specific series of steps that allowed one to construct a
square of the same area as this lune using a straightedge and compass alone.) Mathematicians have since
proved that there are only four other types of lunes
that can also be so squared.
In the figure on the preceding page, the areas of the
two shaded lunes sum to the area of the triangle. Here,
each circular arc is a semicircle drawn on the side of a
right triangle. This claim follows from the generalized
version of PYTHAGORAS’S THEOREM, which states that
the areas of two semicircles constructed on the two
shorter sides of a right triangle sum to the area of the
semicircle constructed on the hypotenuse. (Consequently, in our diagram, areas A 1 + A 2 and A 3 + A 4
sum to A 2 + A 4 + A 5 , yielding A 1 + A 3 = A 5 .)
In the study of spherical geometry, the part of the
surface of a sphere bounded by two great circles is also
called a lune. If the angle between the two great circles
is θ degrees, then the surface area of this
slice of the sphere is 4πr
2 ·
units squared, where r
is the radius of the sphere.
θ
––
360
lune 323