Many geometry problems ask the solver to calculate the radius of a circle from given information about the circle, such as arc lengths, chord lengths and arc angles. While there are explicit solutions to many of these problems, in cases where the length of a chord and its corresponding arc are known and the radius is unknown, only an implicitly defined solution is possible. You can still use the Taylor series approximation of the sine of an angle to get an approximate answer, however.
- Skill level:
- Moderately Challenging
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Things you need
- Pencil and paper
Draw a straight line from the centre of the circle to the exact midpoint of the chord and arc using your ruler. The length of this line is equal to the radius of the circle and is perpendicular to the chord. Label one of the angles formed by this line and one of the arc's radii "x."
Set up equations relating the angle x, the radius length "R," the chord length "C" and the arc length "A." The sine of x is equal to the opposite over hypotenuse of the triangle formed by the line you drew, the arc's radius and half of the chord length, so sin x = C/2R. The arc angle is equal to the arc length divided by the radius, so the angle x, which is half the arc angle's measure, is equal to half of the arc length divided by the radius, or A/2R.
Combine the two equations derived in the previous step to eliminate x, leaving only the variable R and the known lengths A and C. You know that R * sin x = C/2, and you can substitute "A/2R" for x to get the equation R * sin (A/2R) = C/2. This is the implicit solution for the radius of the circle, given the arc length and chord length.
Approximate sin (A/2R) using the first two terms of Taylor series approximation of the sine: x - (x^3/6). Sin (A/2R) is approximately equal to A/2R - 1/6 * (A/2R)^3.
Plug the value from the previous step into the implicit solution from Step 3 to get the equation R * (A/2R - 1/6 * (A/2R)^3) = C/2. This simplifies to (A/2 - C/2) * R^2 = A^3/48. Solving for R yields the equation R^2 = (A^3/(24 * (A - C)). If the arc length is 4 and the chord length is 3.5, the radius squared is equal to 4 cubed divided by 12, or 5.33. The square root of 5.33 is 2.31, which is the length of the radius.
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