What is the name of a rectangle with two rounded ends, like a circle
with top and bottom evenly truncated? For example, an 'oval' racetrack
with two straightaways on opposite sides, parallel to each other and
both of the same length. I know that's not really an oval, but I'm
not sure what else to call it!

I am trying to calculate the midpoint between cases of legionella and
their nearest neighbor. How can I calculate the distance between two
points below which they can be treated as if they were in a plane
rather than on a sphere?

I know that an equation like 2x + y + z = 3 represents a plane in
three dimensions. How can I sketch that plane on the xyz axes? Also,
how can I sketch a system of such equations to find the solution
geometrically?

I tried to derive the formula for the surface area of a cone by taking
the integral the circumference of the solid of revolution, but it didn't
work. What did I do wrong? Can the formula be derived using this method?

A vessel in a plant where I work is the frustum of a cone on its side.
A liquid is contained in this section and pours out the end of the
cone section, therefore the liquid only takes up a certain portion of
the cone's volume. How can I compute the volume of the liquid?

Explain the derivation of the formula V = |a.(b x c)| (the volume of a
parallelepiped is equal to the magnitude of the scalar triple product
of the vectors that determine the parallelepiped; where a, b, and c
are those vectors).