specific series of steps that allowed one to construct a

square of the same area as this lune using a straight-

edge 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 P

YTHAGORAS

’

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. (Conse-

quently, in our diagram, areas A1+ A2and A3+ A4

sum to A2+ A4+ A5, yielding A1+ A3= A5.)

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 θdegrees, then the surface area of this

slice of the sphere is 4πr2· units squared, where r

is the radius of the sphere.

θ

––

360

lune 323