How to Calculate Trigonometric Rings on a Sphere
You can draw two kinds of rings on a sphere: great circles and small circles. Either kind defines a plane passing through the sphere. For great circles, this plane passes through the center of the sphere; for small circles, it does not. Treating Earth as a sphere, the lines of longitude that run between the North and South Poles are great circles. The lines of latitude which run perpendicular to the lines of longitude are small circles, except for the equator, which is a great circle. You can calculate the circumference of any great or small circle.
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Instructions
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Great Circle Circumference
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Find either the diameter or the radius of the sphere. For Earth, the diameter is 7,926 miles.
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2
If you have the radius instead of the diameter, multiply it by two to get the diameter.
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3
Multiply the diameter by pi. For Earth, the circumference of a great circle is 7,926 miles X 3.142 = 24,900 miles. All great circles on the same sphere have the same circumference.
Small Circle Circumference
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Calculate the great circle circumference of the sphere. For Earth, this is 24,900 miles.
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5
There is only one great circle with a plane parallel to the plane of the small circle you are calculating. A line drawn from the circumference of the small circle to the center of the sphere makes an angle with this great circle's plane. Find this angle. For latitude lines on Earth, this angle is the designated latitude. As an example, take the small circle on Earth at latitude 38 degrees, 45 minutes and 59 seconds north.
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If the angle is not in degrees or fractions of a degree, convert it. There are 60 minutes in a degree and 3,600 seconds in a degree. Latitude 38 degrees, 45 minutes and 59 seconds = 38 degrees + (45 / 60) degrees + (59 / 3,600) degrees = 38.77 degrees.
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Take the cosine of the angle. The cosine of 38.77 degrees is 0.7797.
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Multiply this cosine by the great circle circumference to get the small circle circumference. The example small circle has a circumference of 0.7797 X 24,900 miles = 19,410 miles.
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Tips & Warnings
Latitudes north of the equator are positive, and latitudes south of it are negative. The negative sign does not affect the cosine function and can be dropped.
Use the same number of significant digits for all figures in your calculations and round after each step. The example calculations used four significant digits.
The algebraic formula for any circle is x squared + y squared = radius squared. For the small circle example, x squared + y squared = 376,700,000 square miles.
Figures for Earth's dimensions are approximate. Earth is not a perfect sphere. Not only is it wider around the equator than it is from pole to pole, but it is not smooth and changes over time.
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