Draw Arc of Circle From Cord

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

In this tutorial you'll acquire about:

• names for different parts of a circle
• degrees and radians and how to catechumen between them
• chords, arcs and secants
• sine and cosine
• how to work out the length of an arc and chord
• how to calculate the area of sectors and segments
• the equation of a circle in the Cartesian coordinate organization

What is a Circle?

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

A circumvolve is a single sided shape, but tin also be described as a locus of points where each point is equidistant (the aforementioned altitude) from the centre.

Circumference, diameter and radius

© Eugene Brennan

Angle Formed past 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 frequently "borrow" letters from the Greek alphabet to employ in math and science. So for case nosotros utilise the Greek letter "p" which is π (pi) and pronounced "pie" to correspond the ratio of the circumference of a circumvolve to the diameter.
We also use the Greek letter of the alphabet θ (theta) and pronounced "the - ta", for representing angles.

An angle is formed by two rays diverging from the centre of a circle. This angle ranges from 0 to 360 degrees

Image © Eugene Brennan

360 degrees in a total circle

Image © Eugene Brennan

Parts of a Circumvolve

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 example of a segment, formed when the chord equals the length of the diameter.

Arc, sector, segment, rays and chord

Image © Eugene Brennan

What is Pi (π) ?

Pi represented by the Greek alphabetic character π is the ratio of the circumference to the bore of a circle. It's a non-rational number which means that information technology can't be expressed every bit a fraction in the form a/b where a and b are integers.

Pi is equal to three.1416 rounded to 4 decimal places.

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

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

Then the circumference C = πD

Merely D = 2R

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So in terms of the radius R

C = πD = 2πR

What's the Area of a Circle?

The area of a circle is A = πR ²

But R = D/2

Then the area in terms of the radius R is

A = πR ⁱⁱ = π (D/2)² = πD ²/4

What are Degrees and Radians?

Angles are measured in degrees, only sometimes to make the mathematics simpler and elegant it's better to use radians which is some other 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 past joining two lines from the terminate points of the arc to the eye).

An arc of length R where R is the radius of a circle, corresponds to an angle of 1 radian

So if the circumference of a circle is 2πR i.e 2π times R, the angle for a full circle will be 2π times i radian or 2π radians.

And 360 degrees = 2π radians

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

Image © Eugene Brennan

How to Convert From Degrees to Radians

360 degrees = 2π radians

Dividing both sides past 360 gives

one degree = 2π /360 radians

So multiply both sides by θ

θ degrees = (2π/360) ten 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 past 2π giving

one 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

Then 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 bending 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

Carve up by 360 to detect the arc length for i degree:

1 caste corresponds to an arc length 2πR/360

To find the arc length for an angle θ, multiply the issue in a higher place by θ:

1 x θ = θ corresponds to an arc length (2πR/360) 10 θ

So arc length s for an angle θ is:

s = (2πR/360) x θ = πRθ /180

The derivation is much simpler for radians:

By definition, 1 radian corresponds to an arc length R

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

Arc length south = R x θ = Rθ

Arc length is Rθ when θ is in radians

Image © Eugene Brennan

What are Sine and Cosine?

A right-angled triangle has i 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 of the alphabet θ.

The side a is known as the "opposite" side and side b is the "adjacent" side to the bending 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, then it'due south possible to merely have two lines coming together 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 tin imagine a correct angled triangle superimposed on the purple triangle, from which the opposite and adjacent sides and hypotenuse can be adamant.

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

Remember sine and cosine only depend on the angle, non the size of the triangle. So if the length a changes in the diagram beneath 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

Prototype © Eugene Brennan

How to Calculate the Area of a Sector of a Circumvolve

The total area of a circle is πR ⁱⁱ respective to an bending of 2π radians for the total circle.

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

Then the area of the sector is this fraction multiplied by the total surface area of the circumvolve

or

(θ/2π) x (πR ²) = θR ⁱⁱ/ii with θ in radians.

Area of a sector of a circle knowing the angle θ in radians

Image © Eugene Brennan

How to Calculate the Length of a Chord Produced 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 reverse the angle θ is the chord with length c.

From the Cosine Dominion:

c ^two = R ² + R ² -2RRcos θ

Simplifying:

c ² = R ² + R ² -iiR ⁱⁱcos θ

or c ^two = 2R ² (1 - cos θ)

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

Substituting gives:

c² = iiR ² (1 - cos θ) = 2R ²2sin ² (θ/two) = ivR ^twosin ² (θ/2)

Taking square roots of both sides gives:

c = 2Rsin(θ/two) with θ in radians.

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

The length of a chord

Image © Eugene Brennan

How to Summate the Area of a Segment of a Circumvolve

To calculate the area of a segment bounded by a chord and arc subtended past an angle θ , beginning piece of work out the surface area of the triangle, and then decrease this from the expanse of the sector, giving the area of the segment. (see diagrams below)

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

sin(θ/2) = a/R

And then a = Rsin(θ/2) (string length c = 2a = 2Rsin(θ/two)

cos(θ/ii) = b/R

So b = Rcos(θ/2)

Substituting for a and b gives:

Rsin(θ/2)Rcos(θ/2)

= R ²sin(θ/2)cos(θ/two)

But the double angle formula states that sin(2θ) = 2sin(θ)cos(θ)

Substituting gives:

Area of the triangle XYZ = R ^twosin(θ/2)cos(θ/2) = R ² ((1/2)sin θ) = (one/two)R ^twosin θ

Too, the area of the sector is:

R ²(θ/2)

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

Area of segment = R ²(θ/2) - (one/two)R ²sin θ

= (R ²/2)( θ - sin θ ) with θ in radians.

To calculate the expanse of the segment, first calculate the expanse of the triangle XYZ and then subtract it from the sector.

Image © Eugene Brennan

Area of a segment of a circle knowing the bending

Image © Eugene Brennan

Equation of a Circle in Standard Class

If the centre of a circle is located at the origin, we can take whatever point on the circumference and superimpose a right angled triangle with the hypotenuse joining this signal to the middle.
Then from Pythagoras'south 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 nosotros can write the equation as:

x ^two + y ^two = r ²

This is the equation of a circle in standard course in Cartesian coordinates.

If the circle is centred at the signal (a,b), the equation of the circle is:

(10 - a)² + (y - b)² = r ²

The equation of a circle with a centre at the origin is r² = ten² + y²

Paradigm © Eugene Brennan

Equation of a Circle in Parametric Form

Another way of representing the coordinates of a circle is in parametric form. This expresses the values for the x and y coordinates in terms of a parameter. The parameter is chosen equally the bending betwixt the ten-axis and the line joining the point (x,y) to the origin. If this bending is θ, then:

x = cos θ

y = sin θ

for 0 < θ < 360°

Summary of Equations for a Circle

Circle formulas. θ is in radians.

Quantity	Equation
Circumference	πD
Area	πR²
Arc Length	Rθ
Chord Length	2Rsin(θ/2)
Sector Surface area	R²θ/2
Segment Area	(R²/ii) (θ - sin(θ))
Perpendicular distance from circle centre to chord	Rcos(θ/2)
Angle subtended past arc	arc length / (Rθ)
Angle subtended by chord	2arcsin(chord length / (2R))

Instance

Hither's a practical instance 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 piece of work out the distance from points on the curve to the wall of the edifice (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'due south Theorem.

This commodity is accurate and truthful to the all-time of the author's knowledge. Content is for advisory or amusement purposes only and does not substitute for personal counsel or professional advice in business organisation, financial, legal, or technical matters.

© 2018 Eugene Brennan

Eugene Brennan (author) from Ireland on May 31, 2020:

Thanks Austen.

I worked out D in the diagram to a higher place knowing R and L/ii. In reality that'due south probably not necessary considering you lot may already know the altitude from the centre of the arc to the inside of the wall. Adding this to S gives you D.

Austen on May 31, 2020:

Many thank you for solving the curved wall upshot.

I' racked my brains back 45 plus used the existing formulae on your website to crevice it- simply every bit ever when someone who "knows" tells you "how" - information technology becomes so clear you wonder how you lot couldnt see it before.

Thanks for relighting the noesis thirst.

A

Eugene Brennan (author) from Ireland on Apr 30, 2020:

Hi Austen, I spent hours trying to figure this out using angles, simply it turned out that since the chord length is known between two ends of the curved wall (is this correct?), it can hands be worked out using Pythagoras'southward Theorem. I've drawn it up as an example at the lesser of the commodity, hope it helps.

Suggestion, y'all could put the values into a spreadsheet to do the calculations.

Austen SMITH on Apr 28, 2020:

Hullo I have a simple merely frustrating problem- I want to build a regular curved wall a set up distance from a direct wall - the centre of the circle /arc of the wall falls within the building.

I need to piece of work out distance from the straight wall to mensurate, at regular intervals, to create the perfect curve starting and ending on the chord (2nd) forming the altitude from the straight wall.(1st chord)

Hope you can help.

Eugene Brennan (author) from Ireland on Apr 07, 2020:

If you hateful the chord length, it's 2Rsin(θ/2).

See the derivation in a higher place.

darrell on Apr 06, 2020:

how do i calculate the length of a segment of a circle

Lakshay on September nineteen, 2019:

Adept efforts

Eugene Brennan (author) from Republic of ireland on April 05, 2019:

If yous mean you know the coordinates of the beginning and stop points of the chord, yous can work out the length of the chord using Pythagoras's theorem. And so utilise the equation for length of a chord (2Rsin(θ/2) to find θ.

Mazin Chiliad A on April 01, 2019:

Hi,

How can I summate the bending at the center of an arc knowing radius and center, start, and end points? I know how to do that if I take the length of the arc, but in my instance I don't have it.

Eugene Brennan (author) from Ireland on March nineteen, 2019:

Thanks Troy, I'll keep it in mind. Parabolas volition probably come first though.

Troy Sartain on March 19, 2019:

How almost a similar article for ellipses? Just a thought. Evidently, some other level of complexity, even if not rotated.

Larry Rankin from Oklahoma on May 19, 2018:

Very educational.

Eugene Brennan (author) from Ireland on May 18, 2018:

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

George Dimitriadis from Templestowe on May eighteen, 2018:

Hullo.

A skilful introduction to the basics of circle backdrop.

Diagrams are clear and informative.

Just a couple of points.

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

and A = πR^2 = π (D/ii)2 = πD^ii/two

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

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Source: https://owlcation.com/stem/How-to-Calculate-the-Arc-Length-of-a-Circle-Segment-and-Sector-Area

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