Question

Suppose you made a triangle in your backyard. You used a stretched string to make three geodesics to form the sides of the triangle. You then used a protractor and found that the sum of the angles of the triangle was \(180^{\circ} .\) You know that the surface of the Earth is curved, so why didn't your triangle contain more than \(180^{\circ}\) ?

Short answer

The triangle is too small for Earth's curvature to affect the angle sum.

Step-by-step solution

Understanding Geodesics on Curved Surfaces

Geodesics are straight lines on a curved surface, much like great circles on a sphere. On a two-dimensional plane, the sum of the angles in a triangle is always \(180^{\circ}\), but this can differ on curved surfaces.

Considering the Scale of the Triangle

When forming a triangle on a spherical surface like the Earth, if the triangle is very small, the Earth's curvature has a negligible effect on the angle sum. This means the triangle behaves similarly to one constructed on a flat surface.

Comparing to Larger Triangles on Spheres

For larger triangles covering a significant area on a spherical surface, the angles' sum exceeds \(180^{\circ}\). However, for small triangles, the Earth can be approximated as flat.

Conclusion: Why the Triangle's Angles Sum to 180 Degrees

The triangle you created is small relative to the Earth's size, so the curvature does not affect the angles significantly, resulting in a sum of \(180^{\circ}\) similar to a planar triangle.

Key concepts

Geodesics

Geodesics might sound complex, but they are essentially the shortest paths between two points on a surface. For flat surfaces, geodesics are straight lines.
However, when it comes to curved surfaces, like spheres, geodesics tend to curve with the surface. They are the equivalent of straight lines on these surfaces. A familiar example of a geodesic is a great circle on a sphere, like the equator or lines of longitude on Earth.
These paths are significant in various fields, such as navigation and physics, as they represent the shortest distance. On a small enough scale, even geodesics on curved surfaces can appear straight. This principle is crucial in understanding why small-scale geodesic triangles on Earth can behave just like triangles on a flat plane.

Curved Surfaces

Curved surfaces can be found everywhere around us, from the Earth to a basketball. Unlike flat surfaces, these surfaces lack uniform straight lines.
This introduces unique geometric behaviors not seen in flat geometry. On these surfaces, traditional geometrical rules, like the sum of the angles in a triangle, can change markedly. For small regions of a curved surface, it is easy to mistake it for a flat surface.
This is because the effects of curvature are not noticeable over small distances. Yet, as the scale increases, the curvature becomes more apparent.

• Small Triangles: Curvature effect is minimal; behaves like the flat plane.
• Large Triangles: Curvature effect is significant; angle sums differ from 180°.

Understanding curved surfaces is key to many scientific phenomena, especially in navigation and astronomy.

Spherical Geometry

Spherical geometry deals with figures on the surface of a sphere. It differs greatly from plane geometry, which deals with flat surfaces. One notable aspect of spherical geometry is how it alters the sum of angles in a triangle. Spherical triangles can have angle sums greater than 180°.
Spherical geometry is significant when it comes to understanding planetary shapes and mapping the Earth, which is not a perfect sphere but close enough for this geometry to apply. For instance, the sum of angles in a large triangle on Earth can exceed 180° because of Earth's convex curvature.
This type of geometry alters the way lines are perceived; lines aren't endless but close into loops, demonstrating the distinctive properties of spheres. It provides a fascinating lens to view fields like cosmology and navigation that deal with large-scale spaces and surfaces.