A Visual Proof of Girard's Theorem

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram CDF Player or other Wolfram Language products.

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

Girard's theorem states that the area of a spherical triangle is given by the spherical excess: , where the interior angles of the triangle are , , , and the radius of the sphere is 1.

[more]

Rewriting the formula in terms of the exterior angles ', ', and ' gives the equivalent formula .

You can use the sliders to change the exterior angles of the (empty) spherical triangle, shifting the area of the two equivalent triangular gaps without changing their area. When you move the "align" slider all the way to the right, the colored spherical lunes align. A spherical lunar-shaped gap is formed with angle , whose area is double the area of the original spherical triangle. On the other hand, the area of a spherical lune is when the radius is 1. Therefore the area of the original spherical triangle is .

[less]

Contributed by: Okay Arik (March 2011)
Open content licensed under CC BY-NC-SA


Snapshots


Details



Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.
Send