Area of triangle on sphere problem.

[toughcanesrock]

Homework Statement

What is the area of a triangle on Earth that goes from the North Pole down to the equator, through the prime meridian, across the equator to 30 degrees east longitude, then back up to the equator? The radius of the Earth is about 6378 km.

Homework Equations

alpha + beta + gamma = pi + (A/R)

r' = Rsin(r/R)

dA = R*sin(r/R)*dtheta

The Attempt at a Solution

I am having trouble with knowing how to go about solving this. I'm not 100% sure these equations would even help me get to the answer. I think some of my trouble is with understanding the wording of the problem. But even if I knew exactly what the triangle looked like, I would only know the radius of the earth, the length of equator line portion of the triangle, and possibly the spherical length from the north pole to the equator.

 

[SteamKing]

Homework Statement

What is the area of a triangle on Earth that goes from the North Pole down to the equator, through the prime meridian, across the equator to 30 degrees east longitude, then back up to the equator? The radius of the Earth is about 6378 km.

Read this problem statement carefully. Given the directions, I don't think you can draw a spherical triangle as described.

 

[toughcanesrock]

Read this problem statement carefully. Given the directions, I don't think you can draw a spherical triangle as described.

Ok, after staring at this for an hour, I think all I need help with is drawing it. It goes from NP to the equator, through PM, across equator to 30 degrees east longitude, then back up to equator... That does not make any sense to me on how to draw that. I uploaded the only way that I can think to draw it. But how does that go "through" the PM? And how can it go back up to the equator? Wouldn't it go back up to the NP?

 

[Ray Vickson]

Homework Statement

What is the area of a triangle on Earth that goes from the North Pole down to the equator, through the prime meridian, across the equator to 30 degrees east longitude, then back up to the equator? The radius of the Earth is about 6378 km.

Homework Equations

alpha + beta + gamma = pi + (A/R)

r' = Rsin(r/R)

dA = R*sin(r/R)*dtheta

The Attempt at a Solution

I am having trouble with knowing how to go about solving this. I'm not 100% sure these equations would even help me get to the answer. I think some of my trouble is with understanding the wording of the problem. But even if I knew exactly what the triangle looked like, I would only know the radius of the earth, the length of equator line portion of the triangle, and possibly the spherical length from the north pole to the equator.

I think there is a "typo". The triangle can only be a triangle if it goes from NP --> Eq. Pt A --> Eq. Pt. B --> NP. Surely you can draw that, or look on-line for appropriate diagrams.

 

[Mark44]

I think there is a "typo". The triangle can only be a triangle if it goes from NP --> Eq. Pt A --> Eq. Pt. B --> NP. Surely you can draw that, or look on-line for appropriate diagrams.

I too believe that there's a typo in the problem, either as it was originally stated or as it was written here. In either case, I believe the intent was to describe a triangular region on the Earth's surface as Ray describes it. It should be straightforward to calculate the area of such a triangular region without the need for anything very complicated.

 

[verty]

Hint: Consider the slice that reaches all the way down to the south pole and is 30° degrees wide at the equator.

 

[Merlin3189]

Looking back at your diagram I see your confusion.
You are thinking of the correct triangle (allowing for the obvious errors in the wording of the question), but you have only one right angle.
Any NS line (any line from the N pole to the equator) crosses the equator at 90^o, so your triangle should have two right angles.
That is of course difficult to draw on flat paper, but you can show it on a sketch of a sphere.

 

[Nidum]

https://en.wikipedia.org/wiki/Spherical_geometry

Enlarge the picture top right hand side of page .