Geometric warm-up

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Problem 1: triangle on a sphere

What is maximum sum of angles in a triangle on a sphere?

Problem 2: circle on a sphere

$^*$ Consider the sphere of radius $R$. A circle is drawn on the sphere which has radius $r$ as measured along the sphere. Find the circumference of the circle as a function of $r$.

Problem 3: density on a sphere

Suppose that galaxies are distributed evenly on a two-dimensional sphere of radius $R$ with number density $n$ per unit area. Determine the total number $N$ of galaxies inside a radius $r$. Do you see more or fewer galaxies out to the same radius, compared to the flat case?

Problem 4: angular sizes in spaces of constant curvature

An object of size $A$ is situated at distance $B$. Determine the angle at which the object is viewed in flat space and in spaces of constant (positive and negative) curvature.