Circular segment relative to circle

Circle From Chord

I was looking at some curved edging pavers online at the ‘orange box-store’ site, and saw a comment from “acer”: Info needed on circle sizes and how many needed to make different size circles so as to make a round circle without ‘weaving’.
My curiosity aroused, a google Internet search turned up some rather difficult pages (like this one). Most people I know are not truly proficient at mathematics, so for ‘acer’, and other seekers, here is my simplified method for:

How to calculate a circle from a segment or chord:

Units of Measure
While I am using inches and degrees, you can use meters and radians, as you prefer.
Circular segment relative to circle

R is the circle’s radius,
θ is the central angle,
c is the chord length,
s is the arc length,
h is the height of the segment, and d is the height of the triangular portion.

To show you how to do this, we’ll construct an example.

Let us presume that we have 5 bricks laid out in an arc (‘s’ in the diagram) around a small bush at the center; Θ’ [theta].

How many bricks do we need to completely circle the bush?

We measure the distance (length) between the inside corners of our arc (‘c’ in the diagram) to be 15½ inches (39.37 cm), and measure the inside height of our arc (‘h’ in the diagram) to be 4¼ inches (10.80 cm).

This is all the information we need.

We calculate the radius (‘R’) of the circle from (‘c’),  the length of the ‘base‘ and (‘h’), the height of our ‘chord’.

Calculate Circle Radius from Included Chord
Radius  R = h/2 + (c^2)/(8h)
Where c = 15.5 inches, and h = 4.25 inches, so …
Radius R = 4.25/2 + 15.5^2/8*4.25 
Radius R = 4.25/2 + 240.25/34
Radius R = 2.125 + 7.066=9.191 inches
So Circle Diameter = 2*R=2*9.191=18.382 inches
Simplify by Rounding
For these bricks, let’s just round that radius R to 9.25 inches (23.5 cm),  which makes the Diameter ( 2R )  18.5 inches (47 cm).

The next step is to calculate the central angle of the arc segment, Θ’ [theta], which is the angle of the circle used by our 5 bricks.  Looking at the diagram (above) we see that this can be solved using the Trigonometry Law of Cosines: knowing 3 sides of a triangle, find an opposite angle.

Triange: Angles and Opposite sides.

Triangle Angles A, B, and C are opposite Sides a, b, and c.

Triangle sides a and = Radius R, and side c = chord length c.

We now know all three sides of our triangle, and we want to know the angle opposite the long side, so let’s call the long ‘side c‘, which makes the opposite angle C, which is our ‘Θ’ [theta] segment angle.

Cosine Rule 'C'
cos(C) = (a^2 + b^2 - c^2 )/ (2ab)
Where sides  a, b = Radius (9.25 inches)
Where c = chord segment length (15.5 inches)
So cos(C)=(9.25^2+9.25^2-15.5^2)/(2*9.25*9.25)
So cos(C)=(85.56+85.56-240.25)/171.125
So cos(C)=69.125/171.125=-0.4040
And angle theta Theta=angle C=arccos(-0.4040)=113.83 degrees
Simplify by Rounding
These are bricks here, so our measurements aren’t going to be that critical! Let’s just call this 114 degrees.

Since we used 5 bricks to create the arc, each brick is:
(114 degrees)/(5 bricks)=22.8 degrees per brick
So the total number of bricks we need is:
(360 degrees)/(22.8 degrees)=15.79 bricks in the circle.

So we should get 16 bricks (for a single layer).

Note that the ‘missing’ 0.21 (one-fifth) brick in the circumference can be rounded to the nearest full brick. This slight error will not visibly affect our circle due to the natural variance of the bricks, so it can be ignored (in this case).

Accuracy Warning!
In this ‘landscaping’ example, our little circle of bricks can tolerate our ‘sloppy’ calculation and rounding some numbers. In most cases we would not do this.
REMEMBER: Mathematics is only as accurate as your data input, so always measure as accurately and as carefully as you can.

If there is enough interest, I may set a page to automatically calculate these.

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