Semicírca

Semicírca is a geometric figure formed when a circle is divided by a diameter into two equal parts. The boundary of a semicircle consists of the curved arc, which represents half of the circle’s circumference, and the straight line segment of the diameter connecting the arc’s endpoints. The region bounded by these elements is often called a semicircular region or a half-disk.

Key properties and measures: if the circle has radius r, then the diameter is d = 2r. The area of the semicircle is A = (1/2)πr^2. The length of the curved arc is πr, while the diameter itself has length 2r. Therefore, the perimeter of the closed semicircular region (arc plus diameter) is P = πr + 2r. The full circle’s circumference is 2πr, which is twice the arc length of the semicircle.

Coordinate and construction notes: in a Cartesian plane, a semicircle with diameter on the x-axis from (-r, 0) to (r, 0) consists of the points on the circle x^2 + y^2 = r^2 with y ≥ 0 (upper semicircle) or y ≤ 0 (lower semicircle). A semicircle can be viewed as a circular sector of 180 degrees, including the diameter as its straight boundary.

Applications and variations: semicircles appear in geometry problems, design, architecture, and engineering, often as building blocks for areas, probabilities along an arc, or as components of arches and domes. In some contexts, an open semicircle refers only to the arc, excluding the diameter. See also semicircular arc, half-disk, and circle.