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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]
A right trapezoid (also called right-angled trapezoid) has two adjacent right angles. [13] Right trapezoids are used in the trapezoidal rule for estimating areas under a curve. An acute trapezoid has two adjacent acute angles on its longer base edge. An obtuse trapezoid on the other hand has one acute and one obtuse angle on each base.
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 f ( x ) {\displaystyle f(x)} as a trapezoid and calculating its area.
7.1 Heron-type formula for the volume of a ... Area calculator a: 3: ... Heron's formula is also a special case of the formula for the area of a trapezoid or ...
Interpolation with polynomials evaluated at equally spaced points in [,] yields the Newton–Cotes formulas, of which the rectangle rule and the trapezoidal rule are examples. Simpson's rule, which is based on a polynomial of order 2, is also a Newton–Cotes formula.
The use of trapezoidal rule in AUC calculation was known in literature by no later than 1975, in J.G. Wagner's Fundamentals of Clinical Pharmacokinetics. A 1977 article compares the "classical" trapezoidal method to a number of methods that take into account the typical shape of the concentration plot, caused by first-order kinetics. [8]
Simpson's rules are used to calculate the volume of lifeboats, [6] and by surveyors to calculate the volume of sludge in a ship's oil tanks. For instance, in the latter, Simpson's 3rd rule is used to find the volume between two co-ordinates. To calculate the entire area / volume, Simpson's first rule is used. [7]
By this usage, the area of a parallelogram or the volume of a prism or cylinder can be calculated by multiplying its "base" by its height; likewise, the areas of triangles and the volumes of cones and pyramids are fractions of the products of their bases and heights. Some figures have two parallel bases (such as trapezoids and frustums), both ...