How Do I Find the Radius of an Arc With the Delta Angle?

Arc-based calculations are important in civil engineering applications

The delta angle, as it is commonly known in trigonometric problems involving arc length, is the angle between the two lines that are tangent to each side of the arc, on the outward side of the crossing. It frequently shows up in civil engineering courses. The delta angle is actually equivalent to the central angle of the problem, or the angle between the two radii that connects the end points of the arc to the center of the circle upon which the arc lies. You can use several different calculations to determine the radius using the delta angle.

Instructions

1. Draw the problem. It should look like a wedge of pie. Label the delta angle as the angle at the vertex of the wedge. The round perimeter opposite the delta angle is the arc, and the other two sides are the radii whose length you are calculating. Draw a straight line connecting the two radii at the top--severing the arc at the points where it connects to the radii--as the triangle that this forms will be useful.

2. Label each of the angles formed by the intersections of lines and the arc. The two angles between the radii and the base of the triangle drawn at the end of Step 1 are equal to the delta angle divided by 2. The two angles formed by the triangle base and the sides of the arc are also equal to delta over 2.

3. Write what the value of delta is in radians. If it is given in degrees, convert to radians by multiplying the value by 2(pi) and dividing by 360.

4. Write down the length that is known and label the appropriate line in the problem with this value. You must know at least one length to calculate the length of the radius. Remember, angles are simply useful values that will help us scale lengths as required in trigonometric problems--there is no way to find a distance from an angle value alone.

5. Use trigonometry to calculate the radius. If the arc length is known, simply plug that value into the equation r = s/delta, where s is the arc length, to determine r. If the length of the triangle base is known, r will be that length scaled by cos(delta/2). If the line connecting the base of the triangle to the top-most point of the arc (middle ordinate M) is known, r will be equal to M(cos(delta/2))/tan(delta/2). If the perpendicular length L from the middle of the triangle base to the vertex of the wedge is known, r will be equal to L/sin(delta/2).

References

• Iowa State University Civil Engineering: Horizontal Curves
• TheMathPage: Arc Length--Topics in Trigonometry