Problem-Solving with Circular Measure

The accessible marks in circular measure are the formulas; the discriminating marks are composite problems, two arcs sharing a chord, a circle inscribed in a sector, belts and badges and overlapping discs. They all yield to one strategy.

The four-step strategy

1. Decompose and name. Break the figure into sectors, triangles, segments and straight edges, and label them on the diagram (sector OAB, triangle OPQ…). This single habit converts an intimidating picture into a checklist, and the labelled decomposition is creditable method.
2. Hunt the hidden angle or radius. Composite questions rarely hand you every $\theta$ and $r$. They hide one inside geometry: an equilateral triangle ($\theta = \frac{\pi}{3}$) formed by equal radii of touching circles; a right triangle giving an angle by trigonometry; symmetry halving an angle. The hunt is usually one trig step, and it’s where most students stall, so do it first, before any formula.
3. Compute each piece with its own $r$ and $\theta$: arcs, sector areas, triangles by $\tfrac{1}{2}ab \sin C$, chords by $2r \sin(\theta/2)$.
4. Assemble per the question: perimeter = walk the boundary (only boundary pieces count); area = add and subtract per the shading.

A classic configuration

Two circles of radius $r$ touch externally; each centre is joined to the contact point and to the other tangent points… the standard exam versions all start the same way: join the centres. The line of centres has length $r_1 + r_2$ (external touch) or $r_1 - r_2$ (internal), and the triangle it forms with radii is where the hidden angle lives. “Join the centres, mark the radii” unlocks nearly every two-circle figure in past papers.

Working that earns the marks

State the plan in fragments as you go: “Area required $=$ sector $OAB -$ triangle $OAB$”, ”$\theta = \frac{\pi}{3}$ since triangle $OPQ$ is equilateral (all sides $r$)”. Each stated structural fact is a mark-bearing line. Then carry precision and round once: composite answers chain several computed values, and early rounding compounds.

Common mistakes

• Formulas attempted before the hidden angle is found
• Internal lines (radii, chords) included in perimeters
• The equilateral/isosceles structure of equal radii unnoticed, the hidden angle stays hidden
• Each piece computed with the figure’s $r$ and $\theta$ instead of its own
• Final assembly adding a piece the shading subtracts

Full topic context: Circular Measure notes · the unit discipline that underwrites it all: radians.