Sphere-Cut Rectangle Calculator

Calculations for what is referred to here as a sphere-cut rectangle. This shape is formed when a rectangle with side lengths a and b is passed through a larger sphere of radius r such that the rectangle's center moves toward the center of the sphere. Much like a rectangular cookie cutter placed centrally on the round surface. The sphere-cut rectangle is one of the two identical sections cut out from the sphere's surface. The rectangle side a generates the sphere-cut side m, the rectangle side b generates the sphere-cut side n. Such a sphere-cut rectangle has corners of less than 90 degrees, even though this is not apparent in the top view due to perspective distortion. The corners of the more common spherical rectangle have more than 90 degrees.
Please enter the radius of the sphere and the side lengths of the cutting rectangle. The other values ​​will be calculated.

	Sphere radius (r):
	Rectangle side a:
	Rectangle side b:
	Radius small circle rm:
	Radius small circle rn:
	Sphere cut side m:
	Sphere cut side n:
	Arc diagonal (d):
	Vertex angle (α):
	Perimeter (p):
	Area (A):
	Round to   0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15   decimal places.

Radius, lengths, diagonal and perimeter have a one-dimensional unit (e.g. meter), the area has this unit squared (e.g. square meter). A sphere-cut rectangle is two-dimensional but extends into three dimensions. It is part of spherical geometry.

The sides of the sphere-cut rectangle do not lie on great circles but on small circles with a radius smaller than r. Together with the corresponding sides of the rectangle, they form a circular segment with radius r_m and r_n. The radii of the small circles are derived using the Pythagorean theorem. The center of a small circle is located at a distance from the sphere's center equal to half the length of the other side of the rectangle. The length of a side of the sphere-cut rectangle corresponds to the length of the circular arc defined by the corresponding chord of the small circle. The two diagonals of the sphere-cut rectangle are of equal length and lie along great circles of the sphere.
The formula for the area of ​​the sphere-cut rectangle is derived from the area element of the spherical surface and accounts for the curvature of the sphere. For small rectangles, the area approaches the area a*b of a planar rectangle.
These formulas, derived with the aid of AI, were verified using several methods. To this end, the geometric relationships were calculated independently using coordinates and the properties of the sphere and its circles of intersection. Additionally, the area was determined numerically using two different surface integrals. The results were validated against various values ​​for the radius and side lengths, as well as through the analysis of limiting cases.

Derivation of the area

The spherical surface above the xy-plane can be represented as $$z^2=r^2-x^2-y^2$$ and thus, for the upper hemisphere, as $$z=\sqrt{r^2-x^2-y^2}$$ The limits applicable to the sphere-cut rectangle are $$-\frac{a}{2}\le x\le \frac{a}{2},-\frac{b}{2}\le y\le \frac{b}{2}$$ The area element of a surface z = f(x,y) is given by $$dA=\sqrt{1+{\frac{\partial z}{\partial x}}^2+{\frac{\partial z}{\partial y}}^2}dxdy$$ which, for the spherical surface, yields $$dA=\frac{r}{\sqrt{r^2-x^2-y^2}}dxdy$$ Due to symmetry, it suffices to perform the calculation over one-quarter of the area. Therefore, $$A=4r{\int }_0^{\frac{a}{2}}{\int }_0^{\frac{b}{2}}\frac{dydx}{\sqrt{r^2-x^2-y^2}}$$ The inner integral can be evaluated directly, yielding $$A=4r{\int }_0^{\frac{a}{2}}arcsin\left(\frac{b}{2\sqrt{r^2-x^2}}\right)dx$$ The remaining integral is evaluated using integration by parts. With $$u=arcsin\left(\frac{b}{2\sqrt{r^2-x^2}}\right),dv=dx$$ simplification and the substitution of the limits yield the aforementioned closed-form formula for the area of ​​the sphere-cut rectangle.

Calculation of area and other dimensions: Sphere-Cut Rectangle

Last updated on 08/12/2026. Author: Jürgen Kummer

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