The surface area of a shape is the total area of all of the shape's faces . It can be considered the total extent of all of the shape's 2-subfacets in 2-dimensional space. The surface area of a polygon is typically just called its area .
A surface area has dimensions of [length ]2 .
Surface Area Formulae [ ] 2-Dimensional [ ]
Shape
Area Formula
Variables
Square
a
b
{\displaystyle ab}
a, b = edge lengths
Disk
π
4
d
2
{\displaystyle \frac{\pi}{4} d^2}
d = diameter of disc
Regular P-gon
p
4
cot
(
π
p
)
a
2
{\displaystyle \frac{p}{4} \cot{\left( \frac{\pi}{p} \right)} a^2}
p = number of edges, a = edge length
Regular P/Q-gon
p
4
cot
(
π
q
p
)
a
2
{\displaystyle \frac{p}{4} \cot{\left( \frac{\pi q}{p} \right)} a^2}
p = number of edges, q = winding number, a = edge length
Triangle
3
4
a
2
{\displaystyle \frac{\sqrt{3}}{4} a^2}
a = edge length
Pentagon
1
4
5
(
5
+
2
5
)
a
2
{\displaystyle \frac{1}{4} \sqrt{5 \left( 5 + 2 \sqrt{5} \right)} a^2}
a = edge length
Hexagon
3
3
2
a
2
{\displaystyle \frac{3 \sqrt{3}}{2} a^2}
a = edge length
Octagon
2
(
1
+
2
)
a
2
{\displaystyle 2\left(1 + \sqrt{2}\right) a^2}
a = edge length
Decagon
5
2
5
+
2
5
a
2
{\displaystyle \frac{5}{2} \sqrt{ 5 + 2 \sqrt{5} } a^2}
a = edge length
Dodecagon
3
(
2
+
3
)
a
2
{\displaystyle 3\left(2 + \sqrt{3}\right) a^2}
a = edge length
Pentadecagon
15
4
7
+
2
5
+
2
3
(
5
+
2
5
)
a
2
{\displaystyle \frac{15}{4} \sqrt{7 + 2\sqrt{5} + 2 \sqrt{3 \left( 5 + 2 \sqrt{5} \right) } } a^2}
a = edge length
Hexadecagon
4
(
1
+
2
+
2
(
2
+
2
)
)
a
2
{\displaystyle 4 \left( 1 + \sqrt{2} + \sqrt{ 2 \left( 2 + \sqrt{2} \right) } \right) a^2}
a = edge length
Icosagon
5
(
1
+
5
+
5
+
2
5
)
a
2
{\displaystyle 5 \left( 1 + \sqrt{5} + \sqrt{5 + 2\sqrt{5}} \right) a^2}
a = edge length
Icositetragon
6
(
2
+
6
+
5
+
2
6
)
a
2
{\displaystyle 6 \left( 2 + \sqrt{6} + \sqrt{5 + 2\sqrt{6}} \right) a^2}
a = edge length
Triacontagon
15
2
23
+
10
5
+
2
3
(
85
+
38
5
)
a
2
{\displaystyle \frac{15}{2} \sqrt{23 + 10\sqrt{5} + 2 \sqrt{3 \left( 85 + 38\sqrt{5} \right)} } a^2}
a = edge length
Triacontadigon
8
2
+
2
+
2
+
2
2
−
2
+
2
+
2
a
2
{\displaystyle 8 \sqrt{\frac{ 2 + \sqrt{2 + \sqrt{2 + \sqrt{2}}} }{ 2 - \sqrt{2 + \sqrt{2 + \sqrt{2}}} }} a^2}
a = edge length
Tube
π
a
d
{\displaystyle \pi ad }
a = edge length, d = diameter of circle
Torus
1
16
π
2
d
1
d
2
{\displaystyle \frac{1}{16} \pi^2 d_1 d_2}
d1 , d2 = diameters of circles
Sphere
π
d
2
{\displaystyle \pi d^2}
d = diameter of sphere
3-Dimensional [ ]
Shape
Area Formula
Variables
Cube
2
(
a
b
+
a
c
+
b
c
)
{\displaystyle 2\left(ab + ac + bc\right) }
a, b, c = edge lengths
Cylinder
π
d
(
1
2
d
+
a
)
{\displaystyle \pi d \left( \frac{1}{2} d + a \right) }
a = edge length, d = diameter of disk
Ball
π
d
2
{\displaystyle \pi d^2}
d = diameter of ball
Tetrahedron
3
a
2
{\displaystyle \sqrt{3} a^2}
a = edge length
Octahedron
2
3
a
2
{\displaystyle 2\sqrt{3} a^2}
a = edge length
Dodecahedron
3
5
(
5
+
2
5
)
a
2
{\displaystyle 3 \sqrt{ 5 \left( 5 + 2 \sqrt{5} \right) } a^2}
a = edge length
Icosahedron
5
3
a
2
{\displaystyle 5\sqrt{3} a^2}
a = edge length
Regular P,Q-hedron
p
q
cot
(
π
p
)
4
−
(
p
−
2
)
(
q
−
2
)
a
2
{\displaystyle \frac{ pq \cot{\left( \frac{\pi}{p} \right)} } { 4-\left( p-2 \right) \left( q - 2 \right) } a^2}
a = edge length
Solid torus
1
16
π
2
d
1
d
2
{\displaystyle \frac{1}{16} \pi^2 d_1 d_2}
d1 , d2 = diameters of circle & disk
Surface Areas [ ]
Surface Area
Object/Unit
2.6×10-70 m2
Planck area
9.0×1012 m2
Australia
1.0×1013 m2
Europe
1.4×1013 m2
Antarctica
1.8×1013 m2
South America
2.5×1013 m2
North America
3.0×1013 m2
Africa
4.2×1013 m2
America
4.4×1013 m2
Asia
5.4×1013 m2
Eurasia
8.4×1013 m2
Afro-Eurasia
1.5×1014 m2
Earth's land area
5.1×1014 m2
Earth
6.1×1018 m2
the Sun
2.0×1032 m2
the Oort Cloud
7.0×1041 m2
Disk of the Milky Way
2.4×1054 m2
Observable universe
See Also [ ] 
