To me this sound like theories of geometry...I acknowledge that the sky looks circular and maybe spherical. But can anyone cite an example of a larger than 180 degree triangle being drawn on the ground. Like maybe a particular instance used to teach the principal itself.

This is easy enough to see on a globe. On the earth itself it's still true, but far more subtle at scales a person can work in (see later in this post for why).

Start at the North Pole and follow the Prime Meridian due south to the Equator.

At the Equator, take a 90° turn to the west and travel 90° along the Equator to the 90° West meridian.

At the 90° Meridian, take a 90° turn to the north and follow the 90° West meridian to the north pole.

You will approach the pole 90° from the direction you followed when starting out, closing the spherical triangle that has three 90° interior angles for a sum of 270°.

In general, the sum of the interior angles of spherical triangles is

S = 180° (1 + 4a/A)

S is the sum of the interior angles, a is the area enclosed by the triangle, and A is the surface area of the sphere.

In the example above, the area of the triangle is 1/4 the area of the northern hemisphere (this should be obvious from inspection), so it's 1/8 the area of the whole sphere.

S = 180° (1 + 4(1/8))

= 180° (1 + 1/2)

= 180° (1.5)

= 270°, as expected.

Since the surface area, A, of a sphere with radius R is

A = 4 pi R

^{2}then, for the earth, with R = 4000 miles approximately,

A = 4 pi (4000 mi)

^{2} = 4 pi 16,000,000 mi

^{2}A = 201,061,930 mi

^{2} (call it 200 million square miles).

How big would a surveyed triangle have to be to cause the interior angles to add up to 181°?

181° = 180° (1 + a/200,000,000 mi

^{2})

181° - 180° = 180° x a/200,000,000 mi

^{2}1° / 180° = a/200,000,000 mi

^{2}200,000,000 mi

^{2} / 180 = a

a = 1,111,111 mi

^{2}, about twice the area of Alaska. If this were an equilateral spherical triangle, the sides would be roughly 1600 miles long!

That's how big your survey would have to be to produce a 1° excess interior angle, so seeing this "on the ground" will not be easy even though it's real, and there.

Land surveys have much higher precision than 1°, and, as a result, when the area being surveyed exceeds 100 square miles or so, plane surveying (which approximates a limited area as being a plane) may not be accurate enough for some purposes, and much more complex geodetic surveying (which makes no such approximation, at the expense of much more difficult data reduction) is required.