Exercises 15 (Length of an arc and area of a sector)

  1. Convert the following to radians:

    1. \(65^\circ\)
    2. \(125^\circ\)
    3. \(420^\circ\)
    4. \(1120^\circ\)

  2. Convert the following to degrees

    1. \(0.94\,\mathrm{rad}\)
    2. \(1.72\,\mathrm{rad}\)
    3. \(7.325\,\mathrm{rad}\)
    4. \(12.5\,\mathrm{rad}\)

  3. Calculate the length of an arc of a circle whose radius is \(1.15\,\mathrm{m}\) when the angle subtended at the centre is \(160^\circ\).

  4. An arc subtends an angle of \(1.732\,\mathrm{rad}\) at the centre of a circle whilst the length of the arc is \(258\,\mathrm{mm}\). Determine the circle’s diameter.

  5. In a belt drive system, \(300\,\mathrm{mm}\) are in contact with a pulley of \(400\,\mathrm{mm}\) diameter. Determine the angle of lap in both degrees and radians.

  6. Calculate the diameter and circumference of a circle if the area of a sector which subtends an angle of \(1.45\,\mathrm{rad}\) is \(620\,\mathrm{mm}^2\).

  7. Calculate the area of the shaded potion of the sketch given below and its percentage compared to the complete sector.

  8. The exhaust ports of an engine consist of 4 ring sectors (i.e. each part has a shape like the shaded area in the above picture) with outer radii \(50\,\mathrm{mm}\), inner radii \(22\,\mathrm{mm}\) and angle \(30^\circ\). Determine the total area of the ports.

Solutions: 1. (i) \(1.13\); (ii) \(2.18\); (iii) \(7.33\); (iv) \(19.55\); 2. (i) \(53.86^\circ\); (ii) \(98.55^\circ\); (iii) \(419.69^\circ\); (iv) \(716.20^\circ\); 3. \(3.2\,\mathrm{m}\); 4. \(d=298\,\mathrm{mm}\); 5. \(1.5\,\mathrm{rad}\), \(85.94^\circ\); 6. \(58.5\,\mathrm{mm}\), \(183.7\,\mathrm{mm}\); 7. \(573.75\,\mathrm{m}^2\), \(51\%\) 8. \(2100\,\mathrm{mm}^2\);