Area calculator
Area formulas by shape: rectangle is length × width, circle is πr², triangle is half base × height, trapezoid is half the sum of the parallel sides × height, and ellipse is π × the two semi-axes multiplied.
How to calculate area
The one that catches people out is the ellipse perimeter, which has no exact closed form at all: it needs an elliptic integral. Ramanujan published an approximation in 1914 that is accurate to better than one part in ten million for anything but the most extreme ellipse, and that is what is used here. It is a nice reminder that "simple shape" and "simple formula" are not the same thing: the area of an ellipse is trivial and its perimeter is genuinely hard.
Questions
π times the radius squared. A radius of 5 gives 78.54.
Half the base times the perpendicular height. For three known sides, use Heron formula in the triangle calculator.
Half the sum of the two parallel sides, times the perpendicular height between them.
It has no exact elementary formula. Ramanujan approximation is used here and is accurate to about one part in ten million.
Split it into rectangles, triangles and circle segments, calculate each, and add them.