📚 Area of Triangles and Parallelograms | 三角形与平行四边形的面积
In KS3 Cambridge Mathematics, area is the amount of surface inside a 2D shape. This article focuses on two formulas you will use constantly: the area of a parallelogram and the area of a triangle. You will also see how these formulas are connected to the rectangle formula and how to apply them to compound shapes.
在 KS3 剑桥数学中，面积是二维图形内部的表面大小。本文将重点介绍两个你会经常用到的公式：平行四边形面积和三角形面积。你还会看到这些公式如何与长方形公式联系，以及如何将它们应用到复合图形中。
1. Area vs Perimeter | 面积与周长的区别
Area measures the space inside a closed 2D figure, while perimeter measures the total length of the boundary. Always start a question by deciding whether it asks for square units (area) or linear units (perimeter).
面积衡量的是封闭二维图形内部的空间，而周长衡量的是边界线的总长度。解题时首先要判断题目要求的是平方单位（面积）还是线性单位（周长）。
The area of a rectangle is found by multiplying its length by its width: A = l × w. All area formulas in this topic build on that idea.
长方形的面积等于长乘以宽：A = l × w。本主题中的所有面积公式都基于这一思想。
| Quantity 量 | Meaning 含义 | Units 单位 |
|---|---|---|
| Perimeter | Distance around the shape | cm, m |
| Area | Surface inside the shape | cm², m² |
2. Rectangle Area Review | 回顾长方形面积
The rectangle area formula is A = l × w, where l is length and w is width. For example, a rectangle with length 8 cm and width 5 cm has area 40 cm² because 8 × 5 = 40.
长方形面积公式是 A = l × w，其中 l 是长，w 是宽。例如，一个长 8 cm、宽 5 cm 的长方形面积是 40 cm²，因为 8 × 5 = 40。
Perimeter of the same rectangle is P = 2(l + w) = 2 × (8 + 5) = 26 cm. Do not confuse this with area.
同一个长方形的周长是 P = 2(l + w) = 2 × (8 + 5) = 26 cm。不要把它和面积混淆。
3. Area of a Parallelogram | 平行四边形的面积
A parallelogram is a quadrilateral with two pairs of parallel sides. Its area is given by multiplying the base b by the perpendicular height h.
平行四边形是有两组平行边的四边形。它的面积等于底边 b 乘以垂直高度 h。
A = b × h
In a parallelogram, the height must be perpendicular to the base. If a parallelogram has base 9 cm and perpendicular height 4 cm, then A = 9 × 4 = 36 cm².
在平行四边形中，高度必须垂直于底边。如果一个平行四边形的底边为 9 cm，垂直高度为 4 cm，那么 A = 9 × 4 = 36 cm²。
4. Why the Parallelogram Formula Works | 为什么平行四边形面积公式成立
If you cut a right-angled triangle off one end of a parallelogram and move it to the other end, the shape becomes a rectangle with the same base and height. Therefore, the area is the same as base × height.
如果你从平行四边形的一端切下一个直角三角形并移到另一端，图形会变成一个底和高都与原平行四边形相同的长方形。因此，面积等于底乘高。
This is why the slant side length is not used in the area formula. Only the perpendicular distance between the parallel sides matters.
这就是为什么面积公式中不使用斜边长度。只有平行边之间的垂直距离才参与计算。
5. Area of a Triangle | 三角形的面积
Every triangle is exactly half of a parallelogram with the same base and perpendicular height. Therefore, the triangle area formula is half of base times height.
每个三角形都恰好是与它同底同高的平行四边形的一半。因此，三角形面积公式是底乘高的一半。
A = ½ × b × h
For example, a triangle with base 10 cm and height 6 cm has A = ½ × 10 × 6 = 30 cm².
例如，一个底为 10 cm、高为 6 cm 的三角形面积为 A = ½ × 10 × 6 = 30 cm²。
6. Why the Triangle Formula Works | 为什么三角形面积公式成立
Draw a rectangle around a triangle by adding two copies of the triangle, or draw a diagonal in a rectangle. The diagonal divides the rectangle into two congruent triangles, so each triangle has half the rectangle’s area.
通过在三角形周围补上两个相同的三角形，或在一个长方形中画一条对角线，你可以看到：对角线将长方形分成两个全等三角形，所以每个三角形的面积是长方形面积的一半。
If the original rectangle has length b and width h, its area is b × h. Each triangle therefore has area ½ × b × h.
如果原长方形的长是 b，宽是 h，它的面积是 b × h。因此每个三角形的面积是 ½ × b × h。
7. Choosing the Correct Base and Height | 正确选择底边与高
The base is not always the bottom side. You may choose any side as the base, but the height must be perpendicular to that chosen base. In a triangle, the height may fall inside or outside the triangle.
底边并不总是底部的边。你可以选择任意一条边作为底边，但高度必须垂直于所选择的底边。在三角形中，高度可能落在三角形内部或外部。
- In a right-angled triangle, the two legs are perpendicular; one leg can be the base and the other the height. 在直角三角形中，两条直角边互相垂直；一条直角边可作底，另一条作高。
- In an obtuse triangle, the perpendicular height may need to be drawn outside the triangle. 在钝角三角形中，垂直高度可能需要画在三角形外部。
8. Units of Area | 面积单位
Area is always measured in square units. If the lengths are given in cm, the area is in cm²; if in m, the area is in m². Converting area units involves squares of the conversion factor.
面积总是以平方单位计量。如果长度以 cm 给出，面积单位就是 cm²；如果以 m 给出，面积单位就是 m²。换算面积单位时需要使用换算因子的平方。
For example, 1 m = 100 cm, so 1 m² = 100 × 100 = 10,000 cm². This is a common exam trap.
例如，1 m = 100 cm，所以 1 m² = 100 × 100 = 10,000 cm²。这是常见的考试陷阱。
9. Compound Shapes and Problem Solving | 复合图形与问题解决
To find the area of a compound shape, split it into rectangles, parallelograms, and triangles. Find each part separately, then add or subtract as needed.
求复合图形的面积时，可以把它分解为长方形、平行四边形和三角形。分别求出每一部分的面积，然后按需相加或相减。
For example, a shape formed by a rectangle and a triangle on top can be calculated as: total area = rectangle area + triangle area.
例如，一个由长方形和其上方三角形组成的图形可以这样计算：总面积 = 长方形面积 + 三角形面积。
When a region is missing, subtract the area of the missing part from the larger rectangle. Word problems often require this subtraction method.
当图形中有缺失区域时，用大长方形的面积减去缺失部分的面积。应用题经常需要这种减法方法。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
Mistake 1: using the slant side instead of the perpendicular height. Always draw or identify a right angle between the base and height.
错误 1：使用斜边长度代替垂直高度。一定要画出或确定底边与高度之间成直角。
Mistake 2: forgetting the ½ in the triangle formula. Remember that a triangle is half a parallelogram.
错误 2：在三角形公式中忘记乘 ½。记住三角形是平行四边形的一半。
Mistake 3: mixing up perimeter and area. Perimeter has linear units; area has square units.
错误 3：混淆周长和面积。周长使用线性单位，面积使用平方单位。
- Show all steps, including substituting numbers into the formula. 写出所有步骤，包括将数值代入公式。
- Write the correct unit after every area answer. 每个面积答案后都要写明正确单位。
- Check whether a question gives base and height in different units; convert them first. 检查题目中底边和高的单位是否一致；如果不一致，要先换算。
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