Quick questions on Arc length and sector area explained: O-Level E-Maths

A sector is a slice of a circle bounded by two radii and an arc. The fraction of the circle it covers is the central angle over $360^\circ$:

The sector area is the same fraction of the full circle's area:

A segment is the region between a chord and its arc, and its area is the sector area minus the triangle formed by the two radii and the chord. The triangle is found with the trigonometric area rule, $\tfrac{1}{2}r^2\sin\theta$, using the same central angle. So the segment area is $\tfrac{\theta}{360^\circ} \times \pi r^2 - \tfrac{1}{2}r^2\sin\theta$. For a sector of radius $10$ and angle $90^\circ$, the sector area is a quarter circle and the triangle is $\tfrac{1}{2}(10)^2\sin 90^\circ = 50$, so the segment is the difference.

The fraction is the central angle over $360^\circ$; using $180^\circ$ or the radius by mistake gives a wrong result.

Keep full accuracy through the working and round only the final answers.

A sector has radius $12\ \text{cm}$ and angle $90^\circ$. State the fraction of the circle it covers. [1 mark]

Find the arc length of a sector with radius $5\ \text{cm}$ and angle $72^\circ$, taking $\pi = 3.142$. [2 marks]

Find the area of a sector with radius $8\ \text{cm}$ and angle $45^\circ$, taking $\pi = 3.142$. [2 marks]