An annular (ring-shaped) disc has outer and inner radii of 12 cm and 8 cm. What percentage of the full circle is this ring?

The correct option is 55.56%.

To find what percentage of the full circle is occupied by the annular (ring-shaped) disc, we need to calculate the area of the annular ring and compare it to the area of the full (outer) circle.

Let:
- The outer radius of the disc be R = 12 cm
- The inner radius of the disc be r = 8 cm

Step 1: Calculate the area of the full circle (outer circle).
The formula for the area of a circle is:

${\text{Area}}_{\text{full}}=\pi R^2$

Substituting R = 12 cm:

${\text{Area}}_{\text{full}}=\pi \times {12}^2=144\pi {\text{ cm}}^2$

Step 2: Calculate the area of the annular ring.
The area of the ring is the difference between the area of the outer circle and the inner empty circle:

${\text{Area}}_{\text{ring}}=\pi R^2-\pi r^2=\pi (R^2-r^2)$

Substituting R = 12 cm and r = 8 cm:

${\text{Area}}_{\text{ring}}=\pi ({12}^2-8^2)=\pi (144-64)=80\pi {\text{ cm}}^2$

Step 3: Calculate the percentage of the full circle.
The percentage is given by the ratio of the ring area to the full circle area, multiplied by 100:

$\text{Percentage}=\left(\frac{{\text{Area}}_{\text{ring}}}{{\text{Area}}_{\text{full}}}\right)\times 100$

$\text{Percentage}=\left(\frac{80\pi }{144\pi }\right)\times 100=\left(\frac{80}{144}\right)\times 100$

Simplifying the fraction:

$\frac{80}{144}=\frac{5}{9}\approx \mathrm{0.55555...}$

$\text{Percentage}=\mathrm{0.55555...}\times 100\approx \mathrm{55.56\%}$

Thus, the ring accounts for 55.56% of the full circle.