semicircle

A semicircle is the figure formed by dividing a circle along a diameter. It consists of the curved boundary, the semicircular arc, and, if the interior is included, the half-disk bounded by the arc and the diameter. The radius r is half the diameter, and the circle’s center lies on the dividing line.

Formulas and definitions. The area of a semicircular region is A = (1/2)πr^2. The length of the curved boundary (the arc) is πr. The full boundary length, including the straight diameter, is πr + 2r. In coordinate form, the upper semicircle is the set of points (x,y) with x^2 + y^2 ≤ r^2 and y ≥ 0.

Centroid and symmetry. The semicircle is symmetric about the diameter. The centroid of the semicircular area lies on the symmetry axis at a distance 4r/(3π) from the diameter, i.e., toward the curved edge. The center of the parent circle lies on the diameter, and the centroid is offset from it by this same distance along the axis.

Other properties. The semicircle represents half of the circle, so its arc length is half the circumference and equals πr; its full boundary length, including the diameter, equals πr + 2r. Thales’ theorem states that an angle subtended by the diameter in a semicircle is a right angle.

Applications and variants. Semicircles appear in geometry problems, architectural arches, and various design contexts as a simple, symmetric shape. Variants include semicircular segments cut from a circle or semicircular arches integrated into structures.