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CalcMax

Central Angle Calculator

Range: 0 cm – 1,000,000,000 cm

Range: 0 cm – 1,000,000,000 cm

Result

57.2958 °

Central angle

Central angle in radians
1.0000 rad

A central angle calculator takes the radius of a circle and the length of an arc on it and returns how large the angle at the centre is — the angle subtended by that arc — in degrees and in radians. The two are the same angle written two ways, and the interesting thing about this page is which of the two comes first. An angle at the centre measured in radians is defined as the arc length divided by the radius — that is not a formula anyone derived, it is what a radian is — so the radians reading is one division with no π anywhere in it, and the degrees reading is that number multiplied by 180/π. The pair this page loads with shows it: a radius of 5 with an arc of 5 gives exactly one radian, because the arc is exactly as long as the radius, and one radian is 57.2958 degrees. Every other calculation here is the same division with different numbers in it. That is also why this page and the arc length page are two pages rather than one. They are the same equation read in opposite directions: one takes the angle and gives the arc, the other takes the arc and gives the angle. Both are worth having, because the questions arrive the other way round — someone measuring a curved wall has a length and wants to know what angle it turns through, while someone drawing a sector has an angle and wants to know how long the rim will be. The one warning worth carrying over is about the compact formula, L = rθ. In that form θ is in radians, always, because the form only works that way. Feed degrees into it and the answer comes out about 57 times too large, and it will not look wrong — it will just look long. This page gives you both readings, so the conversion is done for you; the warning matters when you meet the compact form somewhere else.

The angle at the centre from an arc length and the radius of its circle

Radius (cm)Arc length (cm)Central angle (degrees)Central angle (radians)
5557.29581
10528.64790.5
510114.59162
4114.32390.25
3357.29581
6000
531.4159359.99976.2832
212.5664360.00086.2832

Eight arcs and four columns — two measurements in, the same angle out twice. The first row is the pair the page loads with and the definition of a radian in one line: an arc as long as the radius gives exactly 1, which is 57.2958 degrees. The second and third rows take that same division up and down — an arc of half the radius gives half a radian and an arc of twice the radius gives two — so reading them together shows the radians column behaving like the plain division it is. The fourth row is a quarter of a radian: an arc of 1 laid on a radius of 4 gives 0.25, which is 14.3239 degrees, so the degrees column carries on being the radians figure multiplied by 57.2958. The last two rows are the same thing in opposite directions and they are the point of the table: a radius of 5 fed its own printed circumference gives 359.9997 degrees and a radius of 2 fed its own gives 360.0008, because one rounding went down and the other went up. A reading just over 360 is therefore correct, and both rows are here to say so.

Formula

θ(radians) = L ÷ r, θ(degrees) = θ(radians) × 180 ÷ π

Arc length
The distance along the curve, in centimetres — the length of the piece of the circle, not the straight distance between its two ends
Radius
The distance from the centre out to the arc, in centimetres. It must be greater than zero, because it is what the arc length gets divided by
θ (theta)
The angle at the centre, in radians — the arc length divided by the radius, which is what a radian is. A full turn is 2π of them, about 6.2832
180 ÷ π
The number of degrees in one radian, about 57.2958. Multiplying the radians reading by it gives the degrees reading, and it is the only place π enters the calculation
Degrees
The same angle written the way most people read angles, out of 360 for a full turn. The reading shown first, because it is the one a protractor would show
L = rθ
The compact form of the relationship, with θ in radians. Rearranged it is the division this page performs; used as written with degrees in place of radians it gives an answer about 57 times too large
Four decimal places
How wide both readings are written. A full turn is 6.2832 radians or 360 degrees, and near that point the four decimals of the input start to show through the answer

Anything that has been measured along a curve and now needs to be turned into an angle. A machine shop is the clearest case: a curved slot, a bent tube, a section of a rolled ring arrive with a length, and the drawing calls out an angle — this is the division that gets you from one to the other. Fabrication and building are the same shape of problem: a curved handrail measured along its length, a bent kerb, an arched opening, an arc of brickwork, all of which need the angle at the centre before anything can be set out. Navigation and astronomy use it in the other direction, taking an arc measured on the ground or across the sky and converting it into degrees of latitude, of longitude or of a bearing. There is also a classroom use that is genuinely the clearest way to understand the unit. Draw a circle, cut a piece of string the length of the radius, and lay it along the circle: the angle it covers at the centre is one radian, and a little over six of them go all the way round. Do it once and radians stop being an arbitrary unit that has to be memorised. And there is a practical use that comes up in software and in engineering repeatedly — given a radius and a length, work out what fraction of a circle has been used. That fraction is the radians reading divided by 2π, and it is what this page's two readings are for.

Worked examples

  1. A radius of 5 and an arc of 5

    1. Divide the arc by the radius: 5 ÷ 5 = 1 radian
    2. Convert to degrees: 1 × 180 ÷ π = 57.2958

    The pair the page loads with, and the definition of a radian in one line: an arc as long as the radius subtends an angle of one radian. The degrees figure, 57.2958, is the number to remember if you remember only one thing about radians. Notice that the radians reading came out exactly — no rounding was needed — because this pair was chosen to make the division exact.

  2. A radius of 10 and an arc of 5

    1. Divide the arc by the radius: 5 ÷ 10 = 0.5 radians
    2. Convert to degrees: 0.5 × 180 ÷ π = 28.6479

    The same arc on a bigger circle, so the angle is smaller — an arc of 5 is half the radius here rather than all of it. Half a radian, 28.6479 degrees. This is the row that shows the angle depends on the ratio rather than on the arc alone, which is the same fact the arc length page states from the other end.

  3. A radius of 4 and a quarter of the circumference

    1. The whole circumference: 2 × π × 4 = 25.1327
    2. A quarter of it: 25.1327 ÷ 4 = 6.2832 = π
    3. Divide the arc by the radius: 12.5664 ÷ 4 = 3.1416 radians
    4. Convert to degrees: 3.1416 × 180 ÷ π = 180.0004

    Half a turn, and the row that shows where the four decimals of the input end up. The angle really is 180 degrees, but the arc was entered as the rounded quarter-circumference 12.5664, and dividing a rounded number by 4 carries that rounding into the answer as 0.0004 of a degree. The radians reading is π to four places, which is the other way to see that this is a half turn.

  4. A radius of 2 and the printed circumference

    1. The whole circumference: 2 × π × 2 = 12.5664, printed to four decimals
    2. Divide the arc by the radius: 12.5664 ÷ 2 = 6.2832 radians
    3. Convert to degrees: 6.2832 × 180 ÷ π = 360.0008

    A whole turn, and a reading slightly over 360 degrees — which is correct rather than an error. The circumference of a circle of radius 2 is 12.5663706, which the page prints as 12.5664, rounded up. Enter that rounded figure back in and the angle comes out a fraction of a degree over a full turn. The rounding in the input is what shows through, not a fault in the arithmetic.

  5. A radius of 6 and an arc of 0

    1. Divide the arc by the radius: 0 ÷ 6 = 0 radians
    2. Convert to degrees: 0 × 180 ÷ π = 0

    No arc at all, so no angle: the two radii have closed onto each other and the arc has shrunk to a point. Zero is a legal input here and the answer is a real zero. The radius is the box that cannot be zero — the arc length is divided by it, so a radius of zero has no answer at all.

Limitations

This page returns the angle and nothing else: not the arc, not the radius, not the area of the sector. Given an angle and a radius, the arc length is the other page. The radius must be greater than zero — the arc length is divided by it, so a radius of zero has no answer, and the page reports that rather than printing a blank. The arc length may be zero, which gives an angle of zero. The arc must not be longer than the whole circumference of that circle: a circle of radius 5 has a circumference of 31.4159, so an arc of 32 is refused. The limit is checked against the circumference as the page prints it, to four decimals, rather than against the full-precision value, and that has a visible consequence: a reading slightly above 360 degrees is possible and is correct. If the circumference of your circle ends in a digit that is rounded up when printed, entering that printed figure back in gives an angle such as 360.0008 degrees, because the rounding was already in the number you typed. It is not the page exceeding its own limit. Both inputs are taken in the same units and nothing is converted; the two outputs are always in degrees and radians regardless of the dropdowns.

Frequently asked questions

Why are there two readings?
Because the angle has two standard units and the page does the conversion for you. The radians reading is the arc length divided by the radius, which is the definition of a radian; the degrees reading is that number multiplied by 180/π. They are the same angle, and having both means you do not have to convert by hand or look up which one a formula wants.
My answer is 360.0008 degrees. Is that wrong?
No, it is correct, and it happens when the arc you entered was already a rounded figure. The circumference of a circle of radius 2 is 12.5663706, which the page prints as 12.5664 — rounded up. Type that printed value back in and the angle comes out a fraction of a degree over a full turn. What you are seeing is the rounding in your input, not an error in the division.
Can the radius be zero?
No. The angle is the arc length divided by the radius, so a radius of zero has no answer at all — there is no circle to measure an angle on. The page says so rather than printing a blank or an infinity. The arc length, by contrast, may be zero, and that gives an angle of zero, which is a real answer.
What is the largest the arc can be?
The circumference of that circle. An arc is a piece of the circle, so it cannot be longer than the whole thing, and the page refuses anything beyond it — for a radius of 5 the limit is 31.4159. Beyond that there is no angle at the centre to find, because the arc would have gone round more than once.
How does this relate to the arc length calculator?
They are the same equation read in opposite directions. This page takes the radius and the arc and gives the angle; that page takes the radius and the angle and gives the arc. Both exist because the questions arrive both ways round — measuring a curve gives you a length, drawing a sector gives you an angle — and each page's table has a row the other one also prints.
Is a radian just another unit, like a degree?
It is a unit, but it is not arbitrary in the way a degree is. A degree is one three-hundred-and-sixtieth of a turn, a division inherited from Babylonian arithmetic; a radian is the angle you get when the arc is exactly as long as the radius. That is why the division on this page has no π in it: π only appears when you convert the result into degrees.

References

Related calculators

Arc Length CalculatorThe length of the arc between two points on a circle.Sector Area CalculatorThe slice of a circle cut by two radii: area, arc length and chord from the angle.Segment Area CalculatorArea between a chord and its arc, from radius and angle.Circle CalculatorRadius, diameter, circumference and area from any one of them.