Skill: Calculate the surface area of cones

To find the surface area of a right circular cone, add:

• the area of the circular base: $\pi r^2$
• the curved surface area: $\pi r\ell$

So the formula is

$$S=\pi r^2+\pi r\ell=\pi r(r+\ell),$$

where $r$ is the radius and $\ell$ is the slant height. The slant height runs along the side of the cone; it is not the perpendicular height.

Example: A cone has radius $5$ cm and perpendicular height $12$ cm. Find its total surface area.

Step 1: Find the slant height.

The radius, perpendicular height, and slant height form a right triangle. Use the Pythagorean theorem:

$$\ell^2=r^2+h^2$$ $$\ell^2=5^2+12^2=25+144=169$$ $$\ell=13\text{ cm}$$

Step 2: Substitute into the surface area formula.

$$S=\pi r(r+\ell)$$ $$S=\pi(5)(5+13)$$ $$S=90\pi\text{ cm}^2$$

Step 3: Give an approximate answer if needed.

$$90\pi\approx 282.7\text{ cm}^2$$

The surface area of the cone is therefore $\boxed{90\pi\text{ cm}^2}$, or approximately $\boxed{282.7\text{ cm}^2}$.