sector and segment areas, and perimeters of shapes built from circles, triangles and rectangles. The reports show that students know the formulae. The marks go missing in the plan: which lengths and areas to include, and which angle to use. 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

• Perimeters. Doubling all the radii, summing only the arc lengths, using a wrong length for a straight edge, or omitting one or more lengths. Others left out the two radii that make up the straight sides of a sector. The reports say candidates should make a clear plan, derived from careful study of the diagram.
• Areas. Leaving out the ½ in the area formula, not halving r² in all three areas of one question, forgetting to square an expression such as 16 − 4.5x, or finding the arc length when the area was needed.
• Combining areas. Subtracting the sector from the square and not adding the triangle, or summing two segments and a sector and subtracting that from the circle. The reports describe an incorrect plan, even when the individual areas were right.
• The wrong angle. Errors in the segment area often involved an incorrect angle. In other questions the wrong angle was used in the sine or cosine rule, or an angle (such as BOC) that should have been found with the inverse tangent was not found correctly. Some students could not find a reflex angle in terms of π and α.
• The wrong lengths. Confusion over which lengths to use, such as a in the cosine rule in place of OC, a chord length not used from an earlier part, forgetting to take the square root of AC² before adding it to another length, and an impossible value for a length (1, 2, 4 or 5 in one question).
• Radius and ratio. Taking the wrong value of the radius (r = 15 in one question), and not using the ratio 7 : 10 correctly.
• Degrees. Making an unnecessary conversion to degrees, which introduced errors. The reports say that finding an angle in degrees first was rarely successful, and that candidates who worked in degrees were more likely to make an error. Working in radians is the efficient route, but those who used degrees were not penalised when radians were not required.
• Rounding and showing a value. Using harshly rounded working values (14.4 for AC in one question), stating the given value of θ (0.885) without showing a more accurate value, or showing a more accurate decimal and not concluding that it rounds to 2.21 to 3 significant figures.
• Formulas. A small number used an incorrect form of the sine or cosine rule or Pythagoras' theorem, and were advised to use the list of formulas on page 2.
• Missing the link to an earlier part. Very few gave correct answers to one part because they did not see the link from the previous part. Others did not give the information provided in the question where possible, using values they had calculated instead.

What examiners advise

• Work in radians, and keep working values to at least 4 significant figures, with a final answer correct to 3 significant figures.
• Plan before you calculate: list the lengths or areas that make up the shape, and check that each appears once.
• Label your calculations clearly. Candidates who did not find the correct answer often confused themselves through a lack of clear labelling.
• Show all key method steps, and use values given in the question where possible.
• Use the list of formulas for the cosine and sine rules.
• Give your answer in the form requested.

Habits that help

• Draw the shape again, large, and write each length and angle on it before you start.
• For an area, write "½r²θ" or "½r²(θ − sin θ)" in full, then substitute.
• Check an angle by seeing whether it is between 0 and 2π and fits the diagram.

Questions students ask

Should I work in degrees or radians in circular measure?

Radians. Unnecessary conversion to degrees introduced errors, and reports say finding an angle in degrees first was rarely successful. Keep working values to at least 4 significant figures and give the final answer to 3.

How do I find the perimeter of a shape made of arcs and straight edges?

List every length that makes up the shape and check each appears once. Students doubled all the radii, summed only the arc lengths, used a wrong straight edge or left out the two radii of a sector.

How do I find the area of a combined shape?

Plan which areas to add and subtract, for example sector plus triangle, and check you have not missed a part. Students subtracted the sector from the square and did not add the triangle. Also remember the ½ in ½r²θ.

Which formulas can I look up in the exam?

The sine and cosine rules are in the list of formulas on page 2. Reports advise using it rather than risking an incorrect form.

Why do I need the answer to an earlier part?

Because later parts often use it. Very few gave correct answers because they did not see the link from the previous part. Use values given in the question where possible, and show a more accurate value of a given angle.