How to Find the Surface Area of a Frustum

Finds the slant height as √((bottom radius − top radius)² + height²) and the lateral area as π × (top radius + bottom radius) × slant height. The total surface area adds the two circular ends.

Explanation

Cut the side of a frustum open and it unrolls into a band, a large sector with a smaller one removed. Everything starts from the slant height $\ell$ , which comes straight from Pythagoras.

\ell = \sqrt{(b - a)^2 + h^2}

$a$ — the top radius
$b$ — the bottom radius
$h$ — the height
$\ell$ — the slant height, measured along the side

The difference $b - a$ is how far the rim juts out when seen from above, and it forms one leg of a right triangle with the height. The lateral area is $\pi (a + b) \ell$ , and the total surface area adds the two circular ends.

S = \pi (a + b) \ell + \pi a^2 + \pi b^2

Example

With the defaults, a top radius of $a = 3$ , a bottom radius of $b = 5$ and a height of $h = 4$ .

\ell = \sqrt{(5 - 3)^2 + 4^2} = \sqrt{20} \approx 4.4721

The lateral area is $\pi \times (3 + 5) \times 4.4721 \approx 112.3970$ , and the ends are $\pi \times 3^2 \approx 28.2743$ and $\pi \times 5^2 \approx 78.5398$ . Together the surface area is about 219.2112.

Watch out

Do not mix up the slant height with the height. The height is the straight distance between the two circles; the slant runs along the side and is always the longer of the two. The volume needs the height, the surface area needs the slant.

For an open container such as a bucket, subtract the top circle $\pi a^2$ to get the area that has to be painted.