Semicircle
GeometryA 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. It is formed by cutting a circle in half along a diameter. The flat side is the diameter and the curved side is a half-arc.
Example
The letter D has a semicircle shape. A protractor is shaped like a semicircle. The top of a Roman arch is a semicircle. If a circle has radius $4$, the semicircle area $= 16\pi/2 = 8\pi$ approximately $25.1$.
Key Insight
Thales' theorem: if you put any point on the curved part of a semicircle and connect it to both ends of the diameter, you always get a right angle at that point. This always-$90^\circ$ property is a beautiful and surprising fact about semicircles.
Definition
A semicircle is half of a circle: the region bounded by a diameter and its subtended arc. Area $= \pi r^2/2$. Perimeter $= \pi r$ (curved arc) $+ 2r$ (diameter) $= r(\pi + 2)$. Thales' Theorem: any angle inscribed in a semicircle (vertex on the arc, endpoints on the diameter) equals $90^\circ$.
Example
Semicircle with radius $5$: area $= 25\pi/2$ approximately $39.27$. Perimeter $= 5(\pi+2)$ approximately $25.71$. If $C$ is any point on the arc and $AB$ is the diameter, angle $ACB = 90^\circ$ by Thales' theorem.
Key Insight
Thales' theorem provides a practical use: to find the center of a circle, draw any two diameters - they intersect at the center. Or, to construct a right angle, find a semicircle and place the vertex on its arc. These constructions work in compass-and-straightedge problems.
Definition
The semicircle is the locus of points from which a fixed segment (the diameter) subtends a right angle - this is the content of Thales' theorem, a special case of the inscribed angle theorem (central angle $= 180^\circ$, inscribed angle $= 90^\circ$). The semicircle is also a fundamental domain for the hyperbolic plane in the upper half-plane model $H = \{z \in \mathbb{C} : \text{Im}(z) > 0\}$, where the boundary is the real line and geodesics are semicircles perpendicular to it.
Example
In the upper half-plane model of hyperbolic geometry, geodesics are semicircles with center on the real axis (or vertical lines, degenerate semicircles with infinite radius). The hyperbolic distance between two points can be computed using cross-ratios along these semicircular geodesics.
Key Insight
The semicircle's dual role - as a geometric figure in Euclidean geometry (Thales' theorem) and as a geodesic in the Poincare half-plane model of hyperbolic geometry - illustrates how the same shape carries completely different meaning depending on context. In hyperbolic geometry, these semicircular "lines" satisfy all of Euclid's axioms except the parallel postulate.