Find the area of the rectangle by adding together the lengths AB, BC and CD, and multiplying it by DE.

$Area (rectangle) = (r + 2 r \sin 0.75 + r) \times r$

Simplify.

$= (2 r + 2 r \sin 0.75) r$

[1]

Write in terms of $r^{2}$

$= 2 r^{2} + 2 r^{2} (\sin 0.75) = r^{2} (2 + 2 \sin 0.75) = 3.36 r^{2}$

Find the area of the segment by subtracting the area of triangle BOC from the area of the sector.

$\begin{array}{rcl} Area (sector) & = & \frac{1}{2} r^{2} θ \\ = & \frac{1}{2} r^{2} (1.5) \end{array}$

$\begin{array}{rcl} Area (triangle) & = & \frac{1}{2} a b \sin C \\ = & \frac{1}{2} r^{2} (\sin 1.5) \end{array}$

$Area (segment) = \frac{1}{2} r^{2} (1.5) - \frac{1}{2} r^{2} (\sin 1.5)$

Factorise by pulling out a factor of $\frac{1}{2} r^{2}$ and simplify to write in terms of $r^{2}$

$\begin{array}{rcl} Area (segment) & = & \frac{1}{2} r^{2} (1.5 - \sin 1.5) \\ = & 0.251 r^{2} \end{array}$

[1]

Subtract the area of the segment from the area of the rectangle.

$\begin{array}{rcl} Area (shaded) & = & 3.36 r^{2} - 0.251 r^{2} \end{array}$

[1]

$3.11 r^{2}$ [1]

Circular Measure (Cambridge O Level Additional Maths)

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