Determining the surface area on a 5 sided lunar esque shape

[GeometryIsHARD]

Homework Statement

I'm trying to figure out the surface area on a 5 sided shape where the sides can all be modeled by "lunes". The shape will end up looking like a banana peel. We are modeling the sides of the shape as lunes with varying angles on a sphere of radius 3 inches. I'm trying to figure out how I would determine the surface area of the banana peel.

Homework Equations

The Attempt at a Solution

Let me begin by saying I am quite confused... I have done a lot of thinking on this problem but seemingly to no avail. If I have 5 sides, could I do a length x width for each side and multiply it by 5? I need one of you genius's on this forum to help me understand what's going on here!

 

[SteamKing]

Homework Statement

I'm trying to figure out the surface area on a 5 sided shape where the sides can all be modeled by "lunes". The shape will end up looking like a banana peel. We are modeling the sides of the shape as lunes with varying angles on a sphere of radius 3 inches. I'm trying to figure out how I would determine the surface area of the banana peel.

Homework Equations

The Attempt at a Solution

Let me begin by saying I am quite confused... I have done a lot of thinking on this problem but seemingly to no avail. If I have 5 sides, could I do a length x width for each side and multiply it by 5? I need one of you genius's on this forum to help me understand what's going on here!

What makes you think that the area of one lune = length multiplied by the width? How closely do your lunes resemble rectangles?

 

[GeometryIsHARD]

Ah, i suppose not very close, was just taking a shot in the dark. Is there a similar equation for this situation? It must be a bit more complicated because it would include angles and radius of the sphere I suppose.

 

[SteamKing]

Ah, i suppose not very close, was just taking a shot in the dark. Is there a similar equation for this situation? It must be a bit more complicated because it would include angles and radius of the sphere I suppose.

It's not clear what this shape looks like, but here is more information on the geometry of spherical lunes:

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

 

[GeometryIsHARD]

so the surface area of a spherical lune is given by 2*θ*R^2, and since there are five lunes would it just be this multiplied by 5?

 

[SteamKing]

so the surface area of a spherical lune is given by 2*θ*R^2, and since there are five lunes would it just be this multiplied by 5?

As long as each lune is identical in having the same angle θ.

You've never attached a picture of this figure, so I can't say for certain.

 

[GeometryIsHARD]

a banana peel where each side can be modeled as a lune