Circular measure — practice questions

Let $r = 8$ cm and $\theta = \frac{2\pi}{3}$ radians.

(a) To find the arc length, we use the formula $s = r\theta$. The angle is already in radians, so we can substitute directly. $s = 8 \times \frac{2\pi}{3}$ $s = \frac{16\pi}{3}$ cm

(b) To find the sector area, we use the formula $A = \frac{1}{2}r^2\theta$. $A = \frac{1}{2} \times 8^2 \times \frac{2\pi}{3}$ $A = \frac{1}{2} \times 64 \times \frac{2\pi}{3}$ $A = 32 \times \frac{2\pi}{3}$ $A = \frac{64\pi}{3}$ cm$^2$

Given $r = 10$ cm and $\theta = 1.2$ radians.

(a) The perimeter of the segment is the sum of the arc length AB and the chord length AB.

Arc length AB = $r\theta = 10 \times 1.2 = 12$ cm.

To find the chord length AB, we can use the cosine rule on triangle AOB: $AB^2 = OA^2 + OB^2 - 2(OA)(OB)\cos(1.2)$ $AB^2 = 10^2 + 10^2 - 2(10)(10)\cos(1.2)$ $AB^2 = 200 - 200\cos(1.2) \approx 200 - 200(0.36236) = 127.528$ $AB = \sqrt{127.528} \approx 11.293$ cm.

Perimeter = Arc length + Chord length Perimeter = $12 + 11.293 = 23.293$ cm. Perimeter $\approx 23.3$ cm (3 s.f.)

(b) The area of the shaded segment is given by the formula $A = \frac{1}{2}r^2(\theta - \sin\theta)$. Ensure calculator is in RADIAN mode. $A = \frac{1}{2}(10^2)(1.2 - \sin(1.2))$ $A = 50(1.2 - 0.93204...)$ $A = 50(0.26796...)$ $A = 13.398...$ cm$^2$ Area $\approx 13.4$ cm$^2$ (3 s.f.)