?????

Three shapes on a square grid — The combined area of these three shapes is approximately 15.57 squares.

百度但是，杨银付认为，仅有文件是不够的，他表示，“营造良好教育生态是一个系统工程，需要做的还有很多。

???????? ????????????????? ?????????????? ( two-dimensional figure) ???? ??????????? ???? ????????? planar lamina ????? ??????? ??????????

????????? ???????? ?????????????????????? ???????????? ???????????? ?????????????????^[?] ????????????????????????? (SI) ???? ?????? ?????????? ??????????? (m² ?????????) ???????? ??????????????????? ????????? ???????????? ? ???? ????????^[?] ? ??????????? ??????????? ????????????????? ????????????? ? ??????????? ??????????? ???????? ?????????? ?????????????????????????? ????????????????? unit square ????????? ???????? ? ??????????????? ??????????????????????????? ?????????????????? dimensionless real number ??????????????

?????? ???????????????????? ????????????????.???????????????????????? ??????????? ??????????????????????? ??????????????????? ??????????????????? ????????? ??????? ?????????????? ??????????? ???????? ?????????????????? ???????????????????? ????????????^[?] ?????????????????? ???????????????????????????? ???????????? ??????????????? ??????????? ???????????? Indeed, ????????? ?????????? ????????????????????????????? ?????????????????????????? ??????????????????????????????????^[?]

?????????

A square made of PVC pipe on grass — A square metre quadrat made of PVC pipe.

???????????????????? ??????? ?????? ?????????????????? ?????????? ???????? ??????????? (m²)? ???????????????? (cm²)? ??????????????? (mm²)? ???????????????? (km²)? ????????? (ft²)? ???????????? (yd²)? ???????????? (mi²) ???????????????????? ????????????????? ??????????????????? ??????????????? ?????????? ??????????????? ????????????????????? ???????????????

?????? SI ???????? ?????????????????????????? SI ?? ???????????? ??????? ???????????????

???????????????????????

?????????? ??? ? ?????????? ??????????????????? ???????????????????:

? ???? x ? ???? = ? ??????????? (m2)

??????????? ???????? ???????????? ???????? ???????? ?????????????? ????????????????:

? ???? x ? ???? = ? ??????????? (m²)

????????? ???? ? ?????? ???????????????????? ???????? ???????? ???????????????????????????? ?????????????????????-

? ??????? ????????? = ?,???,??? ???????????
? ??????? ???? = ??,??? ????????????????= ?,???,??? ???????
? ??????? ????????? = ??? ??????? ????????

??????????????????? ?????????

??????????????????? ????????????? ???????????????????????????????????????? ???????????? ???????????????????? ??????????????????????????????????

? ?? = ?? ????

?????????????? ???????????????? ?????????????? ??????????????????????

? ????????? = ??? ???????????

????? ??? = ??^? = ?? × ??? ?????????:

? ???????????? = ? ?????????
? ???????????? = ?,???,??? ???????????? = ??,???,??? ?????????

????????????????????? ????????????? ?????????????????????? ??????????:

? ??????????? = 6.4516 ????????????????
? ????????? = 0.09290304 ???????????
? ???????????? = 0.83612736 ???????????
? ???????????? = 2.589988110336 ????????????????

??????????????

???????????

???????(Heron (or Hero) of Alexandria)??? ???????????? ????????? ?????????? ????????? ??????????? ????????????????? Heron's formula ??? ?????????????? ????????????????? ?? ??????????????????? ???????????? ????? ?????????????? Metrica ???? ????????????? ??????????????? ??????????????? ??????????????????? ??????????????????????????????^[?] Metrica ??? ????????? ????????????? ?????????????? ???????????????? ?????????????????????? ??????????????? ?????????????????? ?????????? ????????????????????? ?????????????^[?]

Classical age? ??? ?????????? ?????????????? ??????????????????? ???????????????????? Aryabhata ??? ?????? ???????? ????? ??????????????????? ??????????????????????????? Aryabhatiya (section 2.6) ???? ?????????????

Heron ? ????????????????????? ???????????? ?????????? ????????????????? ?????????????? ???????????????? Qin Jiushao ?????????" Mathematical Treatise in Nine Sections" (??????????????: 数书九章; ???????????: 數書九章; ??????: Shùshū Jiǔzhāng; Wade–Giles: Shushu Chiuchang) ?????????????????

??????????

? ???????? ???????????(Brahmagupta)??? ?????????? ????????????????????(Brahmagupta's formula)????????????? ??????????????? ???????????? ?????????(cyclic quadrilateral)? ???????? ??????????????????????????? ??????????????? ???? ?????????? ????????????????????????????????? Carl Anton Bretschneider ????? Karl Georg Christian von Staudt ??????? ?????????? ?????????? ???????????? ??????????????? ???????????? ????????? ?????????????? ????????? ???????????????? Bretschneider's formula ?? ??????????????

?????? ????? ?????

?? ???????? ???? ??????? ????????????????????(Cartesian coordinates) ????????????????? ?? ????? ?????? ?????????????(vertex)? ??????? ????????????????????????? ?????????? ????????? ???????????????????? surveyor's formula ??????? ??????????????

??????????????????

?????

For a non-self-intersecting (simple) polygon, the Cartesian coordinates $(x_{i},y_{i})$ (i=0, 1, ..., n-1) of whose n vertices are known, the area is given by the surveyor's formula:^[?]

A={\frac {1}{2}}|\sum _{i=0}^{n-1}(x_{i}y_{i+1}-x_{i+1}y_{i})|,

where when i=n-1, then i+1 is expressed as modulus n and so refers to 0.

???????????????

A rectangle with length and width labelled — ???????????????????????? ???????? lw ????????

????????????? ????????????????? ???????????????? ?????????????????????? ????? l ????? ??? w ????????? ???????????????? ???????? ???????????????????:^[?]^[?]

A = lw (???????????????)

???????????? ??????????????? ???????? ?????????? ??????????????????????? ?????????????????????? $l = w$ ??????? ??????????????? ??????? s ?????? ???????? ???????? ???????????????????????? ???????:^[?]

A = s² (???????)

???????????????? ????????????????? ?????? ????????????????????? ????????????????????????? ??????? ???????????? ???? ?????????? ?????????????? ???????? ????????????????? ????? ??????????? ??????????????? ????????? ???????????????????????????? ????????????? ????????????? ??????????????? ???????????? ???????????????????

A diagram showing how a parallelogram can be re-arranged into the shape of a rectangle — ?????????????????? ???????

?????????????????????????? ???????????????????? ?????????

????????????????? ?????? ???????????????? ?????????????????????????? ?????????? ?????????? ?????????? ????????????????????? ????????????? ???????????????????? ???????? ??????? ????????? ??????????????????? ?????????

???????????? ??????? ???????????????????????? ?????????(????????????????)????? ???????????????????? ????????????????????????????? ??????????????????? ???????? ????????????????? ???????????????????????????????? ????????????????????? ????????????????? ????????? ?????????????????? ???????????????? ???????? ??????????????? ?????????? ?????????????????:

A = bh (???????????????)

A parallelogram split into two equal triangles — ??????? ?????????

???????????????????????? ??????????????????????????? ?????????????????? ?????????????????? ?????????? ?????????? ?????????????????????????? ????????? ?????????????????? ????????????????????? ??????????????????? ?????????????????????????????:

A={\frac {1}{2}}bh

(?????)

????????????? ????????????????????? ????????? ????????????^[??]????? ?????? ???????????? ?????????? ???????????? ???????????????^[??]

????????????????????? ???????? ????????? ???????????

?????????

A circle divided into many sectors can be re-arranged roughly to form a parallelogram — A circle can be divided into sectors which rearrange to form an approximate parallelogram.

?????????????? ???????????????????? (???????? ???????????????? ??????????????????? ??????????? ????? ???? ??????????? ?????)??? ???????????????????? ???????? ????????????????? ?????????????? ?????????????????????? ????????? ?????????? ???????????? $r$ ????????????? ??????????????? ?????????????????????? ???????????????????????????? ??????????????????????? ????????????????????????????? ???????????????????????? ??????????????????????? ????????????????????????????? ?????????????????????? ?????????? ????????????????? ??????????????????? ????????? $r$ ???????? ???????? ????????????? ???? ???? $π r$ ???????? ?????????? ?????????? ?????????????????? $r \times π r$ , ???? $π r 2$ ???????:

A = πr² (?????????)

?????????????? ???????????????????????????? ??????????????????? ???????????????????????? ???????????? ????????????????? ???????????????????? ????????????????????????? ?????????????? ???????????????????? ????????????? ????? ?????????????? ?????????????? $π r 2$ ?????????????????

?????????????????? ??????????? ???????? ??????????? ???????????? ??????????????????????? ???????? ?????????? ?????????????? method of exhaustion ??????? ?????????????? ?????????????????? ?????? ??? method of exhaustion ??????? ????????? ???????? ????????????? ???????????????????? ?????????????????????? definite integral ??????? ????????? ?????????? ???????? ??????????????????????????????:

A\;=\;2\int _{-r}^{r}{\sqrt {r^{2}-x^{2}}}\,dx\;=\;\pi r^{2}.

?????? ?????????

??????????? ??????????? ?????????? ???????????? ??????????????????????? ????????????? semi-major ????? semi-minor axes ???????????? $x$ ????? $y$ ?????? ?????????? ????????????:^[?]

A=\pi xy.

???????

↑ ^?.? ^?.? ^?.? Area Formulas? Math.com? 2 July 2012 ???? ????????????
↑ Bureau International des Poids et Mesures Resolution 12 of the 11th meeting of the CGPM (1960), retrieved 15 July 2012
↑ Mark de Berg (2000)? "Chapter 3: Polygon Triangulation"? Computational Geometry (2nd revised ed.)? Springer-Verlag? pp. 45–61? ISBN 3-540-65620-0?
↑ Boyer? Carl B. (1959)? A History of the Calculus and Its Conceptual Development? Dover? ISBN 0-486-60509-4?
↑ Heath, Thomas L. (1921)? A History of Greek Mathematics (Vol II)? Oxford University Press? pp. 321–323?
↑ Weisstein, Eric W.? Heron's Formula?
↑ Bourke? Paul (July 1988)? Calculating The Area And Centroid Of A Polygon? 16 September 2012 ???? ????????? ????????????????? 6 Feb 2013 ???? ????????????
↑ Area of Parallelogram/Rectangle? ProofWiki.org? 29 May 2016 ???? ????????????
↑ Area of Square? ProofWiki.org? 29 May 2016 ???? ????????????
↑ Averbach? Bonnie; Chein? Orin (2012)? Problem Solving Through Recreational Mathematics? Dover? p. 306? ISBN 978-0-486-13174-0
↑ Joshi? K. D. (2002)? Calculus for Scientists and Engineers: An Analytical Approach? CRC Press? p. 43? ISBN 978-0-8493-1319-6

[AF-1] ?.? ^?.? ^?.? Area Formulas? Math.com? 2 July 2012 ???? ????????????

[B-2] Bureau International des Poids et Mesures Resolution 12 of the 11th meeting of the CGPM (1960), retrieved 15 July 2012

[bkos-3] Mark de Berg (2000)? "Chapter 3: Polygon Triangulation"? Computational Geometry (2nd revised ed.)? Springer-Verlag? pp. 45–61? ISBN 3-540-65620-0?

[4] Boyer? Carl B. (1959)? A History of the Calculus and Its Conceptual Development? Dover? ISBN 0-486-60509-4?

[5] Heath, Thomas L. (1921)? A History of Greek Mathematics (Vol II)? Oxford University Press? pp. 321–323?

[6] Weisstein, Eric W.? Heron's Formula?

[7] Bourke? Paul (July 1988)? Calculating The Area And Centroid Of A Polygon? 16 September 2012 ???? ????????? ????????????????? 6 Feb 2013 ???? ????????????

[8] Area of Parallelogram/Rectangle? ProofWiki.org? 29 May 2016 ???? ????????????

[9] Area of Square? ProofWiki.org? 29 May 2016 ???? ????????????

[10] Averbach? Bonnie; Chein? Orin (2012)? Problem Solving Through Recreational Mathematics? Dover? p. 306? ISBN 978-0-486-13174-0

[11] Joshi? K. D. (2002)? Calculus for Scientists and Engineers: An Analytical Approach? CRC Press? p. 43? ISBN 978-0-8493-1319-6

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