Calculating the Radius of a Curve

The natural world is infused with curved shapes and lines, and these lines often follow the form of a curved circular arc. Understanding how to calculate the various parts of such an arc is an important mathematical tool in design. Given the width and height of an arc, you can compute its radius.

Defining the Terms

Before you can calculate the radius of a curve, you must understand the terms involved in the process. An arc is a curved line constructed as part of a circle. The radius of a circle is the distance from the center of a circle to its edge. The radius of an arc is the radius of the circle of which it is a part. A chord is a straight line constructed inside a circle or arc that connects one side of the circle or arc to the other side.

Given a circular arc connecting point A to point B, and a chord AB, the height of an arc is the length of a perpendicular line drawn from the midpoint of AB to the edge of the arc. The height is sometimes called the sagitta of the arc. The width is the length of the straight line connecting the starting point, A, of an arc to the ending point, B, of the same arc — in the above example, the length of the chord AB.

Basis of the Formula

The width, height and radius of a circular arc are sometime said to be interrelated, because it is possible — knowing any two of the components — to calculate the third. The formula for calculating the radius of an arc is derived from a succession of geometric corollaries, each building upon the previous. In particular, this formula relies on the intersecting chord theorem.

Method to Calculate Radius

To calculate the radius of an arc, take the height — “H” — of the arc and divide it by two. Call the result “C.” Now take the width — “W” — of the arc and square it by multiplying it by itself. Call the result “D.” Next multiply height, “H,” by eight and call this result “E.” Divide “D” by “E” and call the result “F.” The radius of the arc is C plus F.

Real-World Applications

Building designers, engineers, carpenters, cabinetmakers and stringed instrument artisans often need to design and build using circular arcs, so knowing how to calculate the radius of an arc is critical in these occupations and many others that rely on design and construction.