thumb|200px|the spherical cap is the purple section. in geometry, a spherical cap or spherical dome is a portion of a sphere cut off by a plane. if the plane passes through the center of the sphere, so that the height of the cap is equal to the radius of the sphere, the spherical cap is called a hemisphere. ==volume and surface area== if the radius of the base of the cap is a, and the height of the cap is h, then the volume of the spherical cap is :v = \frac{\pi h}{6} (3a^2 + h^2), and the curved surface area of the spherical cap is :a = 2 \pi r h. the relationship between h and r is irrelevant as long as 0 â¤ h â¤ 2r. the blue section of the illustration is also a spherical cap. the parameters a, h and r are not independent: :r^2 = (r-h)^2 + a^2 = r^2 +h^2 -2rh +a^2, :r = \frac {a^2 + h^2}{2h}. substituting this into the area formula gives: :a = 2 \pi \frac{(a^2 + h^2)}{2h} h = \pi (a^2 + h^2). note also that in the upper hemisphere of the diagram, \scriptstyle h = r - \sqrt{r^2 - a^2}, and in the lower hemisphere \scriptstyle h = r + \sqrt{r^2 - a^2}; hence in either hemisphere \scriptstyle a = \sqrt{h(2r-h)} and so an alternative expression for the volume is :v = \frac {\pi h^2}{3} (3r-h). == application == the volume of all points which are in at least one of two intersecting spheres of radii  and  is  : v = v^{(1)}-v^{(2)}, where :v^{(1)} = \frac{4\pi}{3}r_1^3 +\frac{4\pi}{3}r_2^3 is the total of the two isolated spheres, and :v^{(2)} = \frac{\pi h_1^2}{3}(3r_1-h_1)+\frac{\pi h_2^2}{3}(3r_2-h_2) the sum of the two spherical caps of the intersection. if  is the distance between the two sphere centers, elimination of the variables  and  leads to  :v^{(2)} = \frac{\pi}{12d}(r_1+r_2-d)^2[d^2+2d(r_1+r_2)-3(r_1-r_2)^2]. ==generalizations== ===sections of other solids=== the spheroidal dome is obtained by sectioning off a portion of a spheroid so that the resulting dome is circularly symmetric (having an axis of rotation), and likewise the ellipsoidal dome is derived from the ellipsoid. ===hyperspherical cap=== generally, the n-dimensional volume of a hyperspherical cap of height h and radius r in n-dimensional euclidean space is given by li, s. (2011). "concise formulas for the area and volume of a hyperspherical cap". asian j. math. sta
Spheres