BGN-1.1.1

Note that the definition of trapezoid and trapezium here accord to the American definition.

Formulae 公式

triangles 三角形

Representation of symbols—Image 1.1.1-1
字母含义见图1.1.1-1
Area 面积:
S=12bh=absinC=s(sa)(sb)(sc)(sd)=abc4R=12r(a+b+c)\begin{aligned} S&=\frac{1}{2}bh\\ &=ab\sin C\\ &=\sqrt{s(s-a)(s-b)(s-c)(s-d)} \\ &=\frac{abc}{4R}\\ &=\frac{1}{2}r(a+b+c) \end{aligned} The third line is Heron’s formula, r & R are the in-radius and circumradius of the triangle.
第三行是海龙公式,而 r 和 R 分别是三角形的内接圆和外接圆的半徑。
Image 1.1.1-1 图1.1.1-1 Image 1.1.1-1 图1.1.1-1

quadrilaterals 四边形

Representation of symbols—Image 1.1.1-2
字母含义见图1.1.1-2
Square 正方形: s2s^2
Rectangle 長方形: abab
Trapezoid 梯形: 12(a+b)h\dfrac{1}{2} (a+b)h
Parallelogram 平行四边形: bhbh, absinθab\sin \theta
Rhombus 菱形: 12cd\dfrac{1}{2} cd, s2sinθs^2\sin \theta
Image 1.1.1-2 图1.1.1-2 Image 1.1.1-2 图1.1.1-2 P.S. The area of a unit rhombus is equal to sin\sin θ\theta and θ\angle \theta is acute (smaller than 9090^\circ).
注:一个边长为1的菱形的面积等于角 θ\theta正弦 且角 θ\theta 是锐角。

ellipticals 椭圆形

Circle 圆: πr2\pi r^2, where rr is the radius. πr2\pi r^2rr 代表半径。
Oval 椭圆: πab\pi ab, where aa is half the long axis, bb is half the short axis. πab\pi aba,ba,b 分别指长轴的一半和短轴的一半。

Proof 证明

quadrilaterals 四边形

Imagine a rectangle grid (chocolate bar) with aa rows and bb columns of unit grids (squares of side length 1 unit). It will have aa rows of length bb, i.e. aa rows of area bb, so it’s size is b+b+...+ba times=ab.\underset{\text{a times}}{\underbrace{b+b+...+b}}=ab. Same for squares.
想象一个有a行,b列的长方形网格(或植物大战僵尸场地)。它有a个b格长的列,即a个面积为b的列,所以其面积为 b+b+...+ba 个 b=ab.\underset{\text{a 个 b}}{\underbrace{b+b+...+b}}=ab. 正方形同理。

Image 1.1.1-3 图1.1.1-3 Image 1.1.1-3 图1.1.1-3
As for parallelograms (including rhombuses), we can split them into two like this \rightarrow
so it’s area is 12bh\dfrac{1}{2} bh.
对于平行四边形(和菱形),可以拼切成长方形处理(如右图),得面积公式为 12bh\dfrac{1}{2} bh

Two identical trapezoids can be put together to form a parallelogram with base a+ba+b, height hh, so one trapezoid has size 12(a+b)h\dfrac{1}{2} (a+b)h.
两个全等的梯形可以拼合成一个底为 a+ba+b ,高为 hh 的平行四边形,所以一个梯形的大小就是 12(a+b)h\dfrac{1}{2} (a+b)h

The other formulae of quadrilaterals mentioned above will not be specifically explained, but you can check the wiki and solve yourself. 其他四边形的上述公式不在此详解,如有兴趣可自行推导。

triangles 三角形

Most triangle size formulae are given by half the size formulae of parallelograms.

ellipticals 椭圆形

Image 1.1.1-4 图1.1.1-4 Image 1.1.1-4 图1.1.1-4 Split a circle into equal pieces. Arranging, we get a ‘rectangle’. The more pieces the circle is split in, the more close the arranged shape is to a rectangle with base πr\pi r, height rr.
将一个圆形分成全等的几份。重新拼接后,我们可以得到一个近似长方形。当圆形被分成的份数增加,拼接后的图形就会趋向于一个底为 πr\pi r ,高为 rr 的长方形。

Techniques 技巧

For other polygons, you can either split them up or think of them as a few small polygons taken from a big one.
对于其他多边形,我们可以将其视为一些图形的结合或一个被裁减的大多边形。 Image 1.1.1-5 图1.1.1-5 Image 1.1.1-5 图1.1.1-5

Note by Jeff Giff
3 months, 4 weeks ago

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Report to moderators: This happened when I tried to edit my OTHER note but forgot to close a LaTeX bracket. Then the note malfunctioned and no matter how I refreshed my browser it still showed up wrong. :( Please help! 🙏
@Brilliant Mathematics

Jeff Giff - 3 months, 3 weeks ago

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Hi Jeff, this page looks fine by me. Do you still face the same issue?

Brilliant Mathematics Staff - 3 months, 3 weeks ago

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THIS page is fine, but it’s my OTHER page that malfunctioned :)

Jeff Giff - 3 months, 2 weeks ago

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@Jeff Giff Hi Jeff, the other pages look fine by me as well.

Please send me the relevant URL of your notes that have been affected. Thank you.

Brilliant Mathematics Staff - 3 months, 2 weeks ago

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@Brilliant Mathematics I know:

BGN-1.1.2

Yajat Shamji - 3 months, 2 weeks ago

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@Brilliant Mathematics

Yajat Shamji - 3 months, 3 weeks ago

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Put this one and the Bilingual Geometry Notebook in your RadMaths note!

Yajat Shamji - 3 months, 4 weeks ago

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Yup, working on that

Jeff Giff - 3 months, 4 weeks ago

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