How to Calculate the Surface Area of a Dome
A dome is a shape with a flat circular bottom, rounded sides, and a rounded top. Apart from those requirements, domes can vary in steepness and curvature. Three basic varieties of dome shapes that allow you to calculate the surface area via simple formulas are spherical caps, ellipsoids, and paraboloids. To calculate the surface areas of these shapes, you only need to know the radius of the base and the height. A hemispherical dome belongs to both the spherical cap and ellipsoid categories, since it is a special case of both general shapes. Here is a list of geometric formulas for finding the surface area of a dome according to each category.
Surface Area of a Spherical Cap Dome
Let R be the radius of the sphere from which the dome is cut and H be the dome's height. If the dome is in the shape of a spherical cap, then its curved surface area is given by the formula
Surface Area = 2πRH
This formula holds regardless of whether the height is greater than, equal to, or less than the radius! In the case where R = H, the dome is a hemisphere and the surface area is 2πR^2.
If you don't know the radius of the greater sphere from which the dome is cut, but you do know the radius of the dome's base B, then the surface area formula is
Surface Area = π(B^2 + H^2)
The figures below show various spherical cap domes.
Surface Area of an Ellipsoid Dome
An ellipsoid dome is the surface of revolution obtained by revolving half an ellipse about its axis. A general ellipsoid has three axes not necessarily equal to one another. When exactly two are equal, the ellipsoid is also known as a spheroid. (And when all three axes are equal it is called a sphere.) Such domes may either be prolate spheroids (height greater than radius) or oblate spheroids (height less than radius). When the height equals the radius, the dome is half a sphere and the profile is a semicircle.
The surface area of a prolate spheroid dome is given by the formula
= πR^2 + π*sqrt[ R^2 * H^4 / (H^2 - R^2) ]*arcsin( sqrt[ 1 - (R/H)^2 ] )
where the inverse sine, arcsin, takes its argument in radians, not degrees. The surface area of an oblate spheroid dome is given by the formula
= πR^2 + π*sqrt[ R^2 * H^4 / (R^2 - H^2) ]*artanh( sqrt[ 1 - (H/R)^2 ] )
where artanh is the inverse hyperbolic tangent. The oblate formula can be written in an equivalent form involving natural logarithms (Ln):
= πR^2 + π*sqrt[ R^2 * H^4 / (R^2 - H^2) ]*Ln( R/H + sqrt[ (R/H)^2 - 1] )
And in the special case where R = H, the surface area is 2πR^2. Some ellipsoid domes are shown the diagram below.
Surface Area of a Paraboloid Dome
The profile of a paraboloid dome is a parabola, as the name suggests. If the radius of the base is R and the height is H, then the surface area is given by the formula
= (π/6)(R/H^2)[ (R^2 + 4H^2)^(3/2) - R^3]
This formula works regardless of the relative sizes of R and H. A paraboloid dome is shown below.
Which Type of Dome Is It?
If you aren't the one constructing the dome or don't have access do its exact specifications, you can't always know precisely what type of dome shape you're looking at. For example, a short parabolic dome looks very similar to a short spherical cap dome. An ellipsoid dome always has a vertical tangent plane at its base, whereas a paraboloid dome always has a slanted tangent plane at its base. If you're not sure, you can compute the surface area using two different formulas and average the results. Polyhedral domes can be approximated by one of the three types discussed.
Suppose a paraboloid dome has a radius of 10 feet and a height of 9 feet. Since R = 10 and H = 9 in this case, the surface area is
(π/6)(10/81)[ (100 + 324)^(3/2) - 1000 ]
= (10π/486)*[ 8730.69436 - 1000 ]
= 499.726 square feet.
Now suppose a spherical cap dome has a base radius of 10 feet and a height of 4 feet. With B = 10 and H = 4, its curved surface area is
π*(10^2 + 4^2)
= 364.425 square feet.
Finally, suppose an ellipsoid dome has a radius of 10 feet and height of 9 feet. Since the height is less than the radius, it is half of an oblate spheroid. Using the surface area formula for oblate ellipsoid domes, we get
100π + π*sqrt[ 100*6561/(100-81) ]*Ln( 10/9 + sqrt[ (10/9)^2 - 1] )
= 586.875 square feet.
Polygonal Dome or Polyhedral Dome Surface Area
When a dome is comprised of polygonal faces rather than being a curved surface, the area of the dome is the sum of the areas of the polygons. Complexes of triangles, or of hexagons and pentagons are the most common type of polygonal dome. Polyhedra made of pentagons and hexagons are called fullerenes, named after the cage-like allotropes of carbon molecules. Here are some examples spheres approximated by polyhedra. A polyhedral dome can be made by slicing these along any plane.