Circular Measure

Learn, then practise

The original worked lessons remain the main explanations. Use these focused reminders to check the conditions before you start.

Radians, degrees and complete rotations

Convert both ways between degrees and radians.
Use the radian definition and distinguish minor from reflex angles.
Retain full turns for distance, but reduce signed rotations for final direction.

\(180^\circ=\pi\) rad and \(\displaystyle \theta=\frac{s}{r}\) when both lengths use the same unit. A full turn is \(2\pi\) rad. A swept angle may exceed one turn; final direction and total travel are different quantities.

Read the full notes and 3 worked examples · Next: short practice

Arc length, sector perimeter and rolling distance

Choose the requested arc and use compatible length units.
Distinguish curved length, sector perimeter and full-circle boundary.
Translate rolling or rotational descriptions into total swept angles.

Use \(s=r\theta\) with \(\theta\) in radians. A proper sector, \(0<\theta<2\pi\), has \(P=r\theta+2r\). A full disk has no exposed radial edges. For rolling without slipping, distance equals radius times total swept angle.

Read the full notes and 3 worked examples · Next: short practice

Sector area, inverse problems and constraints

Use area, radius, angle and arc length together.
Solve inverse problems and reject geometrically impossible solutions.
Use completing the square for constrained fixed-perimeter problems.

\(A=\tfrac12r^2\theta=\tfrac12rs\). A proper sector has \(r>0\) and \(0<\theta<2\pi\). For fixed perimeter \(L\), \(\displaystyle A= \frac{Lr}{2}- r^{2}\). An algebraic maximum is usable only when its radius and angle satisfy every condition.

Read the full notes and 3 worked examples · Next: short practice

Chords, minor segments and major segments

Use triangle geometry to find chords and central angles.
Subtract the correct triangle from a sector.
Trace the correct arc-plus-chord boundary; interpret tangents and overlap.

For the minor central angle \(0<\theta\le\pi\), chord \(\displaystyle c=2r\sin(\frac{\theta}{2})\) and minor segment area \(\tfrac12r^2(\theta-\sin\theta)\). Use RAD mode. A segment perimeter is arc plus chord, not arc plus two radii. The major segment is the rest of the disk.

Read the full notes and 3 worked examples · Next: short practice

Annular sectors and composite boundaries

Combine or subtract circular and straight-edged regions.
Count only exposed edges, including both arcs of an annular sector.
Handle scaling, inverse geometry and limiting boundary cases.

An annular sector has \(A=\tfrac12(R^2-r^2)\theta\) and \(P=(R+r)\theta+2(R-r)\), for \(0<r<R\), \(0<\theta<2\pi\). A complete annulus has no radial connectors. Internal joins do not contribute to a composite perimeter. Enlargement factor \(k\) multiplies areas by \(k^2\).

Read the full notes and 3 worked examples · Next: short practice