---
tag: 0WQE3G7B88
kind: definition
name: Surface area
msc: [28A75]
defines: 0S29Y4ZW0Y
uses: [05788J51WZ, 0BJDQ2HBH1, 0NKBMFC18R, 0565DC8RMK]
sources:
  - url: https://en.wikipedia.org/wiki/Surface_area
    locator: Definition
  - url: https://en.wikipedia.org/wiki/Parametric_surface
    locator: Surface area
---

## Definition

Let $D \subseteq \mathbb{R}^2$ be a bounded [[Jordan measurable|0565DC8RMK]] closed set, and let $r \colon D \to \mathbb{R}^3$ be a [[parametrised surface|05788J51WZ]] that is injective on the interior of $D$.
The surface area of $S = r(D)$ is
$$\operatorname{area}(S) = \iint_D \| r_u(u, v) \times r_v(u, v) \| \, du \, dv.$$

## Remarks

The integrand is the length of the [[normal vector|0BJDQ2HBH1]]. It is the area of the parallelogram spanned by $r_u$ and $r_v$, which approximates the image of a small parameter rectangle.
For the graph $z = f(x, y)$ over $D$, the area is $\iint_D \sqrt{1 + f_x^2 + f_y^2} \, dx \, dy$.
For the unit sphere, $\| r_\varphi \times r_\theta \| = \sin \varphi$, and the area is $\int_0^{2\pi} \int_0^\pi \sin \varphi \, d\varphi \, d\theta = 4\pi$.
The value does not depend on the parametrisation. A piecewise smooth surface gets the sum of the areas of its pieces. Inscribed polyhedra do not give a good definition: the Schwarz lantern is a counterexample.

## Explanations

### L1 human

To find the area of a curved surface, cut the parameter domain into small rectangles.
Each one maps to a small curved patch, which is nearly a parallelogram, and you add up the areas of these parallelograms.