Let's say I have 8 90° triangles on a sphere, like this, where all the angles are 90° when measured:
I know that the area of one of those triangles will be (4πr2) * 1/8 as each triangle will take up 1 eighth of the sphere's surface area.
As long as each shape's area is equal, and they collectively cover the entire surface area, this should work.
Let's say there was only one of those triangles and I didn't know the percentage of the sphere's surface area it covered, I could still do some work.
If I find the diameter (πd) of a circle straight through the center of the sphere we can find the side length of the triangle on the curved surface. Since it's a sphere the side lengths should be the same on all sides. If I now used bh/2, I should have the area of that triangle minus the circle segment. To the area of that, I can use:
After adding the two, they should be equal to (4πr2) * 1/8, but it won't be.
I'm guessing this has to do with the sphere itself, and how it is curved rather than being a flat surface. The interior angle of the triangle should be (n-2) * 180° (triangle would be (3-2) * 180°). Instead, it's 270° when shown on this sphere. This would happen when stretching an 8 sided die until it becomes a sphere.
Is there a formula I can use to calculate the area of this shape on a sphere, what is it, and how can it apply to calculating areas of other shapes on a sphere?
Best Answer
Area of a spherical triangle enclosed between three geodesics is
$$ R^2 ( A+B+C- \pi) $$
where the quantity in parenthesis is known as the spherical excess. In case of one of the 8 octants you've shown its area $$= R^2( \pi/2+\pi/2+\pi/2-\pi) =\frac{\pi R^2}{2}.$$