In two dimensional space, shapes can be considered to have an area. This is like a two dimensional analogy to length in one dimension. Let's cover the formulas for the areas of a few common shapes.
The circle is one of the most basic shapes, and it has a simple formula for the area given by
\[A = \pi r^2\]
A disc's area can be calculated very easily, by simply removing the smaller discs area
\[A = \pi R^2 - \pi r^2\]
An ellipse is a squashed circle, which has a semi-major axis \(a\) and semi-minor axis \(b\). The area is easily given by
\[A = \pi a \cdot b\]
This, hopefully, should be quite familiar to the circle.
The square has are given by
\[A = a^2\]
In general, rectangles have area length times width, so
\[A = a \cdot b\]
A parallelogram has area base times height. Imagine taking the triangle "extra" from one side and sticking it on the other to make a rectangle!
\[A = b \cdot h\]
Triangles are always half the area of the rectangle that would contain them. This makes their formula very easy despite the complex shapes they may form!
\[A = \frac{1}{2} b \cdot h\]
The trapezoid has area given by
\[A = \frac{1}{2} (a + b) h\]
What I want you to notice about all these formulas is they involve multiplying some length by some other length. Therefore, they all have units of length squared.