Question

Use the fact that the length of an arc intercepted by an angle is proportional to the radius to find the area of the sector given r = 3 cm and Θ =
π
4
.

Answer 1

Sector area = (Central Angle (Degrees) / 360) * PI * radius^2
One way to solve this is that an angle of (PI/4) = 45 degrees.
45 degrees is one eighth of a circle.
Area of ENTIRE circle = PI*radius^2 = PI * 3^2 = 28.2743338823  square centimeters
Area of the sector is one eighth of this =  3.5342917353 square centimeters

Answer 2

Answer:

Let l be the length of an ac and r be the radius of the circle.

Use the fact that the length of an arc intercepted by an angle is proportional to the radius

i.e

$l \propto r$

⇒$l = r \theta$ where, $\theta$ is the angle in radian.

To find the Area of the sector:

Given: r = 3 cm and $\theta = \frac{\pi}{4}$

$\text{Area of the sector A} = \pi r^2 \cdot \frac{\theta}{360^{\circ}}$

where, $\theta$ is the angle in degree.

Use conversion:

1 radian = $\frac{180}{\pi}$ degree

then;

$\frac{\pi}{4}$ = $\frac{\pi}{4 } \times \frac{180}{\pi} = 45^{\circ}$

then;

$\theta =45^{\circ}$ and use $\pi = 3.14$

Substitute the given values we have;

$A= 3.14 \cdot 3^2 \cdot \frac{45}{360}$

⇒$A = 3.14 \cdot 9 \cdot 0.125$

Simplify:

$A = 3.5325$ square cm

Therefore, the area of the sector is, 3.5325 square cm