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