Properties

Circles

A circle drawn on a globe can have any length radius. But the circumference of the circle will be limited by the size of the globe (largest circumference at the equator). A circle centered at the North Pole with radius 1/2 the circumference of the Earth will be a point at the South Pole.

Hmmm...radius is 12,500 miles but circumference is zero miles?

Area of Triangles

The Euclidean area of a triangle
A = bh/2 has three different answers, depending on which side is used as the base. The new formulas are given below, where a, b, and g are the interior angle measures.

Spherical triangle area:

A = r2(a + b + g - p)

Hyperbolic triangle area:

A = p - (a + b + g)

Exterior sum of angles

In Euclidean geometry, the sum of the exterior angles of a triangle is always fixed at 360 degrees.

What about in spherical and hyperbolic geometry?

I think that the exterior sum is not a fixed number in non-Euclidean geometry. What do you think?

Pythagorean Theorem

The Pythagorean Theorem is not valid in spherical and hyperbolic geometries.

In fact, in taxicab geometry, the theorem for sides of a triangle will be the sum of the legs is equal to the hypotenuse --

a + b = c

Check it out and see!