if the height h is known. If the lengths of the three sides are known then Heron's formula can be used: where a, b, c are the sides of the triangle, and is half of its perimeter. If an angle and its two included sides are given, the area is where is the given angle and and are its included sides. If the triangle is graphed on a coordinate plane, a matrix can be used and is simplified to the absolute value of . This formula is also known as the shoelace formula and is an easy way to solve for the area of a coordinate triangle by substituting the 3 points (x1,y1), (x2,y2), and (x3,y3). The shoelace formula can also be used to find the areas of other polygons when their vertices are known. Another approach for a coordinate triangle is to use calculus to find the area. A simple polygon constructed on a grid of equal-distanced points (i.e., points with integer coordinates) such that all the polygon's vertices are grid points: , where i is the number of grid points inside the polygon and b is the number of boundary points. This result is known as Pick's theorem. Area in calculus The area between a positive-valued curve and the horizontal axis, measured between two values a and b (b is defined as the larger of the two values) on the horizontal axis, is given by the integral from a to b of the function that represents the curve: The area between the graphs of two functions is equal to the integral of one function, f(x), minus the integral of the other function, g(x): where is the curve with the greater y-value. An area bounded by a function expressed in polar coordinates is: The area enclosed by a parametric curve with endpoints is given by the line integrals: or the z-component of (For details, see .) This is the principle of the planimeter mechanical device. Bounded area between two quadratic functions To find the bounded area between two quadratic functions, we subtract one from the other to write the difference as where f(x) is the quadratic upper bound and g(x) is the quadratic lower bound. 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