Editing Surface area (section)

== Common formulas ==
{{anchor|List of surface area formulas}}
{{See also|List of formulas in elementary geometry}}

{| class="wikitable"
|+ Surface areas of common solids
|-
!Shape
!Formula/Equation
!Variables
|-
|[[Cube]]
|<math> 6a^2 </math>
|''a'' = side length
|-
|[[Cuboid]]
|<math> 2\left(lb+lh+bh\right) </math>
|''l'' = length, ''b'' = breadth, ''h'' = height
|-
|[[Triangular prism]]
|<math> bh+l\left(p+q+r\right) </math>
|''b'' = base length of triangle, ''h'' = height of triangle, ''l'' = distance between triangular bases, ''p'', ''q'', ''r'' = sides of triangle
|-
|All [[Prism (geometry)|prisms]]
|<math> 2B+Ph </math>
|''B'' = the area of one base, ''P'' = the perimeter of one base, ''h'' = height
|-
|[[Sphere]]
|<math> 4\pi r^2=\pi d^2 </math>
|''r'' = radius of sphere, ''d'' = diameter
|-
|Hemisphere
|<math> 3\pi r^2 </math> 
|''r'' = radius of the hemisphere
|-
|Hemispherical shell
|<math> \pi \left(3R^2+r^2\right) </math>
|''R'' = external radius of hemisphere, ''r'' = internal radius of hemisphere
|-
|[[Spherical lune]]
|<math> 2r^2\theta </math>
|''r'' = radius of sphere, ''θ'' = [[dihedral angle]]
|-
|[[Torus]]
|<math> \left(2\pi r\right)\left(2\pi R\right)=4\pi^2Rr</math>
|''r'' = minor radius (radius of the tube), ''R'' = major radius (distance from center of tube to center of torus)
|-
|Closed [[Cylinder (geometry)|cylinder]]
|<math> 2\pi r^2+2\pi rh=2\pi r\left(r+h\right) </math>
|''r'' = radius of the circular base, ''h'' = height of the cylinder
|-
|Cylindrical [[Annulus (mathematics)|annulus]]
|<math> 2\pi Rh+2\pi rh+2(\pi R^2-\pi r^2) =2\pi (R+r)(R-r+h) </math>
|''R'' = External radius
''r'' = Internal radius, ''h'' = height
|-
|[[Capsule (geometry)|Capsule]]
|<math> 2\pi r(2r+h)  </math>
|''r'' = radius of the hemispheres and cylinder, ''h'' = height of the cylinder
|-
|Curved surface area of a [[cone (geometry)|cone]]
|<math> \pi r\sqrt{r^2+h^2}=\pi rs </math>
|<math> s=\sqrt{r^2+h^2} </math><br/>
''s'' = slant height of the cone, ''r'' = radius of the circular base, ''h'' = height of the cone
|-
|Full surface area of a cone
|<math> \pi r\left(r+\sqrt{r^2+h^2}\right)=\pi r\left(r +s\right) </math>
| ''s'' = slant height of the cone, ''r'' = radius of the circular base, ''h'' = height of the cone
|-
|Regular [[Pyramid (geometry)|Pyramid]]
|<math>B+\frac{Ps}{2}</math>
|''B'' = area of base, ''P'' = perimeter of base, ''s'' = slant height
|-
|[[Square pyramid]]
|<math> b^2 + 2bs = b^2+ 2b\sqrt{\left(\frac{b}{2}\right)^2+h^2} </math>
|''b'' = base length, ''s'' = slant height, ''h'' = vertical height
|-
|Rectangular pyramid
|<math> lb+l\sqrt{\left(\frac{b}{2}\right)^2+h^2}+ b\sqrt{\left(\frac{l}{2}\right)^2+h^2} </math>
|''l'' = length, ''b'' = breadth, ''h'' = height
|-
|[[Tetrahedron]]
|<math> \sqrt{3}a^2 </math>
|''a'' = side length
|-
|[[Surface of revolution]]
|<math>2\pi \int_a^b {f(x) \sqrt{1+(f'(x))^2} dx}</math>
|
|-
|[[Parametric surface]]
|<math>\iint_D \left \vert \vec{r}_u \times \vec{r}_v \right \vert dA</math>
|<math>\vec{r}</math> = parametric vector equation of surface,

<math>\vec{r}_u</math> = partial derivative of <math>\vec{r}</math> with respect to <math>u</math>,<br/>
<math>\vec{r}_v</math> = partial derivative of <math>\vec{r}</math> with respect to <math>v</math>,<br/>
<math>D</math> = shadow region
|}

===Ratio of surface areas of a sphere and cylinder of the same radius and height===
[[Image:Inscribed cone sphere cylinder.svg|thumb|300px|A cone, sphere and cylinder of radius ''r'' and height ''h''.]]
The below given formulas can be used to show that the surface area of a [[sphere]] and [[cylinder (geometry)|cylinder]] of the same radius and height are in the ratio '''2&nbsp;:&nbsp;3''', as follows.

Let the radius be ''r'' and the height be ''h'' (which is 2''r'' for the sphere).

<math display="block">\begin{array}{rlll}
\text{Sphere surface area}   & = 4 \pi r^2      &                    & = (2 \pi r^2) \times 2 \\
\text{Cylinder surface area} & = 2 \pi r (h + r) & = 2 \pi r (2r + r) & = (2 \pi r^2) \times 3
\end{array}</math>

The discovery of this ratio is credited to [[Archimedes]].<ref>{{cite web|first = Chris|last = Rorres|url = http://www.math.nyu.edu/~crorres/Archimedes/Tomb/Cicero.html|title = Tomb of Archimedes: Sources|publisher = Courant Institute of Mathematical Sciences|access-date = 2007-01-02|url-status = live|archive-url = https://web.archive.org/web/20061209201723/http://www.math.nyu.edu/~crorres/Archimedes/Tomb/Cicero.html|archive-date = 2006-12-09}}</ref>
{{Clear}}