Central Angle: From θ = s ÷ r to Pie Slices and Polygons

What a central angle actually is

A central angle is the angle formed at the very middle of a circle by two radii drawn out to the ends of an arc. Its vertex sits at the centre, its two sides are radii, and the piece of the rim caught between them is the arc it “opens up to.” That is the whole idea: a central angle and its arc are two ways of describing the same slice of a circle — one measured as a turn at the centre, the other as a length along the edge.

Because the circle is so regular, those two descriptions are locked together by a single ratio. Fix the radius, and the arc length depends on nothing but the central angle; grow the angle and the arc grows in exact proportion. That tidy relationship is what the central angle calculator runs on: hand it an arc length and a radius and it returns the angle that arc subtends, in degrees, radians and gradians at once. This guide is about where that formula comes from, how to read every number it gives back, and the handful of ways the same angle turns up when the arc isn’t the thing you actually know.

The formula: θ = s ÷ r

The headline relationship is as short as geometry gets:

θ = s ÷ r

where s is the arc length, r is the radius, and θ is the central angle in radians. It looks almost too simple to need deriving, and in a sense it doesn’t: this is the definition of the radian. One radian is precisely the angle whose arc is exactly as long as the radius. So “how many radii long is the arc?” and “how many radians is the angle?” are the same question, and the answer to both is s ÷ r.

Degrees take one more step. A full turn is 2π radians and also 360°, so to convert you multiply the radian value by 180/π (about 57.2958 degrees per radian). Threading that through the formula gives the degree version people often meet first:

θ° = (s × 360) ÷ (2πr)

which is just s ÷ r dressed up — the s ÷ (2πr) part is the fraction of the whole circumference the arc covers, and multiplying a fraction of the circle by 360° gives the angle in degrees. Gradians (or gon) work the same way with 400 in place of 360, so a right angle is 100 gon and a full turn is 400 gon. The central angle calculator reports all three from a single pair of inputs, keeping π at full machine precision and rounding only when it prints the result.

Worked example: an arc of 5 on a radius of 2

Take an arc 5 units long sitting on a circle of radius 2 units. Straight from the definition:

θ = s ÷ r = 5 ÷ 2 = 2.5 radians

To read that in degrees, multiply by 180/π:

2.5 × 180/π ≈ 143.24°

And to see how much of the circle it is, compare the arc to the full circumference, 2π·2 ≈ 12.566 units: 5 ÷ 12.566 ≈ 0.3979, so the arc covers about 39.79% of the circle. Two quick sanity checks keep you honest here. An arc exactly as long as the radius (s = r) gives θ = 1 radian ≈ 57.30° — one radius, one radian, by definition. And an arc as long as the whole circumference (s = 2πr) gives θ = 2π radians = 360°, a full lap. Feed the calculator either of those and anything but 57.30° or 360° means the input slipped.

Reading every number the calculator returns

The single input pair produces four outputs, and each answers a slightly different question.

Radians — the natural unit

Radians come out first because they are s ÷ r, no conversion involved. If your next step is another formula, this is almost certainly the number you want: arc length s = r·θ, sector area A = ½·r²·θ and the small-angle approximations all expect θ in radians. Reach for radians whenever the angle is feeding maths rather than being read by a person.

Degrees — the everyday unit

Degrees are the number to quote a client, a drawing or a classmate. They are the radian value scaled by 180/π, nothing more, and they range from 0° up to 360° for a single sweep around the circle.

Gradians and the fraction of the circle

Gradians (gon) turn up in surveying and some European engineering, where a right angle being a round 100 units is convenient. The fraction of the circle — θ ÷ 2π, printed as a percentage — is the most underrated output: it is exactly the slice size you need for a pie chart or a proportional wheel, without ever converting to degrees at all.

Finding the central angle when you don’t have the arc

Arc length and radius is the cleanest route, but plenty of real problems hand you the angle’s fingerprints in some other form. The angle is the same; only the arithmetic that reaches it changes.

From a fraction or a pie slice

If you already know what share of the circle you want — say a budget category is 15% of spending — the central angle is just that fraction of 360°: 0.15 × 360° = 54°. This is the everyday use hiding inside every pie chart: each slice’s angle is its share of the whole turned into degrees. Going the other way, the calculator’s “fraction of circle” output hands you that share directly from an arc and radius.

From the sector area

A sector’s area and its central angle are tied by A = ½·r²·θ (θ in radians), so if you know the area of the pie slice and the radius, rearrange to θ = 2A ÷ r². A sector of area 10 on a radius-2 circle spans 2·10 ÷ 4 = 5 radians — more than a full circle, which is a useful red flag that the numbers don’t belong to a single simple sector.

From the number of equal pieces

Cutting a circle into n equal parts — the teeth of a gear, the sides of a regular polygon, the spokes of a wheel — gives each piece a central angle of 360° ÷ n. Six equal slices are 60° each; a twelve-tooth sprocket steps 30° per tooth. This is the same relationship read backwards, and it is why regular shapes drawn inside circles come out so evenly.

The angle it keeps getting confused with

The central angle has a near-twin that trips people up constantly: the inscribed angle. Both can stand on the very same arc, but the central angle has its vertex at the centre while an inscribed angle has its vertex out on the circle itself. The inscribed angle theorem says the central angle is exactly twice any inscribed angle subtending the same arc. Miss that factor of two and a layout can be off by 50% in one stroke.

A second, gentler confusion is between the central angle and the “arc measure.” In degrees these two are the same number — the measure of an arc is its central angle — which is exactly why the terms get swapped. The thing to hold onto is that arc measure (in degrees) and arc length (in units of distance) are not the same: length also depends on the radius, measure does not. The central angle calculator works from arc length precisely because that is the quantity you can put a tape measure to.

Common mistakes

Using degrees where the formula wants radians. θ = s ÷ r gives radians, full stop. Plug that straight into a formula expecting degrees — or feed degrees into s = r·θ or A = ½·r²·θ — and the answer is off by a factor of about 57. Convert first: θ_rad = θ_deg × π/180.

Mixing units between the arc and the radius. The formula only cares about the ratio s ÷ r, so the two must be in the same unit. An arc measured in centimetres over a radius in metres produces a ratio a hundred times too big. Match the units and the unit itself then cancels out entirely — which is why the angle is the same whether you work in inches or kilometres.

Confusing the central angle with the inscribed angle. If the vertex you care about is on the circle rather than at the centre, you want half the central angle. This is the single most common slip on the topic, and it hides well because both angles are perfectly valid — just not interchangeable.

Expecting an angle above 360° from a normal arc. A single arc around a circle tops out at the full circumference, giving exactly 360°. If the calculator returns more than that, the arc you entered is longer than the circle it supposedly sits on — mathematically it means more than one wrap, but for a plain sector it means the radius or the arc is wrong.

Where central angles quietly show up

This is one of those geometry ideas that turns out to be everywhere once you notice it.

Charts and dashboards

Every pie chart and doughnut chart is central angles in disguise: each slice’s angle is its share of the total turned into 360°. Get the angles wrong and the picture lies, so the fraction-of-circle output is doing real work behind a lot of everyday reporting.

Polygons, gears and anything evenly spaced

Every side of a regular polygon inscribed in a circle is a chord that subtends the same central angle, 360° ÷ n. That is the thread connecting the central angle to the chord length calculator and the hexagon calculator: a hexagon is six 60° slices, and because a 60° chord equals the radius, a regular hexagon’s side matches the circle it sits inside. The same 360° ÷ n logic spaces gear teeth, clock markings and bolt-hole patterns.

Curves, cams and CAD

Anyone laying out an arc — a cam profile, a curved road, a bent handrail — is working in central angles and arc lengths, because a fabrication drawing usually specifies a sweep. Machine-tool controllers and CAD packages think in these terms natively, describing an arc by its centre, radius and swept angle rather than by a pile of coordinates.

When the central angle is only the start

The central angle is rarely the finish line — it is the bridge between an arc, a sector and a fraction of the whole circle, and knowing it usually unlocks the next number. Once you have the angle, the circle calculator takes you on to circumference and area, the chord length calculator turns the same angle into the straight-line chord and its sagitta, and if the circle is living in a coordinate plane the circle equation calculator writes down its equation. The right triangle that sits under so many of these results is the Pythagorean theorem at work, and the chord length guide and circle calculator guide pick up the rest of the story. The formal statement of everything here — the radian definition, the inscribed angle theorem — is set out cleanly in MathWorld’s Central Angle entry, and the SI treatment of the radian as the unit of plane angle lives in the BIPM SI Brochure. Both are worth a bookmark when you want the definitions rather than the shortcut.