Standard +0.8 This is a multi-step geometric proof requiring identification of sectors, triangles, and careful angle work with radians. Students must recognize the 60° angles from equilateral triangles, calculate areas of circular segments, and combine them algebraically to reach a specific exact form. More demanding than routine sector area questions but uses standard C2 techniques.
2.
\includegraphics[max width=\textwidth, alt={}, center]{e5d62032-84ad-4e0b-9b72-ccfd8f4dbac8-1_588_513_813_593}
The diagram shows a circle of radius \(r\) and centre \(O\) in which \(A D\) is a diameter.
The points \(B\) and \(C\) lie on the circle such that \(O B\) and \(O C\) are arcs of circles of radius \(r\) with centres \(A\) and \(D\) respectively.
Show that the area of the shaded region \(O B C\) is \(\frac { 1 } { 6 } r ^ { 2 } ( 3 \sqrt { 3 } - \pi )\).
2.\\
\includegraphics[max width=\textwidth, alt={}, center]{e5d62032-84ad-4e0b-9b72-ccfd8f4dbac8-1_588_513_813_593}
The diagram shows a circle of radius $r$ and centre $O$ in which $A D$ is a diameter.\\
The points $B$ and $C$ lie on the circle such that $O B$ and $O C$ are arcs of circles of radius $r$ with centres $A$ and $D$ respectively.
Show that the area of the shaded region $O B C$ is $\frac { 1 } { 6 } r ^ { 2 } ( 3 \sqrt { 3 } - \pi )$.\\
\hfill \mbox{\textit{OCR C2 Q2 [6]}}