Surface Area
Surface area is the total area of all the outer faces of a three-dimensional solid, measured in square units.
Formula
SA = \text{sum of all face areas}
Definition
Surface area is the total amount of area covering the outside of a 3-D shape, like measuring all the wrapping paper needed to cover a box with no gaps or overlaps; it is the sum of the areas of all its faces (for polyhedra) or the total outer curved and flat area (for curved solids like cylinders and spheres), always expressed in square units even though the shape is 3-D. Unfolding a solid into a flat net reveals all faces simultaneously. For a smooth surface parameterized by $r(u, v)$, the surface area is $$A = \iint |r_u \times r_v| \, du \, dv,$$ and for a surface given by $z = f(x, y)$, this becomes $A = \iint \sqrt{1 + f_x^2 + f_y^2} \, dA$.
Example
A cube with side length $4$ cm has $6$ square faces, each of area $4 \times 4 = 16$ cm$^2$, giving total surface area $6 \times 16 = 96$ cm$^2$. A rectangular prism $5 \times 4 \times 3$ cm has $SA = 2(5)(4) + 2(5)(3) + 2(4)(3) = 40 + 30 + 24 = 94$ cm$^2$. The surface area of a sphere of radius $r$ can be found by parameterizing as $(r\sin\varphi\cos\theta, r\sin\varphi\sin\theta, r\cos\varphi)$, giving area element $|r_\varphi \times r_\theta| = r^2\sin\varphi$, so $$A = \int_0^\pi\int_0^{2\pi} r^2\sin\varphi \, d\theta \, d\varphi = 4\pi r^2.$$
Key Insight
For cells and organisms, surface area relative to volume determines how efficiently nutrients and oxygen can be exchanged; smaller cells have a higher surface-area-to-volume ratio, which is why cells stay small and divide when they grow. The isoperimetric inequality in 3-D states that among all surfaces enclosing a fixed volume $V$, the sphere has the minimum surface area, $A^3 \ge 36\pi V^2$, explaining why bubbles and droplets are spherical in the absence of external forces.