If a sphere is not Euclidean, why can small triangles on Earth be nearly 180°?

Conditions

• The triangle is small relative to the Earth's curvature scale.
• The region is considered locally.

Reasoning, step by step

1. Recognize that the sphere is globally non-Euclidean.
2. Understand that locally, small patches of the sphere approximate a flat plane.
3. Apply the Euclidean angle-sum rule (180°) to the small local triangle as an approximation.

Example

The video zooms into a small map region showing a red triangle labeled 90°, 50°, and 40°, whose sum is exactly 180°, illustrating the local Euclidean approximation.

Common misconceptions

• Confusing local Euclidean behavior with global Euclidean structure.
• Assuming that because small patches look flat, the whole sphere is Euclidean.

Watch the explanation

Explore next

• What is the difference between global spherical geometry and local Euclidean approximation in this clip?
• Why does the triangle drawn on Earth have angle sum greater than 180°?