The Bowness of a Circular Arc

To create a circular arc in a shapesheet’s Geometry section, requires the specification of where the arc ends and a cell called “A” that holds the measurement of how much the arc differs from a straight line between the end points of the arc. The deflection indicates how much the arc bows. It is possible to fragment a circular shape to get an idea of a value of “A”, but it is a value not a formula.

So what is a formula to describe the content of the “A” cell? To determine a formula, you need to revisit your grade school trigonometry notes. A line that connects two points a circle is called a chord and has a few special properties. The largest chord passes through the center of the circle and is called the diameter. A triangle formed by the chord and the center of the circle forms an Isosceles triangle. The angles at either end of the chord are identical. If you use the half way point on the chord, call it B, to bisect the triangle through the center of the circle, call it C, you end up with two identical right angles triangles.

The length from B to C divided by the hypotenuse of the right angle triangle is the sine of the angle of the right angle triangle at the center of the circle. In this case, the hypotenuse is the radius of the circle and the angle is half the value of the angle formd by Isosceles triangle.

So the length of BC is:  Radius x Cosine (angle/2).  

So the formula for cell “A” is:  Radius – Radius times Cosine (angle/2)  or Radius (1 – Cosine(angle/2))

John… Visio MVP

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