About the Central Angle Calculator
A central angle has its vertex at the centre of a circle and its two sides running out to the circle; it cuts off an arc, a chord and a sector. This central angle calculator finds that angle from the radius together with any one of the other three: the arc length, the chord length, or the sector area. It then fills in the rest — arc, chord, sector and segment area, the fraction of the full circle, and the inscribed angle that subtends the same arc.
Typical uses are geometry and trigonometry homework, laying out curved walls, paths and arches, working out the angle of a pie-chart slice or a pizza wedge, finding how far round a wheel or pulley a belt wraps, and converting a distance travelled along a circular track into an angle.
From a chord the calculator returns the minor angle (180° or less), because a chord alone cannot tell the small arc from the large one; subtract from 360° for the major arc. Arc length and sector area must describe less than a full circle.
With the default inputs, the central angle (degrees) is 85.9437 °. Change any value above to recalculate instantly.
How to use the central angle calculator
- 1Choose what you know besides the radius: arc length, chord length or sector area.
- 2Enter the radius.
- 3Enter the arc, chord or sector value in the same unit.
- 4Read the central angle in degrees and radians, plus the other arc measurements.
Formula and method
A radian is defined as the angle whose arc equals the radius, so the central angle in radians is simply arc length ÷ radius. A sector’s area is ½r²θ, which rearranges to θ = 2A ÷ r².
The chord and the two radii form an isosceles triangle. Splitting it down the middle gives two right triangles with hypotenuse r and opposite side c/2, so sin(θ/2) = c/(2r) and θ = 2·arcsin(c/(2r)); this gives the minor angle. Once θ is known, arc = rθ, chord = 2r·sin(θ/2), sector = ½r²θ and segment = ½r²(θ − sin θ). The inscribed angle theorem says any angle on the circle subtending the same arc is θ/2.
- θ
- Central angle
- r
- Radius
- s
- Arc length
- c
- Chord length
- A
- Sector area
Worked examples
Arc length 15, radius 10
θ = 15 ÷ 10 = 1.5 radians ≈ 85.94°. That arc spans about 23.9% of the circle, the sector has area ½ × 100 × 1.5 = 75 and the chord is 2 × 10 × sin 0.75 ≈ 13.63.
Chord 8, radius 5
sin(θ/2) = 8 ÷ 10 = 0.8, so θ/2 ≈ 53.13° and θ ≈ 106.26°. The minor arc is 5 × 1.8546 ≈ 9.27 long.
Sector area 50, radius 8
θ = 2 × 50 ÷ 64 = 1.5625 rad ≈ 89.52° — just under a right angle — with an arc of 8 × 1.5625 = 12.5.
Frequently asked questions
How do you find the central angle of a circle?+
Divide the arc length by the radius to get radians, then multiply by 180/π for degrees. An arc of 5 on a circle of radius 5 is 1 radian ≈ 57.30°.
How do I find the central angle from a chord?+
Use θ = 2·arcsin(c ÷ 2r). A chord of 6 in a circle of radius 5 gives θ = 2·arcsin(0.6) ≈ 73.74°. This is the minor angle; the major angle is 360° minus it.
What is the relationship between central and inscribed angles?+
An inscribed angle (vertex on the circle) is half the central angle that subtends the same arc. A 100° central angle corresponds to a 50° inscribed angle.
Do the central angles of a circle add up to 360°?+
Yes. Non-overlapping central angles that go all the way around a circle total 360° (2π radians), which is why a pie chart slice angle is its percentage × 3.6°.