The formula for the surface area of a cone is given by:

$S = \pi r (r + l)$

where $S$ represents the total surface area, $r$ is the radius of the base, and $l$ is the slant height of the cone.

To better understand this formula, let’s break it down into two distinct components: the area of the base and the lateral (side) surface area.

1. Base Area: The base of the cone is a circle, and its area can be calculated using the formula:

   $A_{\text{base}} = \pi r^2$

   Here, $r$ is the radius of the base.

2. Lateral Surface Area: The lateral surface area requires a bit more consideration. It is calculated with the formula:

   $A_{\text{lateral}} = \pi r l$

   In this expression, $l$ represents the slant height of the cone, which is the distance measured along the surface from the apex of the cone to any point on the circumference of the base.

To find the total surface area of the cone, you simply add the area of the base to the lateral surface area:

$S = A_{\text{base}} + A_{\text{lateral}} = \pi r^2 + \pi r l$

Factoring out $\pi r$ from both terms, the formula simplifies to:

$S = \pi r (r + l)$

Example Calculation

Consider a cone with a radius of $3 \, \text{cm}$ and a slant height of $5 \, \text{cm}$. Substituting these values into the formula for the surface area yields:

$S = \pi \times 3 \times (3 + 5) = \pi \times 3 \times 8 = 24\pi \, \text{cm}^2$

If you require a numerical approximation, you can use $\pi \approx 3.14$, which gives:

$S \approx 24 \times 3.14 = 75.36 \, \text{cm}^2$

Understanding this formula is essential for tackling problems related to cones in your GCSE Maths exams.