About this chapter
Radians questions are some of the longest on the WMA11 paper. A typical one builds a shape from a sector and one or two triangles, then asks for its area and perimeter, so the sine and cosine rules from Chapter 6 are needed alongside s = rθ and A = ½r²θ.
The chapter covers converting to radians and exact values, arcs, sectors and segments, triangles solved in radian mode, and the graphs of sine and cosine in radians. The later questions design a stage, divide a plot of land in a given ratio, handle an obtuse angle in radians, write one curve as both a sine and a cosine, and find the largest sector possible for a fixed perimeter.
The ten questions
- Q1 (4 marks): write 150° in radians, then the arc length and area of a sector in terms of π.
- Q2 (6 marks): a sector of area 40 cm²: show that r = 8, then the perimeter and the chord.
- Q3 (5 marks): sketch y = cos(x − π/3) for 0 ≤ x ≤ 2π with exact intercepts and turning points.
- Q4 (5 marks): f(x) = 3 sin 2x: its period, an exact value, the maximum points and a solution count.
- Q5 (6 marks): a chord of 12 cm in a circle of radius 10 cm: show the angle is 1.287 radians, then the segment's area and perimeter.
- Q6 (9 marks): a stage made from a sector and two congruent triangles: the radius, an angle, the total area and the perimeter.
- Q7 (9 marks): a fence divides a sector so one region is twice the other: a "show that", then OP and the fence length to the nearest 100 m.
- Q8 (9 marks): a triangle with an obtuse angle in radians, joined to a sector: area and perimeter of the whole shape.
- Q9 (8 marks): y = a sin(x + b) from its maximum point, exact crossings, then the same curve as a translated cosine.
- Q10 (10 marks): a sector from its perimeter and area via a quadratic, rejecting an impossible angle, then the maximum possible area.
Key skills tested
Radians and exact values. π radians is 180°. For example, 150° is 5π/6, sin(π/3) = √3/2 and cos(π/4) = √2/2.
Arc length. s = rθ with θ in radians. For example, r = 5 and θ = 1.2 give an arc of 6.
Area of a sector. A = ½r²θ with θ in radians. For example, r = 4 and θ = π/4 give an area of 2π.
Area of a segment. Take the triangle away from the sector: ½r²θ − ½r² sin θ, which is ½r²(θ − sin θ).
Perimeters. Add every outside edge, including arcs, radii and straight sides, and never an internal line.
Triangles in radians. The sine rule, cosine rule and ½ab sin C all work in radians, as long as the calculator is in radian mode.
Equations from a sector. Write the perimeter and area in terms of r and θ, eliminate one, and check that θ is between 0 and 2π.
Trig graphs in radians. Sine and cosine have period 2π and tan has period π. For example, y = sin(x + π/3) is y = sin x moved π/3 to the left.
Solutions from graphs. Rearrange so one side is a graph you know and the other is a line you can draw, then count where they cross.
Worked example
Question 1 from this chapter. (a) Write 150° in radians as an exact multiple of π. A sector OAB has radius 6 cm and angle 150°. (b) Find, in terms of π, the arc length AB and the area of the sector.
(a) 180° is π radians, so 150° is 150/180 of π, which is 5π/6.
(b) Arc length: s = rθ = 6 × 5π/6 = 5π cm. Area: A = ½r²θ = ½ × 36 × 5π/6 = 15π cm².
This question carries a calculator warning, so the answers stay in terms of π. A decimal such as 47.1 cm² would not earn the final mark.
Seen on real papers
Every question in our booklets is original. These are the recent papers where each question type has appeared.
- January 2023, Q6: a stage-shaped sector joined to two congruent triangles, with the radius found from the area, as in Q6.
- June 2023, Q5: showing an angle to a given accuracy, then arc lengths and areas of a composite shape, as in Q5.
- October 2024, Q5: a sector split so the areas are in a ratio, a "show that", then the cosine rule and an answer to the nearest 100 m, as in Q7.
- January 2025, Q8: an obtuse angle in radians, a sector built on a triangle, and a perimeter that must leave out an internal side, as in Q8.
- June 2024, Q11: coordinates on a sine graph in radians, and a solution that needs an extra 2π, the ideas in Q9.
Where marks are lost
- Working in degrees. In October 2024 it was remarkably common to answer in degrees even though the question said x was in radians, and June 2024 saw coordinates given in degrees too.
- 2π instead of π. The most common mistake in January 2023 was subtracting an angle from 2π when the angles on a straight line add to π.
- The missing half. Leaving out the ½ in ½r²θ was the most common error in the sector area in January 2023.
- An internal side in the perimeter. In January 2025 a surprising number added a radius that sat inside the shape.
- Units and accuracy. In October 2024 the final mark went for missing units or not rounding to the nearest 100 m as asked.
Common questions
Are the arc length and sector area formulae in the formula booklet?
No. s = rθ and A = ½r²θ both have to be learned, and both need θ in radians.
Can I work in degrees and convert at the end?
It is risky. In October 2024 many answered a radians question in degrees and lost marks, and every conversion is another step where accuracy can slip.
How do I find the area of a segment?
Subtract the triangle from the sector: ½r²(θ − sin θ), with the calculator in radian mode.
Where this chapter leads
- Pure 2 Chapter 6: Trigonometric Identities and Equations: trig equations solved in radians.
- Pure 2 Chapter 7: Differentiation: maximum and minimum problems, such as the largest sector for a fixed perimeter, done with calculus.