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Circular arc

A circular arc is a portion of the circumference of a circle that connects two distinct points on the circle, forming a curved segment shorter than the full circle.^[1] It is defined by the circle's radius and the central angle subtended by the arc at the circle's center, distinguishing it from straight lines or other curves.^[2] Circular arcs are fundamental in Euclidean geometry, where they enable the study of angles, sectors, and segments within circles.^[3] Arcs are classified by their central angle measure: a minor arc spans less than 180 degrees, a major arc exceeds 180 degrees, and a semicircle exactly equals 180 degrees.^[1] The measure of an arc is typically expressed in degrees or radians, directly corresponding to its central angle.^[3] Key properties include the arc length, calculated as l = r \theta where r is the radius and \theta is the central angle in radians, and the chord length connecting the endpoints, given by a = 2r \sin(\theta/2).^[1] These elements underpin theorems such as the inscribed angle theorem, where an angle inscribed in a semicircle is a right angle,^[4] and applications in trigonometry, navigation, and engineering design. Circular arcs also appear in advanced contexts like curvature analysis and parametric equations in calculus.^[5]

Definition and Basic Concepts

Definition

A circular arc is the portion of a circle's circumference connecting two points on the circle, defined by a central angle θ subtended at the circle's center.^[6] A circle itself is a plane figure contained by one line such that all straight lines falling upon it from one point among those lying within the figure are equal to one another, with that point serving as the center and the equal distance being the radius r.^[7] The arc represents a connected subset of this circumference, forming a continuous curve between the endpoints. The term "arc" derives from the Latin arcus, meaning "bow," which evokes the curved shape reminiscent of a bent bow.^[8] This concept was first formalized in Euclidean geometry around 300 BCE, as part of the systematic treatment of circles and their properties in Euclid's Elements.^[9] Arcs can be visualized as paths traced along the circle's edge between two points, in contrast to the straight-line chord that directly links those endpoints.^[10] For example, in a unit circle where r=1, an arc extends from one point to another following the curved boundary, rather than the linear path.

Types of Arcs

Circular arcs are classified based on the measure of the central angle they subtend, which determines their position and extent along the circle's circumference. A minor arc is the portion of the circle between two distinct points where the central angle θ is less than 180° (or π radians); it represents the shorter path connecting the endpoints.^[11]^[12] In circle geometry, the minor arc is typically assumed when referring to an arc without further specification.^[11] A major arc, in contrast, spans the longer path between the same endpoints, with a central angle θ greater than 180° (or π radians), corresponding to a reflex angle at the center.^[11]^[13] This term emphasizes the arc's association with the reflex central angle exceeding 180°.^[13] A semicircle is a specific type of arc where θ equals exactly 180° (or π radians), dividing the circle into two equal halves.^[14]^[15] Equal arcs are those with identical measures, meaning they subtend the same central angle within the same circle; such arcs have equal lengths and imply congruent chords connecting their endpoints.^[16]^[17] For instance, on a clock face, the arc from 12 to 3 subtends a minor arc of 90°, whereas the arc from 12 to 9 proceeding clockwise forms a major arc of 270°.^[18] Arc lengths and associated sector areas vary according to these types, with minor arcs yielding shorter lengths and smaller sectors than major arcs.

Geometric Properties

Arc Length

The arc length L of a circular arc is the measure of the curved path subtended by a central angle \theta in a circle of radius r. The standard formula for arc length, when \theta is expressed in radians, is given by

L = r \theta.

This formula arises directly from the definition of the radian as the ratio of arc length to radius, where the radian measure \theta satisfies \theta = L / r for the unit circle (r = 1), and scales proportionally for any radius.^[19] One derivation of this formula uses polygonal approximations: an arc can be approximated by a regular polygon inscribed in the circle, where the perimeter of the polygon consists of chords subtending small central angles \Delta \theta. As the number of sides increases (and \Delta \theta \to 0), the polygon's perimeter approaches the arc length, yielding a total length of r \sum \Delta \theta = r \theta.^[20] An equivalent calculus-based derivation parameterizes the circle in Cartesian coordinates as x = r \cos \phi, y = r \sin \phi for \phi from 0 to \theta. The differential arc length element is ds = \sqrt{(dx/d\phi)^2 + (dy/d\phi)^2} \, d\phi = r \, d\phi, and integrating gives

L = \int_0^\theta r \, d\phi = r \theta.

^[21] When the central angle is measured in degrees, the formula adjusts to account for the full circumference corresponding to 360°:

L = \frac{\theta}{360} \cdot 2\pi r,

where \theta is in degrees. This follows from the proportion of the arc to the full circle's circumference $2\pi r. The arc length L has the same units as the radius r, such as meters or centimeters. For example, with r = 5 cm and \theta = \pi/3 radians (equivalent to 60°), the arc length is L = 5 \cdot (\pi/3) \approx 5.236 cm.^[22] A fundamental property of the circle's rotational symmetry ensures that, for a fixed radius, the arc length is directly proportional to the central angle—a direct consequence of the circle's uniformity, where equal angles subtend equal arcs regardless of position.^[23]

Chord and Sagitta

In geometry, the chord of a circular arc is the straight-line segment joining its two endpoints on the circle.^[24] To derive its length c, consider the isosceles triangle formed by drawing radii from the circle's center to the arc's endpoints, with the central angle \theta (in radians) subtending the chord; the triangle bisects into two right triangles, where the half-chord length is r \sin(\theta/2), yielding the formula c = 2r \sin(\theta/2).^[24] The sagitta h, also known as the arrow height or versine, is the perpendicular distance from the midpoint of the chord to the midpoint of the arc.^[25] Geometrically, it arises from the same isosceles triangle configuration, where the apothem (distance from center to chord midpoint) is r \cos(\theta/2), so h = r - r \cos(\theta/2) = r (1 - \cos(\theta/2)).^[26] An alternative expression is h = r - \sqrt{r^2 - (c/2)^2}.^[26] For small central angles \theta, the chord length closely approximates the arc length, while the sagitta quantifies the arc's deviation from straightness.^[26] For example, with radius r = 10 m and \theta = 60^\circ = \pi/3 rad, the chord length is c = 2 \times 10 \times \sin(\pi/6) = 10 m, and the sagitta is h = 10 (1 - \cos(\pi/6)) \approx 1.34 m.^[24]^[26] In practical applications, such as the design of arches in architecture and bridges, the sagitta determines the rise (or dip) relative to the span, aiding in structural computations for tension and load distribution.

Area Calculations

Sector Area

A circular sector is the portion of a disk enclosed by two radii of a circle and the arc connecting their endpoints, resembling a slice of pie. This geometric figure is fundamental in applications ranging from engineering to physics, where precise area calculations are essential for design and analysis. The area A of a sector with radius r and central angle \theta measured in radians is given by the formula

A = \frac{1}{2} r^2 \theta.

This formula arises from the proportion of the sector to the full circle: the sector occupies a fraction \frac{\theta}{2\pi} of the circle's total area \pi r^2, simplifying to A = \frac{\theta}{2\pi} \cdot \pi r^2 = \frac{1}{2} r^2 \theta. If the central angle is given in degrees, the formula adjusts to

A = \frac{\theta}{360} \pi r^2,

where \theta is the angle in degrees, reflecting the sector as \frac{\theta}{360} of the full circle. For example, consider a sector with radius r = 4 cm and central angle \theta = \frac{\pi}{2} radians (equivalent to 90°). Substituting into the radian formula yields

A = \frac{1}{2} \cdot 4^2 \cdot \frac{\pi}{2} = \frac{1}{2} \cdot 16 \cdot \frac{\pi}{2} = 4\pi \approx 12.566 \ \text{cm}^2.

Using the degree formula confirms the same result: A = \frac{90}{360} \pi \cdot 16 = 4\pi \approx 12.566 \ \text{cm}^2. Notably, when the central angle is a full revolution (\theta = 2\pi radians or 360°), the formula recovers the area of the entire circle, A = \pi r^2, verifying its consistency with the basic circle area formula.

Segment Area

The circular segment, often referred to as the "cap" of the circle, is the region bounded by a circular arc and the chord connecting its endpoints, typically considering the minor segment where the central angle θ is less than π radians (180°). This area represents the portion of the disk between the arc and the chord, distinguishing it from the larger sector that includes the triangular region.^[26] The area A of the circular segment is given by the formula

A = \frac{1}{2} r^2 (\theta - \sin \theta),

where r is the radius of the circle and \theta is the central angle in radians. This exact expression derives from subtracting the area of the isosceles triangle formed by the two radii and the chord, which is \frac{1}{2} r^2 \sin \theta, from the area of the corresponding circular sector, \frac{1}{2} r^2 \theta.^[26]^[27] For small central angles, an approximation for the segment area uses the chord length c and sagitta (height) h:

A \approx \frac{2}{3} c h + \frac{h^3}{2c}.

This provides a simple estimate when \theta is small, though the exact formula remains preferable for precision.^[26] For example, with radius r = 5 cm and \theta = \pi/3 radians (60°), the segment area is

A = \frac{1}{2} (5)^2 \left( \frac{\pi}{3} - \sin \frac{\pi}{3} \right) = 12.5 \left( \frac{\pi}{3} - \frac{\sqrt{3}}{2} \right) \approx 2.26 \, \text{cm}^2.

^[26] In the specific case of a semicircle, where \theta = \pi radians, the formula yields A = \frac{1}{2} r^2 (\pi - \sin \pi) = \frac{\pi r^2}{2}, corresponding to the full semicircular area since the triangular portion degenerates to zero height.^[26]

Relations and Formulas

Central Angle

The central angle subtended by a circular arc is the angle formed at the center of the circle by the two radii extending from the center to the endpoints of the arc.^[28] Denoted as \theta, this angle quantifies the arc's extent and is typically measured in degrees or radians. The measure of the arc itself equals the measure of the central angle in the chosen units.^[29] Conversion between degrees and radians for the central angle follows the relation \theta_{\text{rad}} = \theta_{\deg} \cdot \frac{\pi}{180}, or inversely \theta_{\deg} = \theta_{\rad} \cdot \frac{180}{\pi}.^[30] A complete circle corresponds to a central angle of $360^\circ or $2\pi radians.^[30] An inscribed angle that subtends the same arc measures half the central angle, as stated by the inscribed angle theorem.^[31] When the arc length L and radius r are known, the central angle in radians is given by \theta = \frac{L}{r}. For instance, an arc of length 10 cm on a circle of radius 20 cm yields \theta = \frac{10}{20} = 0.5 radians, or approximately $28.65^\circ. The central angle determines the arc's classification: a minor arc if \theta < 180^\circ (or \pi radians), and a major arc otherwise.^[32]

Radius Determination

Determining the radius of a circular arc involves solving inverse geometric relations using measurable quantities such as arc length, central angle, chord length, or sagitta. These methods rearrange fundamental properties of circles and isosceles triangles formed by the radii and chord.^[26] When the arc length L and central angle \theta (in radians) are known, the radius r is given by the formula r = \frac{L}{\theta}. This directly inverts the standard arc length relation L = r \theta. For example, if L = \pi m and \theta = \frac{\pi}{2} rad, then r = \frac{\pi}{\pi/2} = 2 m.^[33] Given the chord length c and central angle \theta, the radius follows from the chord formula derived from the isosceles triangle: r = \frac{c}{2 \sin(\theta/2)}. This rearranges the relation c = 2 r \sin(\theta/2), where the half-angle accounts for the geometry of the triangle's sides.^[26] For cases involving the sagitta h (the perpendicular distance from the chord's midpoint to the arc) and chord length c, the exact radius is r = \frac{h}{2} + \frac{c^2}{8h}, obtained by solving the quadratic relation from the sagitta definition h = r - \sqrt{r^2 - (c/2)^2}. This formula is particularly useful when direct angle measurement is unavailable, as it relies solely on linear dimensions. For small h relative to c, approximations may suffice, but the exact form ensures precision.^[34] If only the arc length L and chord length c are measured, determining r requires solving a transcendental equation, as no closed-form solution exists. The approach involves setting up L = r \theta and c = 2 r \sin(\theta/2), then numerically solving for \theta (e.g., via Newton's method) before computing r = L / \theta. For small arcs where \theta is modest, an approximation like r \approx \frac{L^2 + 4 h^2}{8 h} can be used after estimating h from c and L, though numerical methods are preferred for accuracy.^[35] In engineering applications, such as dimensional metrology for mechanical components, radius determination from measured arc lengths and chords is common for circle fitting and quality control, often employing robust least-squares algorithms to handle measurement errors.^[36]

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