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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 f ( x ) {\displaystyle f(x)} as a trapezoid and calculating its area.
In geometry, a trapezoid (/ ˈ t r æ p ə z ɔɪ d /) in North American English, or trapezium (/ t r ə ˈ p iː z i ə m /) in British English, [1] [2] is a quadrilateral that has at least one pair of parallel sides. [3] The parallel sides are called the bases of the trapezoid.
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] It is called the shoelace formula because of the constant cross-multiplying for the coordinates making up the ...
The sum of all the internal angles of a simple polygon is π(n−2) radians or 180(n–2) degrees, where n is the number of sides. The formula can be proved by using mathematical induction : starting with a triangle, for which the angle sum is 180°, then replacing one side with two sides connected at another vertex, and so on.
This gives the expected results of 4 π steradians for the 3D sphere bounded by a surface of area 4πr 2 and 2 π radians for the 2D circle bounded by a circumference of length 2πr. It also gives the slightly less obvious 2 for the 1D case, in which the origin-centered 1D "sphere" is the interval [−r, r] and this is bounded by two limiting ...
In geometry, a heptagon or septagon is a seven-sided polygon or 7-gon.. The heptagon is sometimes referred to as the septagon, using "sept-" (an elision of septua-, a Latin-derived numerical prefix, rather than hepta-, a Greek-derived numerical prefix; both are cognate) together with the Greek suffix "-agon" meaning angle.
Suppose that we want to solve the differential equation ′ = (,). The trapezoidal rule is given by the formula + = + ((,) + (+, +)), where = + is the step size. [1]This is an implicit method: the value + appears on both sides of the equation, and to actually calculate it, we have to solve an equation which will usually be nonlinear.
Any non-self-crossing quadrilateral with exactly one axis of symmetry must be either an isosceles trapezoid or a kite. [5] However, if crossings are allowed, the set of symmetric quadrilaterals must be expanded to include also the crossed isosceles trapezoids, crossed quadrilaterals in which the crossed sides are of equal length and the other sides are parallel, and the antiparallelograms ...