How to Calculate Arc Length of a Circle, Segment and Sector Area
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.
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Angle Formed by Two Rays Emanating from the Center of 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.
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.
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
What are 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. ( "Subtended" means produced by joining two lines from the end of the arc to the centre).
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
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
How to Find the Length of an Arc
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
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
How to Calculate the Area of a Sector of a Circle
The total area of a circle is πR2 corresponding to an angle of 2π radians for the full circle.
If the angle is θ, then this is θ/2π the fraction of the full angle for a circle.
So the area of the sector is this fraction multiplied by the total area of the circle
(θ/2π) x (πR2) = θR2/2
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
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)
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.
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:
Substituting for a and b gives:
But the double angle formula states that Sin(2θ) = 2Sin(θ)Cos(θ)
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:
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 )
Equation of a Circle in Standard Form
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:
x2 + y2 = r2
This is the equation of a circle in standard form in Cartesian coordinates.
If the circle is centred at the point (a,b), the equation of the circle is:
(x - a)2 + (y - b)2 = r2
Summary of Equations for a Circle
(R²/2) (θ - Sin(θ))
Perpendicular distance from circle centre to chord
Here's a practical example of using trigonometry with arcs and chords. A curved wall is built in front of a building. The wall is a section of a circle. It's necessary to work out the distance from points on the curve to the wall of the building (distance "B"), knowing the radius of curvature R, chord length L, distance from chord to wall S and distance from centre line to point on curve A. See if you can determine how the equations were derived. Hint: Use Pythagoras's Theorem.
This article is accurate and true to the best of the author’s knowledge. Content is for informational or entertainment purposes only and does not substitute for personal counsel or professional advice in business, financial, legal, or technical matters.
© 2018 Eugene Brennan