formula for the area of any polygon with known vertex locations by Gauss in the 19th century. Areas determined using calculus The development of integral calculus in the late 17th century provided tools that could subsequently be used for computing more complicated areas, such as the area of an ellipse and the surface areas of various curved three-dimensional objects. Area formulas Polygon formulas For a non-self-intersecting (simple) polygon, the Cartesian coordinates (i=0, 1, ..., n-1) of whose n vertices are known, the area is given by the surveyor's formula: where when i=n-1, then i+1 is expressed as modulus n and so refers to 0. Rectangles The most basic area formula is the formula for the area of a rectangle. Given a rectangle with length and width , the formula for the area is: (rectangle). That is, the area of the rectangle is the length multiplied by the width. As a special case, as in the case of a square, the area of a square with side length is given by the formula: (square). The formula for the area of a rectangle follows directly from the basic properties of area, and is sometimes taken as a definition or axiom. On the other hand, if geometry is developed before arithmetic, this formula can be used to define multiplication of real numbers. Dissection, parallelograms, and triangles Most other simple formulas for area follow from the method of dissection. This involves cutting a shape into pieces, whose areas must sum to the area of the original shape. For an example, any parallelogram can be subdivided into a trapezoid and a right triangle, as shown in figure to the left. If the triangle is moved to the other side of the trapezoid, then the resulting figure is a rectangle. It follows that the area of the parallelogram is the same as the area of the rectangle: (parallelogram). However, the same parallelogram can also be cut along a diagonal into two congruent triangles, as shown in the figure to the right. It follows that the area of each triangle is half the area of the parallelogram: (triangle). Similar arguments can be used to find area formulas for the trapezoid as well as more complicated polygons. Area of curved shapes Circles The formula for the area of a circle (more properly called the area enclosed by a circle or the area of a disk) is based on