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Suppose you are on a strange planet and observe, at night, that the stars do not rise and set, but circle parallel to the horizon

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Suppose you are on a strange planet and observe, at night, that the stars do not rise and set, but circle parallel to the horizon. Next, you walk in a constant direction for 8000 miles (Wow! That's a long walk.), and at your new location on the planet, you find that all stars rise straight up in the east and set straight down in the west, perpendicular to the horizon. How could you determine the circumference of the planet without any further observations? What is the circumference, in miles, of the planet? Imagine you are on an asteroid. You walk for 15 miles across the surface and notice the stars seem to shift by an angle of 30 degrees in your sky. What is the circumference of this asteroid?