Question

1.An arc of a circle of radius 5cm subtend an angle of 102°at the center of circle calculate
a. the length of the arc
b. perimeter of the sector

2. The area of the sector which subtends an angle of x at the center of the circle of radius 5.2cm is 21cm square calculate:
a.the value of x
b.length of the arc
c. perimeter of the sector

3. The perimeter of the sector which subtends an angle of 105°at the center of a circle is 17.25m calculate:
a.radius of the circle
b.length of an arc and
c.area of the sector

Answer 1

1a. The length of the circle arc is:

$$L = \pi r\dfrac{\theta}{180\degree} = \pi \cdot 5\dfrac{102\degree}{180\degree} \approx8.9cm$$

1b. Perimeter of the sector is the length of the circle arc and two bounding radii. Thus:

$$P = L + 2r = 8.9+10 = 18.9 cm$$

Answer. a) 8.9 cm, b) 18.9 cm.

2a. The area of sector is given by:

$$A = \pi r^2 \dfrac{x}{360\degree}$$

Then

$$x = \dfrac{360\degree A}{\pi r^2} = \dfrac{360\degree\cdot 21}{\pi \cdot 5.2^2}\approx89\degree$$

2b. The length of the circle arc is:

$$L = \pi r\dfrac{\theta}{180\degree} = \pi \cdot 5.2\dfrac{89\degree}{180\degree} \approx8.1cm$$

2c. Perimeter of the sector is the length of the circle arc and two bounding radii. Thus:

$$P = L + 2r = 8.1+10.4 = 18.5 cm$$

Answer. a) 89 degrees, b) 8.1 cm, c) 18.5 cm.

3a. Perimeter of the sector is the length of the circle arc and two bounding radii:

$$P = L + 2r$$

Substituting the expression for $L$, get:

$$P = L + 2r = \pi r\dfrac{\theta}{180\degree} + 2r = r\left(\pi\dfrac{\theta}{180\degree} + 2 \right)$$

Expressing $r$, obtain:

$$r = \dfrac{P}{\pi\dfrac{\theta}{180\degree} + 2} = \dfrac{17.25}{\pi\dfrac{105\degree}{180\degree} + 2} \approx 4.5m$$

3b. The length of the circle arc is:

$$L = \pi r\dfrac{\theta}{180\degree} = \pi\cdot 4.5\dfrac{105\degree}{180\degree} \approx 8.2m$$

3c. The area of the sector is:

$$A = \pi r^2 \dfrac{\theta}{360\degree} = \pi\cdot 4.5^2 \dfrac{105\degree}{360\degree} \approx 18.6m^2$$

Answer. a) 4.5m, b) 8.2m, c)18.6m^2.