halfcircle

A halfcircle, commonly written as semicircle, is the shape formed when a circle is divided into two equal parts by a diameter. The boundary consists of the curved arc on one side of the diameter and, in many definitions, the straight diameter itself. The center of the circle lies at the midpoint of that dividing line, and the arc subtends 180 degrees.

In Euclidean geometry, a diameter bisects the circle into two congruent semicircles. Thales' theorem states that any angle subtended by a diameter is a right angle; thus, a triangle inscribed in a semicircle is a right triangle.

Key measurements: for a semicircle with radius r, the area is (1/2)πr^2, and the length of the curved arc is πr. If the diameter is included in the boundary, the perimeter is πr + 2r. The diameter is d = 2r, and the arc spans 180 degrees.

Applications and appearances: semicircular shapes occur in architecture (semicircular arches and vaults) and in design, logos, and graphic motifs. They also appear in various engineering contexts where a half-disk boundary is needed.