Semicircle
A semicircle is exactly half of a circle, formed by a diameter and the arc it subtends, with any inscribed angle on the arc being 90 degrees.
Formula
\text{Area} = \pi r^2/2; \text{Perimeter} = \pi r + 2r
Definition
A semicircle is exactly half of a circle, formed by cutting along a diameter, with the flat side being the diameter and the curved side a half-arc. Area $=\pi r^2/2$, perimeter $=\pi r+2r=r(\pi+2)$, and by Thales' Theorem, any angle inscribed in a semicircle (vertex on the arc, endpoints on the diameter) is always $90^\circ$, a special case of the inscribed angle theorem (central angle $180^\circ$, inscribed angle $90^\circ$). The semicircle also serves as a fundamental domain in the upper half-plane model of hyperbolic geometry, where geodesics are semicircles perpendicular to the real-axis boundary.
Example
The letter D and a protractor are shaped like semicircles, as is the top of a Roman arch; a semicircle of radius $4$ has area $16\pi/2=8\pi\approx25.1$. For radius $5$: area $=25\pi/2\approx39.27$, perimeter $=5(\pi+2)\approx25.71$, and for any point $C$ on the arc with $AB$ the diameter, angle $ACB=90^\circ$. In the upper half-plane model of hyperbolic geometry, geodesics are semicircles centered on the real axis (or vertical lines as a degenerate case), and hyperbolic distances along them are computed using cross-ratios.
Key Insight
Thales' theorem, that any point on the curved part of a semicircle connected to both ends of the diameter always forms a right angle, is a beautiful and surprising fact used practically: drawing two diameters of an unknown circle locates its center where they intersect, and placing a vertex on a semicircle's arc constructs a right angle in compass-and-straightedge work. The semicircle's dual role, as a Euclidean figure via Thales' theorem and as a hyperbolic geodesic in the Poincare half-plane model, shows how the same shape can carry entirely different meaning depending on context, satisfying all of Euclid's axioms except the parallel postulate in the latter case.