Subtended angle
In geometry, a subtended angle is the angle formed at a given point by the lines connecting that point to the endpoints of a line segment, arc, or other object, effectively measuring the angular width of the object from the observer's viewpoint.[1] This concept is fundamental in various contexts, such as optics and astronomy, where it quantifies how objects like the Sun or Moon appear in size from Earth— for instance, both the Sun and Moon subtend approximately 0.5 degrees, enabling total solar eclipses.[1] The subtended angle holds particular significance in circle geometry, where it distinguishes between central angles—formed at the circle's center by radii to the arc's endpoints—and inscribed angles (or peripheral angles)—formed at a point on the circumference by chords to the arc's endpoints.[2] A key theorem states that the measure of an inscribed angle is half the measure of the central angle subtended by the same arc, a relationship that underpins many proofs in Euclidean geometry.[2][3] This inscribed angle theorem implies that all inscribed angles subtended by the same arc are equal, regardless of their position on the remaining circumference.[3] Notable applications include Thales' theorem, which posits that an angle inscribed in a semicircle—subtended by the diameter—is always a right angle (90 degrees), as it is half of the central angle spanning 180 degrees.[4] These principles extend to more complex configurations, such as angles formed by intersecting chords or tangents, where theorems equate such angles to half the sum or difference of the subtended arcs, facilitating problem-solving in circle-based constructions and trigonometry.[5]Fundamentals
Definition
In geometry, the angle subtended by a line segment AB at a point P is the angle ∠APB formed by the two rays originating from P and passing through the endpoints A and B of the segment. This configuration arises whenever lines are drawn from an external or arbitrary point to the extremities of the segment, creating the visual angle at that point.[6] Unlike an inscribed angle, which is a specific type of subtended angle where the vertex lies on the circumference of a circle and the sides pass through two points on the circle, a subtended angle can occur at any point in the plane, not restricted to circular geometry. The measure of such an angle is typically expressed in degrees, where a full rotation is 360°, or in radians, where a full rotation is 2π, providing a dimensionless unit based on the ratio of arc length to radius in circular contexts. For illustration, consider a straight line segment AB viewed from point P not on the segment; the subtended angle is the spread ∠APB, which decreases as P moves farther away while keeping AB fixed.Notation
In geometry, the subtended angle is commonly denoted by the Greek letter θ (theta), which serves as a variable for the measure of the angle formed by lines from a vertex to the endpoints of a segment or arc.[7] Subscripts are frequently added for specificity, such as θ_{AB} to indicate the angle subtended by segment AB at a given point, or θ_{P} for the angle at point P.[8] Angles may also be named using three capital letters, with the middle letter as the vertex, such as ∠APB for the angle at P subtended by AB.[9] Specific terminology distinguishes types of subtended angles; for instance, a central angle refers to one subtended by an arc at the center of a circle, often denoted as ∠AOB where O is the center and A, B are points on the circumference.[10] In perceptual and optical contexts, the term visual angle describes the angle subtended by an object at the observer's eye, typically symbolized as θ_v or α.[11] The notation for angles traces its origins to ancient Greek mathematics, particularly in Euclid's Elements, where angles were denoted by three points labeling the rays and vertex, reflecting early conventions for geometric figures without symbolic variables like θ.[9] Notation varies by unit of measure: in degrees, θ is expressed as a numerical value out of 360 for a full circle, while in radians—the dimensionless SI unit—θ equals the ratio of arc length s to radius r (θ = s/r) for a central angle, emphasizing arc proportion over arbitrary division.[12][13]Geometric Applications
In Plane Geometry
In plane geometry, the subtended angle manifests prominently in the study of circles, where an arc or chord forms angles at the center or on the circumference. A central angle is formed by two radii extending from the circle's center to the endpoints of the arc, directly measuring the arc's extent. An inscribed angle, by contrast, has its vertex on the circle's circumference, with its sides as chords connecting to the arc's endpoints.[14][15] The inscribed angle theorem states that the measure of an inscribed angle is exactly half the measure of the central angle subtending the same arc. This relationship holds regardless of the inscribed angle's position on the remaining circumference, as long as it intercepts the identical arc. The theorem underscores the symmetry inherent in circular geometry and is fundamental for solving problems involving arc measures and angle relationships.[14][16] To outline the proof, consider a circle with center O and arc AB. Let \angle AOB = \theta be the central angle. For an inscribed angle \angle ACB subtending the same arc, draw radius OC. Triangles OAC and OBC are isosceles, as OA = OC = OB (all radii). The base angles in these isosceles triangles are equal, and through case analysis considering the position of the center relative to the inscribed angle, the measure \angle ACB = \frac{\theta}{2} is demonstrated. This construction shows the halving property.[14][17] For arcs, the central angle subtended by an arc of length s in a circle of radius r measures \theta = s / r radians, providing a direct proportionality between arc extent and angular measure. This relation facilitates applications in determining arc proportions and sector areas without invoking degrees.[18] As an illustrative example, consider a circle with center O and chord AB subtending central angle \angle AOB = 120^\circ (as depicted in a standard diagram labeling points A, B, and O). An inscribed angle \angle ACB at any point C on the major arc would measure $60^\circ, half the central angle, highlighting the theorem's practical use in angle prediction from chord positions.[19][14]In Solid Geometry
In solid geometry, the concept of a subtended angle extends from the two-dimensional plane angle to the three-dimensional solid angle, which quantifies the extent to which a surface or object occupies the field of view from a specific point in space. A solid angle, denoted Ω and measured in steradians (sr), is defined as the area of the portion of a unit sphere (radius 1) subtended by the projection of the surface onto that sphere from the given point.[20] The total solid angle surrounding a point in three-dimensional space is 4π steradians, analogous to the full 2π radians in a plane.[20] The solid angle relates to plane angles through integration over directions: it represents the integral of infinitesimal projected plane angles across the spherical coordinate system, where the differential solid angle is given byd\Omega = \sin\theta \, d\theta \, d\phi,
with θ as the polar angle and φ as the azimuthal angle, effectively accumulating the "area" in angular space subtended by the surface.[20] This projection ensures that the solid angle accounts for the orientation and distance of surface elements relative to the point, distinguishing it from simple planar subtension by incorporating depth and volume. A representative example is the solid angle subtended by a right circular cone with half-angle α at its apex, which forms a spherical cap on the unit sphere and equals
\Omega = 2\pi (1 - \cos\alpha).
For a sphere of radius R observed from an external point at distance d > R from its center, the subtended solid angle matches that of the tangent cone, again $2\pi (1 - \cos\alpha), where \sin\alpha = R/d, illustrating how the sphere's visible portion projects to a cap defined by the angular radius α.[21] This property highlights the invariance of the solid angle for conical or spherical geometries under radial projection onto the unit sphere.[20]