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In calculus, the trapezoidal rule (also known as the trapezoid rule or trapezium rule) [a] is a technique for numerical integration, i.e., approximating the definite integral: The trapezoidal rule works by approximating the region under the graph of the function as a trapezoid and calculating its area. It follows that.
Shoelace scheme for determining the area of a polygon with point coordinates (,),..., (,). The shoelace formula, also known as Gauss's area formula and the surveyor's formula, [1] is a mathematical algorithm to determine the area of a simple polygon whose vertices are described by their Cartesian coordinates in the plane. [2]
Trapezoidal rule (differential equations) In numerical analysis and scientific computing, the trapezoidal rule is a numerical method to solve ordinary differential equations derived from the trapezoidal rule for computing integrals. The trapezoidal rule is an implicit second-order method, which can be considered as both a Runge–Kutta method ...
Heron's formula. A triangle with sides a, b, and c. In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths Letting be the semiperimeter of the triangle, the area is [1] It is named after first-century engineer Heron of Alexandria (or Hero) who ...
The area of the surface of a sphere is equal to quadruple the area of a great circle of this sphere. The area of a segment of the parabola cut from it by a straight line is 4/3 the area of the triangle inscribed in this segment. For the proof of the results Archimedes used the Method of exhaustion of Eudoxus.
Because each special triangle has area , a polygon of area will be subdivided into special triangles. [ 5 ] The subdivision of the polygon into triangles forms a planar graph , and Euler's formula V − E + F = 2 {\displaystyle V-E+F=2} gives an equation that applies to the number of vertices, edges, and faces of any planar graph.
An illustration of Monte Carlo integration. In this example, the domain D is the inner circle and the domain E is the square. Because the square's area (4) can be easily calculated, the area of the circle (π*1.0 2) can be estimated by the ratio (0.8) of the points inside the circle (40) to the total number of points (50), yielding an approximation for the circle's area of 4*0.8 = 3.2 ≈ π.
In geometry, calculating the area of a triangle is an elementary problem encountered often in many different situations. The best known and simplest formula is where b is the length of the base of the triangle, and h is the height or altitude of the triangle. The term "base" denotes any side, and "height" denotes the length of a perpendicular ...