📚 Area of Triangles and Parallelograms (p157_2) | 三角形与平行四边形的面积（p157_2）
is a typical exercise that asks you to apply area formulas accurately and combine shapes.
在这份 KS3 剑桥数学复习笔记中，我们聚焦一个核心几何技能：计算三角形和平行四边形的面积。标注为 p157_2 的样题是一道典型练习，要求你准确应用面积公式并组合图形。
1. What Area Means | 什么是面积
Area measures the amount of surface inside a flat shape. In KS3 Cambridge Maths, we usually measure area in square units such as square centimetres (cm²) or square metres (m²). Understanding area helps you solve real-life problems such as painting a wall, laying tiles, or comparing floor plans.
面积衡量一个平面图形内部表面的大小。在 KS3 剑桥数学中，我们通常用平方单位来度量面积，例如平方厘米（cm²）或平方米（m²）。理解面积有助于解决现实生活中的问题，例如刷墙、铺瓷砖或比较户型图。
A square with side length 1 cm has an area of 1 cm². If you cover a shape with 20 such squares, the shape has an area of 20 cm². This counting method gives the idea behind the formulas we will use.
边长为 1 cm 的正方形面积是 1 cm²。如果用 20 个这样的正方形覆盖一个图形，该图形的面积就是 20 cm²。这种计数方法正是我们接下来使用的公式背后的思想。
2. Key Formula: Parallelogram | 关键公式：平行四边形
A parallelogram is a four-sided shape with two pairs of parallel sides. Its area is found by multiplying the base by the perpendicular height, not the sloping side. The perpendicular height is the shortest distance from the base to the opposite side.
平行四边形是有两对平行边的四边形。其面积等于底乘垂直高，而不是斜边。垂直高是从底边到对边的最短距离。
A = b × h
Here, b is the base and h is the perpendicular height. If the height is not given directly, you may need to use a dashed line or a right-angle mark to identify it.
这里 b 是底，h 是垂直高。如果高没有直接给出，你可能需要通过虚线或直角标记来识别它。
- Area of parallelogram = base × perpendicular height — 平行四边形面积 = 底 × 垂直高
- Do not use the sloping side length — 不要使用斜边长度
3. Key Formula: Triangle | 关键公式：三角形
A triangle is half of a parallelogram with the same base and height. Therefore, the area of a triangle is half the base multiplied by the perpendicular height.
三角形是与它具有相同底和高的平行四边形的一半。因此，三角形的面积等于底乘垂直高的一半。
A = ½ × b × h
Choose any side as the base, but then the height must be measured at a right angle from that base to the opposite vertex.
可以选择任意一边作为底，但高必须从该底边到对顶点的垂直距离。
4. Worked Example: Parallelogram | 示例：平行四边形
Question: A parallelogram has a base of 9 cm and a perpendicular height of 6 cm. Find its area.
题目：一个平行四边形的底为 9 cm，垂直高为 6 cm。求它的面积。
Solution: Use A = b × h. Substitute b = 9 and h = 6. A = 9 × 6 = 54 cm².
解答：使用公式 A = b × h。代入 b = 9 和 h = 6。A = 9 × 6 = 54 cm²。
Always include the correct unit, cm², because you are measuring two-dimensional space.
始终带上正确的单位 cm²，因为你测量的是二维空间。
5. Worked Example: Triangle | 示例：三角形
Question: A triangle has base 10 cm and perpendicular height 8 cm. Calculate its area.
题目：一个三角形的底为 10 cm，垂直高为 8 cm。计算它的面积。
Solution: Apply A = ½ × b × h. Substitute b = 10 and h = 8. A = ½ × 10 × 8 = 40 cm².
解答：应用公式 A = ½ × b × h。代入 b = 10 和 h = 8。A = ½ × 10 × 8 = 40 cm²。
Notice that the triangle uses half of the rectangle or parallelogram built on the same base and height.
注意，三角形使用的是同底同高的矩形或平行四边形面积的一半。
6. Units of Area | 面积单位
en converting between units, remember that 1 m = 100 cm, so 1 m² = 10000 cm², not 100 cm².
面积单位总是带平方。常见单位包括 mm²、cm²、m² 和 km²。转换单位时，记住 1 m = 100 cm，因此 1 m² = 10000 cm²，而不是 100 cm²。
- 1 cm² = 100 mm² — 1 cm² = 100 mm²
- 1 m² = 10000 cm² — 1 m² = 10000 cm²
- 1 km² = 1000000 m² — 1 km² = 1000000 m²
Always check the unit given in the question and convert if necessary before using the formula.
使用公式前，务必检查题目给出的单位，必要时进行换算。
7. Finding Missing Lengths | 求缺失的长度
Some questions give you the area and one length, and ask you to find the missing base or height. Rearrange the formula to make the unknown the subject.
有些题目给出面积和一个长度，要求你求缺失的底或高。重新整理公式，使未知量成为主项。
For a parallelogram, if A = b × h, then b = A ÷ h and h = A ÷ b.
对于平行四边形，如果 A = b × h，则 b = A ÷ h，h = A ÷ b。
For a triangle, if A = ½ × b × h, then b = 2A ÷ h and h = 2A ÷ b.
对于三角形，如果 A = ½ × b × h，则 b = 2A ÷ h，h = 2A ÷ b。
Example: A triangle has area 36 cm² and base 9 cm. Find its height. h = 2 × 36 ÷ 9 = 8 cm.
示例：一个三角形的面积为 36 cm²，底为 9 cm。求它的高。h = 2 × 36 ÷ 9 = 8 cm。
8. Composite Shapes | 组合图形
In many KS3 exam questions, a shape is made from two or more basic shapes. Work out the
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