Define the discriminant of "f"("x")-"g"("x") as By simplifying the integral formula between the graphs of two functions (as given in the section above) and using Vieta's formula, we can obtain The above remains valid if one of the bounding functions is linear instead of quadratic. The general formula for the surface area of the graph of a continuously differentiable function formula_35 where formula_36 and formula_37 is a region in the xy-plane with the smooth boundary: An even more general formula for the area of the graph of a parametric surface in the vector form formula_39 where formula_40 is a continuously differentiable vector function of formula_41 is:  The above calculations show how to find the areas of many common shapes. The areas of irregular polygons can be calculated using the "Surveyor's formula". The isoperimetric inequality states that, for a closed curve of length "L" (so the region it encloses has perimeter "L") and for area "A" of the region that it encloses,  and equality holds if and only if the curve is a circle. Thus a circle has the largest area of any closed figure with a given perimeter. At the other extreme, a figure with given perimeter "L" could have an arbitrarily small area, as illustrated by a rhombus that is "tipped over" arbitrarily far so that two of its angles are arbitrarily close to 0° and the other two are arbitrarily close to 180°. For a circle, the ratio of the area to the circumference (the term for the perimeter of a circle) equals half the radius "r". This can be seen from the area formula "πr"2 and the circumference formula 2"πr". The area of a regular polygon is half its perimeter times the apothem (where the apothem is the distance from the center to the nearest point on any side). Doubling the edge lengths of a polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the dimension of the space the polygon resides in). But if the one-dimensional lengths of a fractal drawn in two dimensions are all doubled, the spatial content of the fractal scales by a power of two that is not necessarily an integer. This power is called the fractal dimension of the fractal. There are an infinitude of lines that bisect the area of a triangle.