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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.
In this example, the triangle's side lengths and area are integers, making it a Heronian triangle. However, Heron's formula works equally well when the side lengths are real numbers. As long as they obey the strict triangle inequality, they define a triangle in the Euclidean plane whose area is a positive real number.
In the case of a pyramid, the base faces are polygonal and the side faces are trapezoidal. A right frustum is a right pyramid or a right cone truncated perpendicularly to its axis; [ 3 ] otherwise, it is an oblique frustum .
The formula for the surface area of a sphere is more difficult to derive: because a sphere has nonzero Gaussian curvature, it cannot be flattened out. The formula for the surface area of a sphere was first obtained by Archimedes in his work On the Sphere and Cylinder. The formula is: [6] A = 4πr 2 (sphere), where r is the radius of the sphere.
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]
where l i is the length of each surface in contact with the aqueous body. In open channel flow, the wetted perimeter is defined as the surface of the channel bottom and sides in direct contact with the aqueous body. Friction losses typically increase with an increasing wetted perimeter, resulting in a decrease in head. [1]