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A 10 foot circle appears as an ellipse with minor axis of 7 feet and major axis of 10 feet. How do you calculate the viewing angle of the circle to see it an this size ellipse?

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In the below figure, your eye is far off in the $-x$ direction. The major axis of the visible ellipse is perpendicular to the page. You have a right triangle.

enter image description here

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Consider this analogy. A coin is held parallel to the plane of the eye. The diameter is observed as equally along x & y direction. Now, assume the wall to be the plane of the eye. Using a torch if light beamed on the coin the shadow is a circle representing the diameter of the coin. If the coin is tilted towards or away from the light, its shadow appears like an ellipse. The ratio of the shorter axis to the diameter represents the cosine of the angle at which the coin is tilted. In this case cos-inv(7/10) is the angle of the tilt = 45.57 degrees

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