A solid cone has a radius of 5 cm and a vertical height of 12 cm, as shown - Junior Cycle Mathematics - Question 3 - 2018

The total surface area (T.S.A) of a cone is given by the formula:

$TSA = ext{Curved Surface Area} + ext{Base Area}$

1. Curved Surface Area = $\pi r l$ where $l$ is the slant height:

   • $\pi = 3.14$ (approximately)
   • $r = 5$ cm, and we already found $l = 13$ cm.
   • So, $\text{Curved Surface Area} = \pi (5)(13) = 65\pi \approx 204.2 \text{ cm}^2$

2. Base Area = $\pi r^2$:

   • $\text{Base Area} = \pi (5^2) = 25\pi \approx 78.5 \text{ cm}^2$

Now, adding both areas:

$TSA = 204.2 + 78.5 \approx 282.7 \text{ cm}^2$

Thus, the total surface area of the cone is approximately $282.7 \text{ cm}^2$.

The radius of the circle is the same as the radius of the cone:

• Radius of the circle = 5 cm.

The circumference of a circle is calculated using the formula:

$C = 2\pi r$

Substituting the radius:

$C = 2\pi(5) = 10\pi \approx 31.4 ext{ cm}$

The radius of the sector of the circle is equal to the slant height of the cone:

• Radius of the sector = 13 cm.

The length of the arc can be calculated using the formula for the arc length of a sector:

$L = \frac{\theta}{360} \times 2\pi r$

In this case, we have:

• Radius = 13 cm
• Assuming the angle for the sector is 180 degrees (half the circle), we find: $L = \frac{180}{360} \times 2\pi(13) = \pi(13) \approx 40.8 ext{ cm}$