A central angle is an angle whose vertex is the center of a circle and whose sides are two radii. It cuts off an arc, and the arc’s degree measure equals the central angle. A 90° central angle marks off a quarter of the circle, and the central angles around the center add to 360 degrees.
A central angle and its arc have the same degree measure. Move the vertex out to the circle and the angle that sees the same arc is only half as big.
Formal definition and notation
For a circle with center O and points A and B on the circle, ∠ AOB is a central angle. It intercepts minor arc AB, and by definition marc AB = m∠ AOB. The fraction of the circle cut off is θ360°, which gives arc lengthθ360° · 2π r and sector area θ360° · π r2.
In a survey of 240 students, 54 chose soccer. On a pie chart, what central angle should the soccer slice have?
Fraction choosing soccer: 54240 = 940.
The whole chart is 360°.
Angle: 940 × 360° = 81°.
Answer: The soccer slice needs an 81° central angle.
Common mistakes
✗ MistakeSaying an inscribed angle that cuts off a 70° arc is 70°.
✓ InsteadOnly the central angle equals the arc: central 70°, inscribed 35°.
The vertex position matters. At the center the angle equals the arc; on the circle it is half.
✗ MistakeArc length of a 90° angle in a circle of radius 4: 90360 × π × 42 = 4π.
✓ InsteadArc length uses circumference: 90360 × 2π(4) = 2π. 4π is the sector area.
Arc length is a length, so it is a fraction of 2π r. The π r2 version gives the area of the wedge.
✗ MistakeMixing up arc measure and arc length: arc measure = 50 cm.
✓ InsteadArc measure is in degrees (50°); arc length is a distance in cm.
Two arcs with the same degree measure can have different lengths in circles of different sizes.
Not to be confused with
Central Anglevs.Inscribed angle
An inscribed angle has its vertex on the circle, not at the center, and measures half its arc. A 100° arc gives a 100° central angle but a 50° inscribed angle.
The central angle is measured in degrees (or radians); the arc length is the distance along the circle in units such as cm. A 60° angle gives a 2π arc when r = 6 but a π arc when r = 3.
Quick check: 3 questions
Question 1 of 3
A central angle cuts off 15 of a circle. What does it measure?
Show answer and explanation
Answer: A.15 × 360° = 72°. 36° uses 180° as the whole, 20° divides 100 by 5, and 18° uses 90°.
Question 2 of 3
Central angle ∠ AOB = 130°. An inscribed angle cuts off the same arc AB. What does it measure?
Show answer and explanation
Answer: C. The inscribed angle is half the central angle: 130° ÷ 2 = 65°. 130° treats it as central, 260° doubles instead of halving, and 50° is the supplement of 130°.
Question 3 of 3
In a circle of radius 10 m, a central angle of 36° cuts off a sector. What is the sector’s area?
Show answer and explanation
Answer: D.36360 × π(10)2 = 110 × 100π = 10π. 2π is the arc length, 100π is the whole circle, and 36π uses the angle as if it were the area.
Central Angle: frequently asked questions
Is the central angle equal to the arc?
Its degree measure is. A central angle of 75° cuts off an arc that measures 75°, because arcs are measured by the angle they make at the center. The arc’s length, though, depends on the radius as well: in a circle of radius 12, a 75° arc is 75360 × 24π = 5π units long.
What is the difference between a central angle and an inscribed angle?
Where the vertex is. A central angle has its vertex at the center of the circle; an inscribed angle has its vertex on the circle itself, with chords as its sides. When both cut off the same arc, the inscribed angle is exactly half the central angle. That is the Inscribed Angle Theorem.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSG-C.A.2, CCSS HSG-C.B.5
Cite this entry
Effortless Math Team. “Central Angle.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/central-angle/
Flashcard
TermCentral Anglevertex at the center of a circle
DefinitionAn angle formed by two radii, with its vertex at the circle’s center. It equals its arc: a 60° central angle cuts off a 60° arc, one sixth of the circle.
AngleAn angle is formed by two rays that share the same endpoint, called the vertex. It is measured in degrees by how far one ray is turned from the other. A right angle measures 90 degrees, a straight angle measures 180 degrees, and one full turn is 360 degrees.Geometry · Grade 4, Grade 5, Grade 6VertexA vertex is a corner point where two or more lines, rays, or sides meet. It is the point of an angle, a corner of a polygon such as a triangle, a corner of a solid, and the turning point of a parabola. The plural of vertex is vertices: a square has 4 vertices.Geometry · Grade 4, Grade 5, Grade 6CircleA circle is the set of all points in a flat plane that are the same distance from one fixed point, the center. That distance is the radius. A segment through the center from edge to edge is the diameter, twice the radius. The distance around is the circumference, 2π r, and the area inside is π r2.Geometry · Kindergarten, Grade 1, Grade 7Arc LengthArc length is the distance along a curved part of a circle, measured in units of length. An arc with central angle θ in a circle of radius r is θ360 of the circumference: s = θ360 × 2π r when θ is in degrees, or simply s = rθ when θ is in radians.Geometry · Geometry, PrecalculusDegreeA degree is the standard unit for measuring an angle, written with the symbol °. One degree is 1360 of a full turn, so a full turn is 360°, a straight angle is 180° and a right angle is 90°. An angle that turns through 25 one-degree angles measures 25°.Geometry · Grade 4, Grade 5, Grade 7Equal ToEqual to means having exactly the same value or amount, and it is written with the symbol =. Numbers that look different can still be equal to each other: 12 is equal to 0.5. When two amounts are not the same, one is greater than or less than the other.Symbols and Notation · Kindergarten, Grade 1, Grade 2ChordA chord is a line segment whose two endpoints both lie on a circle. The diameter is the longest chord, because it passes through the center. A chord at distance d from the center of a circle with radius r has length 2√r2 − d2, so chords closer to the center are longer.Geometry · Grade 7, GeometryDiameterThe diameter of a circle is the distance straight across it through the center. It is twice the radius, d = 2r, and it is the longest chord any circle has. The word also names the line segment itself: a segment that passes through the center and touches both sides of the circle.Geometry · Grade 6, Grade 7, Grade 8SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2Regular PolygonA regular polygon is a polygon whose sides are all the same length and whose angles are all the same size. The equilateral triangle, the square and the stop-sign octagon are regular. Each interior angle of a regular n-gon is (n − 2) × 180°n, and each exterior angle is 360°n.Geometry · Grade 5, Grade 8, GeometryRadiusThe radius is a line segment from the center of a circle to any point on the circle, or the length of that segment. The radius is half the diameter: if the diameter is 10, the radius is 5. All radii in the same circle have equal length, and r is the radius in C = 2π r.Geometry · Grade 6, Grade 7, Grade 8LengthLength is how long something is: the distance from one end of an object to the other. You measure it by counting equal units laid end to end with no gaps or overlaps, such as inches, feet, centimeters or meters. A new pencil is about 7 inches long.Measurement · Grade 1, Grade 2, Grade 3CircumferenceCircumference is the distance around a circle, found by multiplying the diameter by pi, or the radius by 2 pi: C = π d = 2π r. It is the circle’s perimeter, measured in units of length. A circle with diameter 10 has a circumference of about 31.4, because π ≈ 3.14.Geometry · Grade 6, Grade 7, Grade 8RadianA radian is a unit for measuring angles: one radian is the central angle that cuts off an arc exactly as long as the circle’s radius. A full turn is 2π radians, so π radians = 180° and 1 radian is about 57.3°. To change degrees to radians, multiply by π180.Trigonometry · Geometry, Algebra 2, Precalculus