The diagram shows a shape made from a rectangle \(20\) cm by \(14\) cm with a semicircle of diameter \(20\) cm on top. (a) Calculate the perimeter of the sh...

The shape is a rectangle with a semicircle sitting on the top edge. That top edge of the rectangle is inside the shape, so it counts for the area but not for the perimeter.

(a) Going round the outline: the bottom of the rectangle is \(20\) cm, the two vertical sides are \(14\) cm each, and the curved edge is half the circumference of a circle of diameter \(20\) cm.

\[ P=20+2\times 14+\frac{\pi\times 20}{2} \] [M1]

\[ P=20+28+31.415...=79.415... \]

Correct to 3 significant figures, the perimeter is \(79.4\) cm [A1] (\(79.415...\) is accepted).

(b) For the area, add the rectangle and the semicircle. The semicircle has diameter \(20\) cm, so its radius is \(10\) cm:

\[ A=20\times 14+\frac{\pi\times 10^2}{2} \] [M1]

\[ A=280+157.07...=437.07... \]

Correct to 3 significant figures, the area is \(437\) cm\(^2\) [A1] (\(437.07...\) is accepted).

Notice how the \(20\) cm is used differently in the two parts: as a diameter in the arc length \(\frac{\pi d}{2}\), and halved to a radius of \(10\) cm in the area \(\frac{\pi r^2}{2}\). Mixing these up is the usual source of error in composite circle questions.