Non-Euclidean geometry can be taught to students using astronomical images. By using photographs of the Moon taken with a smartphone through a simple telescope, we were able to introduce these concepts to high-school students and lower-level college students. We teach students how to calculate lengths of mountain ranges or areas of craters on the Moon's surface and introduce ideas of geodesics and spherical triangles. Students can see that accurate measurements cannot be obtained using flat geometry. Instead, by using three-dimensional curved geometry, estimates of lengths and areas can be computed with less than 4% error.