1.4.1 Circular Measure Short Questions (Question 1, 2 & 3)

Question 1:

The figure shows the sector OCB of radius 13 cm at the centre O. The length of the arc CB = 5.2 cm. Find

(a) the angle in radians,

(b) the perimeter of the shaded region.

Solution:
(a)

\begin{array}{l} s = r θ \\ 5.2 = 13 (∠ C O B) \\ ∠ C O B = 0.4 radian \end{array}

(b)

\begin{array}{l} \cos ∠ C O B = \frac{O A}{O C} \\ \cos 0.4 = \frac{O A}{13} (change calculator to Rad mode) \\ O A = 11.97 cm \\ ∴ A B = 13 - 11.97 = 1.03 cm \\ C A = \sqrt{13^{2} - {11.97}^{2}} \\ C A = 5.07 cm \end{array}

Perimeter of the shaded region = 5.07 + 1.03 + 5.2 = 11.3 cm.

Question 2:

The figure shows the sector AOB of a circle, centre O and radius 5 cm. The length of the arc AB is 6 cm. Find the area of:

(a) the sector AOB,

(b) the shaded region.

Solution:
(a) Arc AB = 6cm

s = r θ

6 = 5 θ

θ = 6/5 rad

\begin{array}{l} Area of sector A O B \\ = \frac{1}{2} r^{2} θ = \frac{1}{2} {(5)}^{2} (\frac{6}{5}) = 15 c m^{2} \end{array}

(b)

\begin{array}{l} Area of shaded region \\ = \frac{1}{2} r^{2} (θ - \sin θ) (change calculator to Rad mode) \\ = \frac{1}{2} {(5)}^{2} (\frac{6}{5} - \sin \frac{6}{5}) \\ = 3.35 {cm}^{2} \end{array}

Question 3:

Diagram below shows a sector QOR of a circle with centre O.

It is given that PS = 8 cm and QP = PO= OS = SR = 5 cm.

Find

(a) the length, in cm, of the arc QR,

(b) the area, in cm², of the shaded region.

Solution:

(a) Length of arc QR = r θ = 10 (1.75) = 17.5 cm

(b)

Area of the shaded region

= Area of sector QOR – Area of triangle POS

= ½ (10)² (1.75) – ½ (5) (5) sin 1.75 (change calculator to Rad mode)

= 87.5 – 12.30

= 75.2 cm²