A regular polygon has 24 sides, as shown. Work out the sum of the interior angles of this polygon.

The interior angles of a polygon with \(n\) sides add up to

\((n - 2) \times 180^\circ\)

The reason for the \(n - 2\) is that any \(n\)-sided polygon can be cut by diagonals from one vertex into \(n - 2\) triangles, and each triangle contributes \(180^\circ\). For \(n = 24\):

\((24 - 2) \times 180\) [M1]

\(= 22 \times 180 = 3960^\circ\) [A1]

The fact that the polygon is regular is not needed for this part, since the sum of the interior angles depends only on the number of sides; it would matter only if a single angle were asked for, which would be \(3960 \div 24 = 165^\circ\).

Two errors are common. The first is multiplying by 24 rather than 22, giving \(4320^\circ\), which forgets the two triangles lost in the dissection. The second is confusing this with the exterior angles, which always total \(360^\circ\) regardless of the number of sides. A rough check on the size: 24 interior angles each a little under \(180^\circ\) should total a little under \(24 \times 180 = 4320^\circ\), and \(3960^\circ\) fits.