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When the Angles of a Triangle Don't Add up to 180 Degrees

This article outlines the underlying axioms of spherical geometry giving a simple proof that the sum of the angles of a triangle on the surface of a unit sphere is equal to pi plus the area of the triangle.

Spherical Triangles on Very Big Spheres

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

 

In a previous problem, 'Pythagoras on a sphere', it was shown that for a right-angled spherical triangle with sides a, b and c drawn on a sphere of radius R we have the relationship

$$ \cos\left(\frac{a}{R}\right) = \cos\left(\frac{b}{R}\right) \cos\left(\frac{c}{R}\right) $$

a) By expanding $\cos(x)$ using the approximation $\cos(x) = 1-0.5 x^2$ show that for small triangles on large spheres the usual flat Pythagoras' theorem approximately holds.

b) In a flat salt-plane a large right angled triangle is drawn with shorter sides equal to 10km. Approximating the radius of the Earth to be 6000km, what is the percentage error in the length of the hypotenuse calculated using Pythagoras' theorem?

c) Two radio transmitters A and B are located at a distance x from each other on the equator. A third transmitter C is located at a distance x due north of A. A telecoms engineer who doesn't know about about spherical triangles needs to lay a cable between B and C and calculates the distance using the flat version of the Pythagoras theorem.

Investigate the differences and percentage errors given by the two Pythagoras theorems for different distances x. At what value of x is the percentage error to the true length of cable required greater than 0.1%?

Do you conclude that telecoms engineers need to know about spherical triangles?