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:

$\begin{aligned}
S&=\frac{1}{2}bh\\
&=ab\sin C\\
&=\sqrt{s(s-a)(s-b)(s-c)(s-d)} \\
&=\frac{abc}{4R}\\
&=\frac{1}{2}r(a+b+c)
\end{aligned}$
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

## 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: $s^2$

Rectangle 長方形: $ab$

Trapezoid 梯形: $\dfrac{1}{2} (a+b)h$

Parallelogram 平行四边形: $bh$, $ab\sin \theta$

Rhombus 菱形: $\dfrac{1}{2} cd$, $s^2\sin \theta$

Image 1.1.1-2 图1.1.1-2
**P.S. The area of a unit rhombus is equal to $\sin$ $\theta$ and $\angle \theta$ is acute (smaller than $90^\circ$).**

**注：一个边长为1的菱形的面积等于角 $\theta$ 的正弦 且角 $\theta$ 是锐角。**

## ellipticals 椭圆形

Circle 圆: $\pi r^2$, where $r$ is the radius. $\pi r^2$，$r$ 代表半径。

Oval 椭圆: $\pi ab$, where $a$ is half the long axis, $b$ is half the short axis. $\pi ab$，$a,b$ 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 $a$ rows and $b$ columns of unit grids (squares of side length 1 unit). It will have $a$ rows of length $b$, i.e. $a$ rows of **area** $b$, so it’s size is
$\underset{\text{a times}}{\underbrace{b+b+...+b}}=ab.$
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
$\underset{\text{a 个 b}}{\underbrace{b+b+...+b}}=ab.$
正方形同理。

Image 1.1.1-3 图1.1.1-3

As for parallelograms (including rhombuses), we can split them into two like this $\rightarrow$

so it’s area is $\dfrac{1}{2} bh$.

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 $\dfrac{1}{2} bh$。

**Two identical** trapezoids can be put together to form a parallelogram with base $a+b$, height $h$, so **one** trapezoid has size $\dfrac{1}{2} (a+b)h$.

**两个全等的**梯形可以拼合成一个底为 $a+b$ ，高为 $h$ 的平行四边形，所以**一个**梯形的大小就是 $\dfrac{1}{2} (a+b)h$。

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
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 $\pi r$, height $r$.

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 $\pi r$ ，高为 $r$ 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

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## Comments

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

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Yup, working on that

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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

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@Brilliant Mathematics

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

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

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

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BGN-1.1.2

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