An area is measured in something like cm2, so will always be two lengths multiplied together then times a simple number.
For a rectangle or a square, you just multiply the two lengths together.
For a triangle, it can always be considered as half a rectangle. Or rearranged into half a rectangle. So here the problem is working out which two distances you use, I'm fairly confident it's always the longest side times the distance from that side to the opposite corner, halved.
Circles are a bit smaller than a square. If you imagine a square with a circle inscribed you see that it's the diameter, times the diameter, then a bit smaller as the "corners" are rounded off. It turns out to be half the diameter, times half the diameter, times 3.14. If it actually was a square you'd be multiplying by four, but it's a bit smaller.
Trapeziums are nearly the same as rectangles. You multiply the average of the two parallel lengths by the distance between them. It'll look something like a.0.5.(b+c).
Most other shapes are dealt with by drawing arbitrary lines to break the complicated shape, which you don't know a formula for, into simple ones which you do. You could turn the trapezium into a rectangle and two triangles for example.
You don't need (or want) a calculator for this, just some time drawing pictures
