Suppose you are an alien living on the fictitious warlike...

Suppose you are an alien living on the fictitious warlike planet Myrmidon and you want to measure its size. The Sun is shining directly down a missile silo 1000 miles to your south, while at your location the Sun is $36^{\circ}$ from straight overhead. What is the circumference of Myrmidon? What is its radius?

Key Concepts

Spherical Geometry

Spherical geometry is the study of figures on the surface of a sphere. Unlike flat (Euclidean) geometry, it deals with curved surfaces and involves concepts such as great circles and spherical triangles. Understanding spherical geometry is essential when determining distances and angles on a planetary scale, as well as when relating these angles to the total circumference of the sphere.

Central Angle and Arc Length Relationship

The relationship between a central angle and the arc length it subtends is fundamental in circle geometry. Specifically, the ratio of the arc length to the total circumference of the circle is the same as the ratio of the central angle (in degrees or radians) to the full angle of the circle. This concept allows one to compute the entire circumference of a sphere when the length of an arc and its corresponding central angle are known.

Trigonometric Techniques for Measuring Planetary Size

Trigonometric techniques involve using angles and distances to infer properties about a system. In planetary measurements, observing angles like the Sun’s elevation angle provides information about the relative position of the observer on a curved surface. By linking such observations to the known arc distance, one can apply trigonometric reasoning to calculate the planet's overall size, including its circumference and radius.

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