Surface area

The surface area (symbol A) of a solid object is a way to measure how much space the outside of that object covers. Imagine wrapping something in paper; the amount of paper needed would be close to the object's surface area. For simple shapes with flat sides, like a box, the surface area is just the total area of all the flat parts added together.

For round or curved shapes, such as a sphere, finding the surface area is more complicated. It uses special math called infinitesimal calculus, which helps break the curved surface into very small, flat pieces that are easier to measure.

A long time ago, smart people like Henri Lebesgue and Hermann Minkowski worked on a general way to find the surface area of any shape, even very strange or irregular ones. Their ideas helped create a field of math called geometric measure theory, which looks at how to measure surfaces in many different situations.

Definition

Finding the exact size of a surface can be tricky, especially when the surface is curved. For simple flat shapes, we can easily measure their area, but for curved surfaces like a sphere, we need more advanced math.

One key idea is that the total area of a surface should be the sum of the areas of its parts. This works well for smooth surfaces made of pieces that can be described using special math formulas. However, for very rough or jagged surfaces, it might not be possible to define a clear area at all.

Common formulas

See also: List of formulas in elementary geometry

Ratio of surface areas of a sphere and cylinder of the same radius and height

Here are some simple formulas to help us understand the surface area of common shapes. For example, we can compare the surface area of a sphere and a cylinder that have the same size.

The surface area of a sphere is four times pi times the radius squared. For a cylinder, it’s two times pi times the radius times the height plus the radius. When the height of the cylinder equals the diameter of the sphere, their surface areas are in a ratio of 2 to 3. This interesting discovery is thanks to Archimedes.

Surface areas of common solids

Shape	Formula/Equation	Variables
Cube	6 a 2 {\displaystyle 6a^{2}}	a = side length
Cuboid	2 ( l b + l h + b h ) {\displaystyle 2\left(lb+lh+bh\right)}	l = length b = breadth h = height
Triangular prism	b h + l ( p + q + r ) {\displaystyle bh+l\left(p+q+r\right)}	b = base length of triangle, h = height of triangle, l = distance between triangular bases, p, q, r = sides of triangle
All prisms	2 B + P h {\displaystyle 2B+Ph}	B = the area of one base P = the perimeter of one base h = height
Sphere	4 π r 2 = π d 2 {\displaystyle 4\pi r^{2}=\pi d^{2}}	r = radius of sphere d = diameter
Hemisphere	3 π r 2 {\displaystyle 3\pi r^{2}}	r = radius of the hemisphere
Hemispherical shell	π ( 3 R 2 + r 2 ) {\displaystyle \pi \left(3R^{2}+r^{2}\right)}	R = external radius of hemisphere r = internal radius of hemisphere
Spherical lune	2 r 2 θ {\displaystyle 2r^{2}\theta }	r = radius of sphere θ = dihedral angle
Torus	( 2 π r ) ( 2 π R ) = 4 π 2 R r {\displaystyle \left(2\pi r\right)\left(2\pi R\right)=4\pi ^{2}Rr}	r = minor radius (radius of the tube) R = major radius (distance from center of tube to center of torus)
Closed cylinder	2 π r 2 + 2 π r h = 2 π r ( r + h ) {\displaystyle 2\pi r^{2}+2\pi rh=2\pi r\left(r+h\right)}	r = radius of the circular base h = height of the cylinder
Cylindrical annulus	2 π R h + 2 π r h + 2 ( π R 2 − π r 2 ) = 2 π ( R + r ) ( R − r + h ) {\displaystyle {\begin{aligned}&2\pi Rh+2\pi rh+2\left(\pi R^{2}-\pi r^{2}\right)\\&=2\pi \left(R+r\right)\left(R-r+h\right)\end{aligned}}}	R = External radius r = Internal radius h = height
Capsule	2 π r ( 2 r + h ) {\displaystyle 2\pi r(2r+h)}	r = radius of the hemispheres and cylinder h = height of the cylinder
Curved surface area of a cone	π r r 2 + h 2 = π r s {\displaystyle \pi r{\sqrt {r^{2}+h^{2}}}=\pi rs}	s = r 2 + h 2 {\displaystyle s={\sqrt {r^{2}+h^{2}}}} s = slant height of the cone r = radius of the circular base h = height of the cone
Full surface area of a cone	π r ( r + r 2 + h 2 ) = π r ( r + s ) {\displaystyle \pi r\left(r+{\sqrt {r^{2}+h^{2}}}\right)=\pi r\left(r+s\right)}	s = slant height of the cone r = radius of the circular base h = height of the cone
Regular Pyramid	B + P s 2 {\displaystyle B+{\frac {Ps}{2}}}	B = area of base P = perimeter of base s = slant height
Square pyramid	b 2 + 2 b s = b 2 + 2 b ( b 2 ) 2 + h 2 {\displaystyle b^{2}+2bs=b^{2}+2b{\sqrt {\left({\frac {b}{2}}\right)^{2}+h^{2}}}}	b = base length s = slant height h = vertical height
Rectangular pyramid	l b + l ( b 2 ) 2 + h 2 + b ( l 2 ) 2 + h 2 {\displaystyle lb+l{\sqrt {\left({\frac {b}{2}}\right)^{2}+h^{2}}}+b{\sqrt {\left({\frac {l}{2}}\right)^{2}+h^{2}}}}	l = length b = breadth h = height
Tetrahedron	3 a 2 {\displaystyle {\sqrt {3}}a^{2}}	a = side length
Surface of revolution	2 π ∫ a b f ( x ) 1 + ( f ′ ( x ) ) 2 d x {\displaystyle 2\pi \int _{a}^{b}{f(x){\sqrt {1+(f'(x))^{2}}}dx}}
Parametric surface	∬ D | r → u × r → v | d A {\displaystyle \iint _{D}\left\vert {\vec {r}}_{u}\times {\vec {r}}_{v}\right\vert dA}	r → {\displaystyle {\vec {r}}} = parametric vector equation of surface r → u {\displaystyle {\vec {r}}_{u}} = partial derivative of r → {\displaystyle {\vec {r}}} with respect to u {\displaystyle u} r → v {\displaystyle {\vec {r}}_{v}} = partial derivative of r → {\displaystyle {\vec {r}}} with respect to v {\displaystyle v} D {\displaystyle D} = shadow region

In chemistry

Surface area plays a big role in how fast chemical reactions happen. When you make the surface area bigger, the reaction usually goes faster. For example, iron that is ground into a fine powder can catch fire easily, but the same iron in big solid pieces stays safe and can be used in buildings. Sometimes we want a lot of surface area, and other times we want just a little.

See also: Accessible surface area

In biology

The surface area of living things is very important for how they stay healthy and live. For example, animals use their teeth to break food into smaller pieces, which helps their bodies absorb nutrients better. The lining of the digestive system has tiny finger-like parts called microvilli that make a lot more space for absorbing nutrients. Elephants have big ears that help them stay cool.

Cells also need the right balance of surface area to volume. As cells get bigger, their volume grows faster than their surface area. This makes it harder for materials to move in and out of the cell. For example, a tiny cell with a radius of 1 μm has a surface area to volume ratio of 3, but a larger cell with a radius of 10 μm has a ratio of 0.3, meaning materials move much more slowly.