Question

A sector has a central angle of 135° and a radius of 8 cm. Another sector of the same circle has a central angle of 3∏/4 radians. What is the ratio of the area of the first sector to the area of the second sector?

• A. 1:1
• B. 3:4
• C. 2:3
• D. 5:6

Answer 1

Answer:

A. 1:1

Area of a sector is proportional to its central angle (when radius is the same):

$A = \frac{\theta}{360^\circ}\pi r^2 \quad \text{(in degrees)}$
$\quad \text{or} \quad$
$A = \frac{1}{2} r^2 \theta \quad \text{(in radians)}$

Since both sectors are from the same circle ((r = 8) cm), the ratio depends only on angles.

Step 1: Convert first angle to radians

$135^\circ = \frac{135\pi}{180} = \frac{3\pi}{4}$

So:

• First sector angle = $( \frac{3\pi}{4} )$

• Second sector angle = $( \frac{3\pi}{4} )$

Compare areas

Since both sectors have the same radius and same central angle, their areas are equal.

Step 3: Ratio

$\text{Ratio} = 1 : 1$

Final Answer:

$\boxed{1:1}$