Although a sphere is not globally a Euclidean space, the laws of Euclidean geometry are good approximations locally. On sufficiently small regions of the Earth's surface, the curvature is negligible, so the angle sum of a small triangle is very nearly 180°.
Conditions: The triangle is small relative to the Earth's curvature scale.; The region is considered locally.
Although a sphere is not globally a Euclidean space, the laws of Euclidean geometry are good approximations locally. On sufficiently small regions of the Earth's surface, the curvature is negligible, so the angle sum of a small triangle is very nearly 180°.
Conditions: The triangle is small relative to the Earth's curvature scale.; The region is considered locally.
Globally, spherical geometry applies to the entire surface of a sphere, where triangles have an angle sum greater than 180°. Locally, Euclidean approximation means that on sufficiently small regions of the sphere, the geometry behaves almost like a flat plane, and triangles have an angle sum very nearly 180°.
Conditions: The comparison is between the entire sphere and a small local patch.; The clip uses Earth as the model for the sphere.
Globally, spherical geometry applies to the entire surface of a sphere, where triangles have an angle sum greater than 180°. Locally, Euclidean approximation means that on sufficiently small regions of the sphere, the geometry behaves almost like a flat plane, and triangles have an angle sum very nearly 180°.
Conditions: The comparison is between the entire sphere and a small local patch.; The clip uses Earth as the model for the sphere.
The sum of the angles of a triangle is exactly 180° in Euclidean geometry, greater than 180° in spherical (or elliptical/Riemannian) geometry, and less than 180° in hyperbolic geometry.
Conditions: The figure is a triangle.; The geometry is specified as Euclidean, spherical, or hyperbolic.
The sum of the angles of a triangle is exactly 180° in Euclidean geometry, greater than 180° in spherical (or elliptical/Riemannian) geometry, and less than 180° in hyperbolic geometry.
Conditions: The figure is a triangle.; The geometry is specified as Euclidean, spherical, or hyperbolic.
The triangle drawn on Earth has an angle sum greater than 180° because the Earth's surface is a sphere, which is a positively curved space. In spherical geometry, the sum of the angles of any triangle always exceeds 180°.
Conditions: The triangle is drawn on the surface of a sphere (Earth).; The geometry is spherical geometry.
The triangle drawn on Earth has an angle sum greater than 180° because the Earth's surface is a sphere, which is a positively curved space. In spherical geometry, the sum of the angles of any triangle always exceeds 180°.
Conditions: The triangle is drawn on the surface of a sphere (Earth).; The geometry is spherical geometry.