Spherical triangle A = 45º. B = 90º and C = 90º?

Question

If side a = 45º, side b = 90º and side c = 90º, How can I devide this spherical triangle into 3 equal areas, in relation from angle B in degrees to longitudes in degrees of side b?

Answer 1

Getting into non-Euclidean geometry. Nice. Question looks like Riemann geometry on a sphere.

Slice it into wedges dividing every 15 degrees through vertex A.

Aloha

Answer 2

This is a good question.

Sixto, I'm answering this in some detail for some of your answerers who have no clue what a spherical (Euler) triangle might be.

"A spherical triangle is a figure formed on the surface of a sphere by three great circular arcs intersecting pairwise in three vertices. The spherical triangle is the spherical analog of the planar triangle, and is sometimes called an Euler triangle (Harris and Stocker 1998). Let a spherical triangle have angles A, B, and C and (measured in radians at the vertices along the surface of the sphere) and let the sphere on which the spherical triangle sits have radius R. Then the surface area of the spherical triangle is

Area = R^2 (A + B + C - pi)

The sum of the angles of a spherical triangle is between 180 deg and 540 deg. (Zwillinger 1995, p. 469). The amount by which it exceeds 180 deg is called the spherical excess and is denoted E ...." [See source.]

For your question, simply use the equation above (has to be in radians, not degrees, however). 1 radian = 57.3 deg. Divide both sides of the equation by 3 to get an area 1/3 the size of the total area.

To show those areas on a sphere simply trisect the 45 deg angle (15 deg each sector) and draw a great circle each 15 deg from the vertex to the great circle arc between the other two vertices.

You can do this because, with your angles provided, the above equation reduces to:

Area = (R^2)(pi/4); where pi/4 converts to 45 deg. When we divide both sides of this by 3, we see the 3 divides nicely into the 45 deg vertex represented by pi/4: Area/3 = R^2(pi/4 ~ 45 deg)/3

Answer 3

Side b and side c are both 90, and these must both end at the same great circle. So a is 45/360 of that great circle. Treat the great circle as the equator and the intersection of b and c as the pole. Then any other line from the pole to the equator is equal to the two lines you have already. So you can split this into as many equal isosceles spherical triangles as you want, just by splitting up the 45 degree arc.

Answer 4

It appears that the only response worth looking so far is the first one. I don't recall the brief 2 days I spent sometime in highschool talking about stuff like this but I do remember it is possible to have funky triangles inscribed on the surface of a sphere that have angles adding up to all kinds of crazy things. As the first answer mentioned: "Ahhh, non-euclidean geometry"

Anyone else who is interested should google "spherical triangle" or look at the wiki entry below.

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*edit Well done eyeonthescreen. You have my vote.

Answer 5

How is it possible to have a "spherical triangle?" This makes no sense to me! And as far as I know, the Angles are what have degrees, not the Sides.

Ok, I stand corrected. I guess I'm not that advanced in math yet.

Answer 6

A triangle can not have two 90 degrees angles!

Answer 7

maybe this is beyond my mathematical knowledge, but i have never heard of a spherical triangle. Plus i dont think it is possible to have a triangle with two 90 degree angles. but please inform me if i am wrong.

Answer 8

Sorry boss, The sum of the angles are 180 degree. So pls check your question.