How to Calculate Arc Length of a Circle Segment and Sector Area

Updated on November 7, 2018
eugbug profile image

Eugene is a qualified control/instrumentation engineer Bsc (Eng) and has worked as a developer of electronics & software for SCADA systems.

What is a Circle?

"A locus is a curve or other figure formed by all the points satisfying a particular equation."

A circle is a single sided shape, but can also be described as a locus of points where each point is equidistant (the same distance) from the centre.

Circumference, diameter and radius
Circumference, diameter and radius | Source

Angles in a Circle

An angle is formed when two lines or rays that are joined together at their endpoints, diverge or spread apart. Angles range from 0 to 360 degrees.
We often "borrow" letters from the Greek alphabet to use in math. So the Greek letter "p" which is π (pi) and pronounced "pie" is the ratio of the circumference of a circle to the diameter.
We also use the Greek letter θ (theta) and pronounced "the - ta", for representing angles.

An angle in a circle ranges from 0 to 360 degrees
An angle in a circle ranges from 0 to 360 degrees | Source
360 degrees in a full circle
360 degrees in a full circle | Source

Parts of a Circle

A sector is a portion of a circular disk enclosed by two rays and an arc.
A segment is a portion of a circular disk enclosed by an arc and a chord.
A semi-circle is a special case of a segment, formed when the chord equals the length of the diameter.

Arc, sector, segment, rays and chord
Arc, sector, segment, rays and chord | Source

What is Pi (π) ?

Pi represented by the Greek letter π is the ratio of the circumference to the diameter of a circle. It's a non-rational number which means that it can't be expressed as a fraction in the form a/b where a and b are integers.

Pi is equal to 3.1416 rounded to 4 decimal places.

What's the Length of the Circumference of a Circle?

If the diameter of a circle is D and the radius is R.

Then the circumference C = πD

But D = 2R

So in terms of the radius R

C = πD = 2πR

What's the Area of a Circle?

The area of a circle is A = πR2

But D = R/2

So the area in terms of the radius R is

A = πR2 = π (D/2)2 = πD2/4

Degrees and Radians

Angles are measured in degrees, but sometimes to make the mathematics simpler and elegant it's better to use radians which is another way of denoting an angle. A radian is the angle subtended by an arc of length equal to the radius of the circle. Basically "subtended" is a fancy way of saying that if you draw a line from both ends of the arc to the centre of the circle, this produces an angle with magnitude of 1 radian.

An arc length R equal to the radius R corresponds to an angle of 1 radian

So if the circumference of a circle is 2πR = 2π times R, the angle for a full circle will be 2π times one radian = 2π

And 360 degrees = 2π radians

A radian is the angle subtended by an arc of length equal to the radius of a circle.
A radian is the angle subtended by an arc of length equal to the radius of a circle. | Source

How to Convert From Degrees to Radians

360 degrees = 2π radians

Dividing both sides by 360 gives

1 degree = 2π /360 radians

Then multiply both sides by θ

θ degrees = (2π/360) x θ = θ(π/180) radians

So to convert from degrees to radians, multiply by π/180

How to Convert From Radians to Degrees

2π radians = 360 degrees

Divide both sides by 2π giving

1 radian = 360 / (2π) degrees

Multiply both sides by θ, so for an angle θ radians

θ radians = 360/(2π) x θ = (180/π)θ degrees

So to convert radians to degrees, multiply by 180/π

What's the Length of an Arc of a Circle for a Given Angle?

You can work out the length of an arc by calculating what fraction the angle is of the 360 degrees for a full circle.

A full 360 degree angle has an associated arc length equal to the circumference C

So 360 degrees corresponds to an arc length C = 2πR

Divide by 360 to find the arc length for one degree:

1 degree corresponds to an arc length 2πR/360

To find the arc length for an angle θ, multiply the result above by θ:

1 x θ corresponds to an arc length (2πR/360) x θ

So arc length s for an angle θ is:

s = (2πR/360) x θ = πθR/180

The derivation is much simpler for radians:

By definition, 1 radian corresponds to an arc length R

So if the angle is θ radians, multiplying by θ gives:

Arc length s = R x θ = Rθ

Arc length is Rθ when θ is in radians
Arc length is Rθ when θ is in radians | Source

What are Sine and Cosine?

A right-angled triangle has one angle measuring 90 degrees. The side opposite this angle is known as the hypotenuse and it is the longest side. Sine and cosine are trigonometric functions of an angle and are the ratios of the lengths of the other two sides to the hypotenuse of a right-angled triangle.

In the diagram below, one of the angles is represented by the Greek letter θ.

The side a is known as the "opposite" side and side b is the "adjacent" side to the angle θ.

sine θ = length of opposite side / length of hypotenuse

cosine θ = length of adjacent side / length of hypotenuse

Sine and cosine apply to an angle, not necessarily an angle in a triangle, so it's possible to just have two lines meeting at a point and to evaluate sine or cos for that angle. However sine and cos are derived from the sides of an imaginary right angled triangle superimposed on the lines. In the second diagram below, you can imagine a right angled triangle superimposed on the purple triangle, from which the opposite and adjacent sides and hypotenuse can be determined.

Over the range 0 to 90 degrees, sine ranges from 0 to 1 and cos ranges from 1 to 0

Remember sine and cosine only depend on the angle, not the size of the triangle. So if the length a changes in the diagram below when the triangle changes in size, the hypotenuse c also changes in size, but the ratio of a to c remains constant.

Sine and cosine are sometimes abbreviated to sin and cos

Sine and cosine of angles
Sine and cosine of angles | Source

How to Calculate the Area of a Sector of a Circle

The total area of a circle is πR2

This is for an angle of 2π radians

If the angle is θ, then this is θ/2π the fraction of the full angle for a circle.

So the area of the segment is this fraction multiplied by the total area of the circle

or

(θ/2π) x (πR2) = θR2/2

Area of a sector of a circle knowing the angle
Area of a sector of a circle knowing the angle | Source

How to Calculate the Length of a Chord Subtended by an Angle

The length of a chord can be calculated using the Cosine Rule.

For the triangle XYZ in the diagram below, the side opposite the angle θ is the chord with length c.

From the Cosine Rule:

c2 = R2 + R2 -2RRCos θ

Simplifying:

c2 = R2 + R2 -2R2Cos θ

or c2 = 2R2 (1 - Cos θ)

But from the half-angle formula (1- cos θ)/2 = sin 2 (θ/2) or (1- cos θ) = 2sin 2 (θ/2)

Substituting gives:

c2 = 2R2 (1 - Cos θ) = 2R22sin 2 (θ/2) = 4R2sin 2 (θ/2)

Taking square roots of both sides gives:

c = 2Rsin(θ/2)

A simpler derivation arrived at by splitting the triangle XYZ into 2 equal triangles and using the sine relationship between the opposite and hypotenuse, is shown in the calculation of segment area below.



The length of a chord
The length of a chord | Source

How to Calculate the Area of a Segment of a Circle

To calculate the area of a segment bounded by a chord and arc subtended by an angle θ , first work out the area of the triangle, then subtract this from the area of the sector, giving the area of the segment. (see diagrams below)

The triangle with angle θ can be bisected giving two right angled triangles with angles θ/2.

Sin(θ/2) = a/R

So a = RSin(θ/2) (cord length c = 2a = 2RSin(θ/2)

Cos(θ/2) = b/R

So b = RCos(θ/2)

The area of the triangle XYZ is half the base by the perpendicular height so if the base is the chord XY, half the base is a and the perpendicular height is b. So the area is:

ab

Substituting for a and b gives:

RSin(θ/2)RCos(θ/2)

= R2Sin(θ/2)Cos(θ/2)

But the double angle formula states that Sin(2θ) = 2Sin(θ)Cos(θ)

Substituting gives:

Area of the triangle XYZ = R2Sin(θ/2)Cos(θ/2) = R2 ((1/2)Sin θ) = (1/2)R2Sin θ

Also, the area of the sector is:

R2(θ/2)

And the area of the segment is the difference between the area of the sector and the triangle, so subtracting gives:

Area of segment = R2(θ/2) - (1/2)R2Sin θ

= R2/2( θ - Sin θ )

To calculate the area of the segment, first calculate the area of the triangle XYZ and then subtract it from the sector.
To calculate the area of the segment, first calculate the area of the triangle XYZ and then subtract it from the sector. | Source
Area of a segment of a circle knowing the angle
Area of a segment of a circle knowing the angle | Source

Equation of a Circle in Cartesian Coordinates

If the centre of a circle is located at the origin, we can take any point on the circumference and superimpose a right angled triangle with the hypotenuse joining this point to the centre.
Then from Pythagoras's theorem, the square on the hypotenuse equals the sum of the squares on the other two sides. If the radius of a circle is r then this is the hypotenuse of the right angled triangle so we can write the equation as:


r2 = x2 + y2

The equation of a circle with a centre at the origin is r² = x² + y²
The equation of a circle with a centre at the origin is r² = x² + y² | Source

Summary of Equations for a Circle

Quantity
Equation
Circumference
πD
Area
πR²
Arc Length
Chord Length
2Rsin(θ/2)
Sector Area
θR²/2
Segment Area
(R²/2) (θ - Sin(θ))
Circle formulas. θ is in radians.

© 2018 Eugene Brennan

Comments

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    • Larry Rankin profile image

      Larry Rankin 

      6 months ago from Oklahoma

      Very educational.

    • eugbug profile imageAUTHOR

      Eugene Brennan 

      6 months ago from Ireland

      Thanks George, I should have proof read before publishing, instead of beta testing on the readers !!

    • verdict profile image

      George Dimitriadis 

      6 months ago from Templestowe

      Hi.

      A good introduction to the basics of circle properties.

      Diagrams are clear and informative.

      Just a couple of points.

      You have So C = πD = πR/2, which should be C = πD = 2πR

      and A = πR^2 = π (D/2)2 = πD^2/2

      should be A = πR^2 = π (D/2)2 = πD^2/4

    working

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