how to area of a trapezoid ?

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3 Ways to Calculate the Area of a Trapezoid – wikiHow
28 févr. 2023The formula for the area of a trapezoid is A = ½(b 1 +b 2)h, where b 1 and b 2 are the lengths of the bases and h is the height. X Research source If you only know the side lengths of a regular trapezoid, you can break the trapezoid into simple shapes to find the height and finish your calculation.
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https://www.omnicalculator.com › math › area-of-a-trapezoid

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Area of a Trapezoid Calculator
18 déc. 2022Find the trapezoid’s height ( h ). Substitute these values into the trapezoid area formula: A = (a + b) × h / 2. You can notice that for a trapezoid with a = b (and hence c = d = h), the formula gets simplified to A = a × h, which is exactly the formula for the area of a rectangle.

https://www.khanacademy.org › math › geometry-home › geometry-area-perimeter › geometry-area-trap-composite › v › area-of-a-trapezoid-1

https://www.khanacademy.org › math › geometry-home › geometry-area-perimeter › geometry-area-trap-composite › v › area-of-a-trapezoid-1
Area of trapezoids (video) | Khan Academy
Therefore, the area of the Trapezoid is equal to [(Area of larger rectangle + Area of smaller rectangle) / 2]. You can intuitively visualise Steps 1-3 or you can even derive this expression by considering each Area portion and summing up the parts. Either way, you will get the same answer.

https://www.youtube.com › watch?v=9hISqaDb6XE

https://www.youtube.com › watch?v=9hISqaDb6XE
Area of a Trapezoid | MathHelp.com – YouTube
Students learn that a trapezoid is a quadrilateral with one pair of parallel sides, and the formula for the area of a trapezoid is 1/2 times (base 1 + base 2) times height. For example, the area …

https://www.cuemath.com › measurement › area-of-trapezoid

https://www.cuemath.com › measurement › area-of-trapezoid
Area of Trapezoid – Formula, Examples, Definition, Derivation – Cuemath
The formula for the area of a trapezoid is expressed as, A = ½ (a + b) h. where (A) is the area of a trapezoid, ‘a’ and ‘b’ are the bases (parallel sides), and ‘h’ is the height (the perpendicular distance between a and b) Example: Find the area of a trapezoid whose parallel sides are 32 cm and 12 cm, respectively, and whose height is 5 cm …

https://www.math.net › area-of-a-trapezoid

https://www.math.net › area-of-a-trapezoid
Area of a trapezoid – Math
The area of a parallelogram is A = bh. The parallelogram formed by the two congruent trapezoids has a base b 1 + b 2 and height h. Therefore, the area of this parallelogram is: A = (b 1 + b 2 )h. Since the parallelogram is made up of two congruent trapezoids, halving the above formula gives us the formula for the area of one of the trapezoids:

https://www.youtube.com › watch?v=-Bl3FlzbIfE

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Area of a trapezoid – YouTube
Area of a trapezoid, step by step, example. For more videos on finding the area of a trapezoid, visit http://MathMeeting.com. 180K views searching4math Order of Operations 1.9M views Surface…

https://tutors.com › lesson › how-to-find-the-area-of-a-trapezoid

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How To Find the Area of a Trapezoid – Tutors.com
16 janv. 2023The formula for the area of a trapezoid is the average of the bases multiplied by the altitude. In the formula, the long and short bases are a and b, and the altitude is h: area=frac {a+b} {2}h area = 2a+bh. Multiplying times frac {1} {2} 21 is the same as dividing by 2 2.

https://www.gigacalculator.com › calculators › area-of-trapezoid-calculator.php

https://www.gigacalculator.com › calculators › area-of-trapezoid-calculator.php
Area of a Trapezoid Calculator 📐
The formula for the area of a trapezoid is (base 1 + base 2) / 2 x height, as seen in the figure below: The calculation essentially relies on the fact a trapezoid’s area can be equated to that of a rectangle: (base 1 + base 2) / 2 is actually the width of a rectangle with an equivalent area.

https://en.wikipedia.org › wiki › Trapezoid

https://en.wikipedia.org › wiki › Trapezoid
Trapezoid – Wikipedia
move to sidebarhide (Top) 1Etymology and trapeziumversus trapezoid 2Inclusive versus exclusive definition 3Special cases 4Condition of existence 5Characterizations 6Midsegment and height 7Area 8Diagonals 9Other properties 10Applications Toggle Applications subsection 10.1Architecture 10.2Geometry 10.3Biology 10.4Computer engineering 11See also

This article is about a rule for approximating integrals. For the trapezoidal rule used for initial value problems, see Trapezoidal rule (differential equations) and Heun’s method.
The function f(x) (in blue) is approximated by a linear function (in red).
In calculus, the trapezoidal rule (also known as the trapezoid rule or trapezium rule; see Trapezoid for more information on terminology) is a technique for 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
An animation that shows what the trapezoidal rule is and how the error in approximation decreases as the step size decreases
The trapezoidal rule may be viewed as the result obtained by averaging the left and right Riemann sums, and is sometimes defined this way. The integral can be even better approximated by partitioning the integration interval, applying the trapezoidal rule to each subinterval, and summing the results. In practice, this chained (or composite) trapezoidal rule is usually what is meant by integrating with the trapezoidal rule. Let
be a partition of
such that
and
be the length of the
-th subinterval (that is,
), then
When the partition has a regular spacing, as is often the case, that is, when all the
have the same value
the formula can be simplified for calculation efficiency by factoring
out:.
The approximation becomes more accurate as the resolution of the partition increases (that is, for larger
, all
decrease).
As discussed below, it is also possible to place error bounds on the accuracy of the value of a definite integral estimated using a trapezoidal rule.
Illustration of chained trapezoidal rule used on an irregularly-spaced partition of
.
History[edit]
A 2016 Science paper reports that the trapezoid rule was in use in Babylon before 50 BCE for integrating the velocity of Jupiter along the ecliptic.[1]
Numerical implementation[edit]
Non-uniform grid[edit]
When the grid spacing is non-uniform, one can use the formula
wherein
Uniform grid[edit]
For a domain discretized into
equally spaced panels, considerable simplification may occur. Let
the approximation to the integral becomes
Error analysis[edit]
An animation showing how the trapezoidal rule approximation improves with more strips for an interval with
and
. As the number of intervals
increases, so too does the accuracy of the result.
The error of the composite trapezoidal rule is the difference between the value of the integral and the numerical result:
There exists a number ξ between a and b, such that[2]
It follows that if the integrand is concave up (and thus has a positive second derivative), then the error is negative and the trapezoidal rule overestimates the true value. This can also be seen from the geometric picture: the trapezoids include all of the area under the curve and extend over it. Similarly, a concave-down function yields an underestimate because area is unaccounted for under the curve, but none is counted above. If the interval of the integral being approximated includes an inflection point, the error is harder to identify.
An asymptotic error estimate for N → ∞ is given by
Further terms in this error estimate are given by the Euler–Maclaurin summation formula.
Several techniques can be used to analyze the error, including:[3]
Fourier series
Residue calculus
Euler–Maclaurin summation formula[4][5]
Polynomial interpolation[6]
It is argued that the speed of convergence of the trapezoidal rule reflects and can be used as a definition of classes of smoothness of the functions.[7]
Proof[edit]
First suppose that
and
. Let
be the function such that
is the error of the trapezoidal rule on one of the intervals,
. Then
and
Now suppose that
which holds if
is sufficiently smooth. It then follows that
which is equivalent to
, or
Since
and
,
and
Using these results, we find
and
Letting
we find
Summing all of the local error terms we find
But we also have
and
so that
Therefore the total error is bounded by
Periodic and peak functions[edit]
The trapezoidal rule converges rapidly for periodic functions. This is an easy consequence of the Euler-Maclaurin summation formula, which says that if
is
times continuously differentiable with period
where
and
is the periodic extension of the
th Bernoulli polynomial.[8] Due to the periodicity, the derivatives at the endpoint cancel and we see that the error is
.
A similar effect is available for peak-like functions, such as Gaussian, Exponentially modified Gaussian and other functions with derivatives at integration limits that can be neglected.[9] The evaluation of the full integral of a Gaussian function by trapezoidal rule with 1% accuracy can be made using just 4 points.[10] Simpson’s rule requires 1.8 times more points to achieve the same accuracy.[10][11]
Although some effort has been made to extend the Euler-Maclaurin summation formula to higher dimensions,[12] the most straightforward proof of the rapid convergence of the trapezoidal rule in higher dimensions is to reduce the problem to that of convergence of Fourier series. This line of reasoning shows that if
is periodic on a
-dimensional space with
continuous derivatives, the speed of convergence is
. For very large dimension, the shows that Monte-Carlo integration is most likely a better choice, but for 2 and 3 dimensions, equispaced sampling is efficient. This is exploited in computational solid state physics where equispaced sampling over primitive cells in the reciprocal lattice is known as Monkhorst-Pack integration.[13]
Rough functions[edit]
For functions that are not in C2, the error bound given above is not applicable. Still, error bounds for such rough functions can be derived, which typically show a slower convergence with the number of function evaluations
than the
behaviour given above. Interestingly, in this case the trapezoidal rule often has sharper bounds than Simpson’s rule for the same number of function evaluations.[14]
Applicability and alternatives[edit]
The trapezoidal rule is one of a family of formulas for numerical integration called Newton–Cotes formulas, of which the midpoint rule is similar to the trapezoid rule. Simpson’s rule is another member of the same family, and in general has faster convergence than the trapezoidal rule for functions which are twice continuously differentiable, though not in all specific cases. However, for various classes of rougher functions (ones with weaker smoothness conditions), the trapezoidal rule has faster convergence in general than Simpson’s rule.[14]
Moreover, the trapezoidal rule tends to become extremely accurate when periodic functions are integrated over their periods, which can be analyzed in various ways.[7][11] A similar effect is available for peak functions.[10][11]
For non-periodic functions, however, methods with unequally spaced points such as Gaussian quadrature and Clenshaw–Curtis quadrature are generally far more accurate; Clenshaw–Curtis quadrature can be viewed as a change of variables to express arbitrary integrals in terms of periodic integrals, at which point the trapezoidal rule can be applied accurately.
Example[edit]
The following integral is given:
Use the composite trapezoidal rule to estimate the value of this integral. Use three segments.
Find the true error
for part (a).
Find the absolute relative true error
for part (a).
Solution
The solution using the composite trapezoidal rule with 3 segments is applied as follows.
Using the composite trapezoidal rule formula
The exact value of the above integral can be found by integration by parts and is
So the true error is
The absolute relative true error is
See also[edit]
Gaussian quadrature
Newton–Cotes formulas
Rectangle method
Romberg’s method
Simpson’s rule
Volterra integral equation § Numerical Solution using Trapezoidal Rule
Notes[edit]
^ Ossendrijver, Mathieu (Jan 29, 2016). Ancient Babylonian astronomers calculated Jupiter’s position from the area under a time-velocity graph. Science. 351 (6272): 482–484. doi:10.1126/science.aad8085. PMID 26823423. S2CID 206644971.
^ Atkinson (1989, equation (5.1.7))
^ (Weideman 2002, p. 23, section 2)
^ Atkinson (1989, equation (5.1.9))
^ Atkinson (1989, p. 285)
^ Burden & Faires (2011, p. 194)
^
a b (Rahman & Schmeisser 1990)
^ Kress, Rainer (1998). Numerical Analysis, volume 181 of Graduate Texts in Mathematics. Springer-Verlag.
^ Goodwin, E. T. (1949). The evaluation of integrals of the form. Mathematical Proceedings of the Cambridge Philosophical Society. 45 (2): 241–245. doi:10.1017/S0305004100024786. ISSN 1469-8064.
^
a b c Kalambet, Yuri; Kozmin, Yuri; Samokhin, Andrey (2018). Comparison of integration rules in the case of very narrow chromatographic peaks. Chemometrics and Intelligent Laboratory Systems. 179: 22–30. doi:10.1016/j.chemolab.2018.06.001. ISSN 0169-7439.
^
a b c (Weideman 2002)
^ Euler-Maclaurin Summation Formula for Multiple Sums. math.stackexchange.com.
^ Thompson, Nick. Numerical Integration over Brillouin Zones. bandgap.io. Retrieved 19 December 2017.
^
a b (Cruz-Uribe & Neugebauer 2002)
References[edit]
Atkinson, Kendall E. (1989), An Introduction to Numerical Analysis (2nd ed.), New York: John Wiley & Sons, ISBN 978-0-471-50023-0
Rahman, Qazi I.; Schmeisser, Gerhard (December 1990), Characterization of the speed of convergence of the trapezoidal rule, Numerische Mathematik, 57 (1): 123–138, doi:10.1007/BF01386402, ISSN 0945-3245, S2CID 122245944
Burden, Richard L.; Faires, J. Douglas (2000), Numerical Analysis (7th ed.), Brooks/Cole, ISBN 978-0-534-38216-2
Weideman, J. A. C. (January 2002), Numerical Integration of Periodic Functions: A Few Examples, The American Mathematical Monthly, 109 (1): 21–36, doi:10.2307/2695765, JSTOR 2695765
Cruz-Uribe, D.; Neugebauer, C. J. (2002), Sharp Error Bounds for the Trapezoidal Rule and Simpson’s Rule (PDF), Journal of Inequalities in Pure and Applied Mathematics, 3 (4)
External links[edit]
The Wikibook A-level Mathematics has a page on the topic of: Trapezium Rule
Trapezium formula. I.P. Mysovskikh, Encyclopedia of Mathematics, ed. M. Hazewinkel
Notes on the convergence of trapezoidal-rule quadrature
An implementation of trapezoidal quadrature provided by Boost.Math
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