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Central angle

A central angle is an angle formed by two radii of a circle, with its vertex at the center of the circle and its sides extending to points on the circumference.^[1] This angle subtends an arc on the circle, and its measure in degrees is equal to the measure of that intercepted arc.^[2] Central angles are fundamental in circle geometry, as they define the size of arcs and relate directly to other circle elements like chords and inscribed angles. A central angle less than 180° subtends a minor arc, while one greater than 180° subtends a major arc, and exactly 180° forms a semicircle.^[2] The sum of all central angles around a circle totals 360°.^[2] Additionally, congruent central angles intercept congruent arcs, and the length of a chord subtended by a central angle depends on the angle's measure and the circle's radius.^[1] One key theorem involving central angles is the inscribed angle theorem, which states that an inscribed angle subtending the same arc as a central angle has half the measure of the central angle.^[3] Central angles also determine arc length via the formula: arc length = (central angle measure / 360°) × 2πr, where r is the radius.^[4] These properties make central angles essential for applications in trigonometry, such as calculating sector areas and solving problems in navigation and engineering.

Definition and Properties

Definition

A central angle is an angle whose vertex is at the center of a circle and whose sides are two radii of the circle, connecting to two distinct points on the circumference.^[5] This angle subtends an arc, which is the portion of the circle's circumference between those two points.^[6] Consider a circle with center O and points A and B on its circumference; the central angle ∠AOB is formed by the radii OA and OB, enclosing the arc AB.^[7] Unlike inscribed angles, which have their vertex on the circle's circumference and sides passing through two other points on the circle, central angles originate at the center and directly measure the extent of the subtended arc.^[8] The concept of the central angle originates in Euclidean geometry, as developed by the ancient Greek mathematician Euclid in his treatise Elements, particularly in Book III, where properties of circles and angles at the center are explored.^[9]

Properties

Central angles possess several inherent geometric properties that govern their interaction with the circle's structure. A primary property is additivity, where the measures of contiguous central angles that collectively encompass the full circumference of a circle sum to 360 degrees, or equivalently 2π radians in radian measure.^[6] This additivity arises from the complete rotational symmetry of the circle around its center.^[10] Another key property is the direct proportionality between the measure of a central angle and the length of the arc it subtends. Specifically, the arc length is a direct fraction of the circle's total circumference, scaled by the central angle's measure relative to 360 degrees; for instance, a 90-degree central angle subtends an arc one-quarter the length of the full circumference.^[11] This proportionality holds regardless of the circle's size, as the angle defines the arc's relative extent.^[10] Central angles may also be reflex, measuring greater than 180 degrees but less than 360 degrees, and these subtend the major arcs of the circle. A reflex central angle complements its corresponding minor central angle—formed by the same two radii but subtending the shorter arc—to exactly 360 degrees, ensuring the full circle is accounted for.^[11] Furthermore, the measure of a central angle remains invariant under rotation or translation of the circle, as it is determined solely by the angular separation at the center. The property is independent of the circle's radius, which influences arc length but not the angle itself.^[12]

Formulas and Calculations

Measure in Degrees and Radians

The measure of a central angle is commonly expressed in degrees, a unit originating from the division of a full circle into 360 equal parts.^[13] One degree thus represents $1/360 of the total circumference.^[13] For a central angle \theta subtended by an arc of length s on a circle with circumference C, the degree measure is given by \theta = (s / C) \times 360^\circ.^[14] An alternative unit for measuring central angles is the radian, which provides a more natural relation to the geometry of the circle.^[13] One radian is defined as the central angle subtended by an arc whose length equals the radius r of the circle.^[13] Consequently, a full circle, with arc length equal to the circumference C = 2\pi r, measures $2\pi radians.^[13] The radian measure \theta for an arc of length s is then \theta = s / r.^[13] Conversion between degrees and radians follows from the fact that $180^\circ equals \pi radians.^[13] To convert from degrees to radians, multiply by \pi / 180; to convert from radians to degrees, multiply by $180 / \pi.^[13] For instance, a right angle of $90^\circ converts to \pi/2 radians via $90 \times (\pi / 180) = \pi/2, and conversely, \pi/2 radians equals $90^\circ via (\pi/2) \times (180 / \pi) = 90^\circ.

Arc Length

The arc length s subtended by a central angle \theta in a circle of radius r is given by the formula s = r \theta, where \theta is measured in radians.^[15] This formula arises from the proportional relationship between the arc and the full circumference of the circle, which is $2\pi r corresponding to a central angle of $2\pi radians; thus, the arc length is the fraction \frac{\theta}{2\pi} of the circumference, simplifying to s = r \theta.^[16] When the central angle is given in degrees, the arc length can be computed using s = \frac{\theta}{360} \times 2\pi r, where \theta is in degrees.^[15] For practical computation, convert the degree measure to radians by multiplying by \frac{\pi}{180}, yielding \theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180}, and then apply the radian formula s = r \theta_{\text{rad}}.^[15] The radian measure serves as the natural unit for this calculation due to its direct proportionality to arc length.^[16] For example, if the radius r = 5 units and the central angle \theta = \frac{\pi}{3} radians, the arc length is s = 5 \times \frac{\pi}{3} = \frac{5\pi}{3} units.^[15] The units of the arc length s are consistent with those of the radius r, such as meters or centimeters, ensuring dimensional homogeneity in the formula.^[16]

Sector Area

A circular sector is the region of a circle bounded by two radii and the arc subtended by the central angle between them.^[17] The area A of a circular 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 derives from the fact that the sector represents a proportion \frac{\theta}{2\pi} of the full circle's area \pi r^2, yielding A = \frac{\theta}{2\pi} \cdot \pi r^2 = \frac{1}{2} r^2 \theta.^[18] When the central angle \theta is given in degrees, the area formula becomes

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

reflecting the proportion \frac{\theta}{360} of the full circle. To use the radian formula instead, convert degrees to radians via \theta_{\text{rad}} = \theta_{\deg} \cdot \frac{\pi}{180}, then substitute into A = \frac{1}{2} r^2 \theta_{\text{rad}}.^[19] For example, consider a sector with radius r = 4 units and central angle \theta = 90^\circ (or \frac{\pi}{2} radians). Using the radian formula,

A = \frac{1}{2} \cdot 4^2 \cdot \frac{\pi}{2} = \frac{1}{2} \cdot 16 \cdot \frac{\pi}{2} = 4\pi

square units. Using the degree formula yields the same result: A = \frac{90}{360} \pi \cdot 16 = \frac{1}{4} \cdot 16\pi = 4\pi square units.^[20] Sectors are classified as minor or major based on the central angle. A minor sector has \theta < 180^\circ (or \theta < \pi radians), enclosing the smaller arc, while a major sector corresponds to the reflex angle $360^\circ - \theta (or $2\pi - \theta radians), enclosing the larger arc; the major sector area is the full circle area minus the minor sector area.^[21]

Applications

Regular Polygons

In a regular polygon with n sides inscribed in a circle of radius r, the vertices divide the circle into n equal arcs, each subtended by a central angle of \frac{360^\circ}{n} or \frac{2\pi}{n} radians.^[22] This central angle forms the vertex angle of an isosceles triangle with two sides equal to the radius r and base equal to the polygon's side length s. Dropping a perpendicular from the center to the base bisects the central angle into two right triangles, each with a half-angle of \frac{\pi}{n} radians. The half-base length is then r \sin\left(\frac{\pi}{n}\right), yielding the side length formula s = 2r \sin\left(\frac{\pi}{n}\right).^[23] The apothem, defined as the perpendicular distance from the center to a side, is the adjacent side in one of these right triangles, given by a = r \cos\left(\frac{\pi}{n}\right).^[23] For example, in a regular pentagon (n=5) inscribed in a circle of radius r, the central angle is $72^\circ, and the side length is s = 2r \sin(36^\circ).^[22] Ancient Greek mathematicians, particularly Euclid in his Elements (Book IV), employed circle division based on central angles to construct regular polygons such as the equilateral triangle, square, pentagon, and hexagon inscribed in a given circle, using compass and straightedge methods.^[24]

Relation to Inscribed Angles

In circle geometry, the inscribed angle theorem establishes a fundamental relationship between central angles and inscribed angles that subtend the same arc. Specifically, the measure of an inscribed angle is half the measure of the central angle that subtends the identical arc; if the central angle has measure \theta, then the inscribed angle measures \frac{\theta}{2}.^[3]^[25] The proof of this theorem relies on properties of isosceles triangles and the proportionality of arcs in Euclidean geometry. Consider a circle with center O and an arc AB. The central angle \angle AOB = \theta forms an isosceles triangle AOB where radii OA and OB are equal, so base angles are each \frac{180^\circ - \theta}{2}. For an inscribed angle \angle ACB subtending the same arc AB, draw radii to C and use the straight angle at C along the circumference or auxiliary lines like diameters to divide the angles, showing through angle sums that \angle ACB = \frac{\theta}{2}. This holds across cases where C lies on the major or minor arc by applying the base case iteratively.^[3]^[26] For example, if a central angle subtends an arc of $120^\circ, any inscribed angle subtending the same arc measures $60^\circ. A notable application occurs with a semicircular arc, where the central angle is $180^\circ, making the inscribed angle $90^\circ, as in Thales' theorem.^[25] An extension of this theorem states that all inscribed angles subtending the same arc and lying in the same segment of the circle are equal, since each is independently half the corresponding central angle.^[25]^[27]

References

[1]
6.12: Chords and Central Angle Arcs
### Summary of Central Angle, Arcs, and Chords from 6.12: Chords and Central Angle Arcs
[2]
Intro to arc measure (video) | Circles - Khan Academy
Aug 10, 2016 · When a central angle is formed by two radii of a circle, it intercepts an arc on the circle. The measure of this intercepted arc is directly determined by the ...
[3]
Inscribed angle theorem proof (article) - Khan Academy
An angle is a central angle ‍ when its vertex is the center of the circle, such as θ ‍ shown in the image. An angle is an inscribed angle ‍ when its vertex is ...
[4]
Arc length from subtended angle (video) - Khan Academy
Aug 9, 2016 · Finding the length of an arc using the degree of the angle subtended by the arc and the perimeter of the circle.Missing: definition | Show results with:definition
[5]
Central Angle -- from Wolfram MathWorld
A central angle is an angle with endpoints and located on a circle's circumference and vertex located at the circle's center (Rhoad et al. 1984, p. 420). A ...
[6]
Central Angles - Varsity Tutors
A central angle is an angle with its vertex located at the center of a circle and its sides containing two radii of the circle.
[7]
Central Angles in Circles - MathBitsNotebook(Geo)
A central angle of a circle is an angle formed by two radii with the vertex at the center of the circle. In circle O, at the right, ∠AOB is a central angle. ...
[8]
[PDF] Circle Definitions and Theorems
the is. Page 2. Central angle- an angle with vertex at the center of the. circle. Arc – part of the circumference (edge) of the circle.Missing: textbook | Show results with:textbook
[9]
Book III - Euclid's Elements - Clark University
In a circle the angle at the center is double the angle at the circumference when the angles have the same circumference as base. Proposition 21. In a ...
[10]
Circle theorems | Lesson (article) - Khan Academy
A central angle in a circle is formed by two radii. This angle lets us define a portion of the circle's circumference (an arc) or a portion of the circle's area ...
[11]
Central Angle in Geometry - Definition, Formulae, Examples
Central Angle is the angle formed by two arms and having the vertex at the center of a circle. The two arms form two radii of the circle intersecting the arc ...
[12]
Central Angles and Arcs - Geometry - CliffsNotes
A central angle is formed by two radii at a circle's center. An arc is a continuous portion of a circle with two endpoints. There are minor and major arcs.
[13]
[PDF] Math 1330
The measure of the central angle thus formed is one radian. θ = 1 arc length = r r r. Radian measure of an angle:.
[14]
Arc Length and Degree Measure - MathBitsNotebook(Geo)
Arc Measure: In a circle, the degree measure of an arc is equal to the measure of the central angle that intercepts the arc. definition. Arc Length: In a circle ...
[15]
https://math.libretexts.org/Bookshelves/Precalculus/Elementary_Trigonometry_(Corral)/04:_Radian_Measure/4.02:_Arc_Length
[16]
MFG Arc Length
Arc length is defined as the length of an arc, s, s , along a circle of radius r r subtended (drawn out) by an angle θ. θ . arc length. The length of the ...
[17]
[PDF] Math 1330 - Section 4.2 Radians, Arc Length, and Area of a Sector
Page 8. 8. A sector of a circle is the region bounded by a central angle and the intercepted arc. Sometimes, you'll need to find the area of a sector. The ...
[18]
[PDF] 7.4 | Area and Arc Length in Polar Coordinates
Here we derive a formula for the arc length of a curve defined in polar coordinates. In rectangular coordinates, the arc length of a parameterized curve.
[19]
[PDF] Arc Length and Sector Area Discovery Activity Teacher Resource ...
... use your equations from step 2 and step 5 to write a formula to find the area of any sector of a circle (in degrees). Sector area = 𝑥. 360. ⋅ 𝜋𝑟2. 2 mi.<|control11|><|separator|>
[20]
[PDF] MATH 1330 - Section 4.2 - Radians, Arc Length, and Area of a Sector
Dividing both sides by 2, we get 180° = 𝜋 radians. Dividing this last equation by 180° or 𝜋 gives the conversion rules that follow.
[21]
[PDF] Maximizing the Area of a Sector With Fixed Perimeter
a circle (both minor and major) and the associated sectors. A quick and easy Geom- eter's Sketchpad construction can illustrate the two possible sectors ...
[22]
Regular Polygons | Brilliant Math & Science Wiki
Here is the proof or derivation of the above formula of the area of a regular polygon. ... central angle is labeled θ \theta θ. The apothem is the distance from ...
[23]
Regular Polygon -- from Wolfram MathWorld
The numbers of sides for which regular polygons are constructible are those having central angles corresponding to so-called trigonometry angles.
[24]
Euclid's Elements, Book IV
### Summary of Construction of Regular Polygons Using Circles and Central Angles
[25]
Inscribed Angle Theorem - Definition, Theorem, Proof, Examples
This theorem states that: The measure of the central angle is equal to twice the measure of the inscribed angle subtended by the same arc. OR. An inscribed ...
[26]
How to Prove the Inscribed Angle Theorem - House of Math
A central angle is twice the size of an inscribed angle if they span the same arc. Discover the inscribed angle theorem, its uses, and how to prove it.
[27]
Angles in the Same Segment Are Equal - Third Space Learning
Here we will learn about the circle theorem: angles in the same segment, including its application, proof, and using it to solve more difficult problems.