What is the formula for the surface area of a cone?

The formula for the surface area of a cone is \( \pi r (r + l) \).

To break this down, the surface area of a cone consists of two parts: the base and the lateral (side) surface. The base of the cone is a circle, and its area is given by \( \pi r^2 \), where \( r \) is the radius of the base. The lateral surface area is a bit trickier to calculate. It is given by \( \pi r l \), where \( l \) is the slant height of the cone. The slant height is the distance from the top of the cone to any point on the edge of the base, measured along the surface of the cone.

So, to find the total surface area of the cone, you add the area of the base to the lateral surface area. This gives you the formula \( \pi r^2 + \pi r l \). By factoring out \( \pi r \) from both terms, you get \( \pi r (r + l) \).

For example, if you have a cone with a radius of 3 cm and a slant height of 5 cm, you would substitute these values into the formula to get the surface area. This would be \( \pi \times 3 \times (3 + 5) = \pi \times 3 \times 8 = 24\pi \) square centimetres. If you need a numerical answer, you can approximate \( \pi \) as 3.14, giving you \( 24 \times 3.14 = 75.36 \) square centimetres.

Understanding this formula is crucial for solving problems related to cones in your GCSE Maths exams.