# 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:**

**How to calculate a circle from a segment or chord:**

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’.

Where c = 15.5 inches, and h = 4.25 inches, so …

Radius R =

Radius R =

Radius R = inches

So Circle Diameter = inches

*bricks*, let’s just round that radius

**R**to 9.25 inches (23.5 cm), which makes the Diameter ( 2

**R**) 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.

Triangle sides ** a** and

**b**= Radius R, and side

**= chord length c.**

*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.

Where sides a, b = Radius (9.25 inches)

Where c = chord segment length (15.5 inches)

So

So

So

And angle theta degrees

*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:

degrees per brick

So the total number of bricks we need is:

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*).

*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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