IGCSE Additional Mathematics (0606)

Arc length and sector area (radians)

What this topic tests, the mistakes that cost marks, and one exam-style question worked through step by step.

Key ideas and common mistakes

When a sector is rolled into a cone, the curved surface area of the cone is equal to the area of the original sector, not the area of a full circle or a different formula.
Common mistake: Students often mistakenly recalculate using the cone's base or height.
Because the angle is already in radians, find θ first, then use arc length = rθ directly, with no conversion to degrees.
Common mistake: Students often mistakenly convert θ to degrees or use degrees in the formula, leading to an incorrect answer.
To find the area of a sector, use (1/2) × radius squared × angle in radians.
Common mistake: A common mistake is to omit the 1/2 or use the formula for triangle area instead, leading to an answer that's too large.
To find the angle at the centre of an arc from a point outside a square, use the vertical distance from the centre to the square's edge (not the full distance from the base), and apply right triangle trigonometry to half the angle.
Common mistake: A common error is using the full distance or the cosine rule unnecessarily.
The perimeter of a region bounded by arcs and straight lines requires adding the lengths of all boundary segments.
Common mistake: Including both arcs and straight sides—where students often omit the straight sides or misapply the arc length formula.
When a sector is rolled into a cone, the arc length becomes the cone's base circumference and the sector's radius becomes the cone's slant height.
Common mistake: Students often confuse which length corresponds to which part of the cone.

Worked example

Find the curved surface area of this cone.

Try it first, then show the working

Use the formula for the area of a sector to find the curved surface area of the cone.

Area = (1/2) \(r^2\) \(\theta\) = (1/2) \(\times\) 1\(2^2\) \(\times\) \(\frac{4\pi}{3}\)
Calculate the curved surface area as 96\pi cm^2, equivalent to 302 cm^2 to 3 significant figures.

\[96\pi = 301.59... = 302 \text{ cm}^2 \text{ to 3 s.f.}\]

Answer: 96pi cm^2 (= 302 cm^2 to 3 s.f.)

Try one yourself

Work out the length of arc AB.Answer it and get it marked → Work out the area of sector ACB.Answer it and get it marked → Show that theta = 1.29 correct to 3 significant figures.Answer it and get it marked →

12 exam-style questions on Arc length and sector area (radians), each marked to the scheme.

Arc length and sector area (radians)

Key ideas and common mistakes

Worked example

Try one yourself

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