Surface Area of a Cone

Finds the slant height as √(radius² + height²), the lateral area as π × radius × slant height, and the total surface area as π × radius × (radius + slant height).

Explanation

The surface of a cone consists of a curved side, which unrolls into a sector, and a circular base. Everything begins with the slant height: the sloping line from the apex to the rim, obtained from the radius and the height by the Pythagorean theorem.

\ell = \sqrt{r^2 + h^2}

With the slant height in hand, the lateral area $A$ and the total surface area $S$ follow.

A = \pi r \ell \qquad S = \pi r (r + \ell)

$\ell$ — the slant height
$A$ — the lateral area
$S$ — the total surface area, side plus base
$r$ — the radius of the base
$h$ — the height

Example

With the defaults, $r = 3$ and $h = 4$ :

slant height $\ell = \sqrt{3^2 + 4^2} = \sqrt{25} = 5$
lateral area $A = \pi \times 3 \times 5 = 15\pi \approx 47.1239$
total surface area $S = \pi \times 3 \times (3 + 5) = 24\pi \approx 75.3982$

The base contributes $9\pi$ , and $9\pi + 15\pi = 24\pi$ , which matches the total.

Watch out

The height and the slant height are not the same thing. This calculator asks for the height, the perpendicular distance from the base to the apex, and derives the slant height from it; the slant height is always the longer of the two. Unrolled, the curved side becomes a sector of radius $\ell$ whose arc equals the base circumference $2\pi r$ . The area of a sector is half the arc times the radius, giving $\dfrac{1}{2} \times 2\pi r \times \ell = \pi r \ell$ , which is the lateral area. Both the radius and the height must be positive.