Area of a Triangle | 三角形的面积

📚 Area of a Triangle | 三角形的面积

The area of a triangle is one of the most fundamental topics in IGCSE Edexcel Mathematics. It appears in Paper 1 and Paper 2, in both non-calculator and calculator sections, and is often combined with trigonometry, Pythagoras’ theorem and coordinate geometry. A clear understanding of the different area formulas will help you choose the correct method quickly and avoid losing easy marks.

三角形的面积是 IGCSE Edexcel 数学中最基础的主题之一。它出现在 Paper 1 和 Paper 2 中，既可能出现在非计算器部分，也可能出现在计算器部分，并且经常与三角函数、勾股定理和坐标几何结合考查。清楚理解不同的面积公式，能帮助你快速选择正确的方法，避免丢分。

In this revision guide, we will cover the standard area formula, the sine-based formula, Heron’s formula, special triangle cases, and exam techniques that will boost your confidence.

在本复习指南中，我们将涵盖标准面积公式、基于正弦的公式、海伦公式、特殊三角形情形以及能提升你信心的考试技巧。

1. The Basic Formula | 基本公式

For any triangle, the most commonly used formula is: Area = ½ × base × height. The base can be any one side of the triangle, while the height is the perpendicular distance from that base to the opposite vertex. This perpendicular distance is sometimes called the altitude.

对于任何三角形，最常用的公式是：面积 = ½ × 底 × 高。底边可以是三角形的任意一条边，而高是从该底边到对顶点的垂直距离。这个垂直距离有时也被称为高线。

Area = ½ × b × h

Why is there a factor of ½? Because any triangle is exactly half of a rectangle or parallelogram with the same base and height. Imagine drawing a diagonal across a rectangle: it splits the rectangle into two congruent right-angled triangles. For non-right triangles, a similar parallelogram argument applies.

为什么会有 ½ 这个系数？因为任何三角形都恰好是与它同底等高的矩形或平行四边形的一半。想象一下在矩形中画一条对角线：对角线将矩形分成两个全等的直角三角形。对于非直角三角形，也可以用类似的平行四边形论证。

Example: A triangle has a base of 10 cm and a perpendicular height of 6 cm. Its area is ½ × 10 × 6 = 30 cm².

例题：一个三角形的底边长为 10 cm，垂直高为 6 cm。其面积为 ½ × 10 × 6 = 30 cm²。

Always check that the base and height use the same units before multiplying. If they are given in different units, convert one of them first.

在相乘之前，务必检查底和高的单位是否一致。如果单位不同，先进行换算。

2. Identifying Base and Height | 识别底边和高

Choosing the correct height is the single most important skill in this topic. In a right-angled triangle, the two shorter sides are perpendicular, so either one can be the base while the other is the height. In an acute triangle, the perpendicular height falls inside the triangle.

选择正确的高是本主题中最重要的技能。在直角三角形中，两条较短的边互相垂直，因此其中任意一条可作为底边，另一条作为高。在锐角三角形中，垂直高落在三角形内部。

However, in an obtuse triangle, the height drawn from an acute vertex may fall outside the triangle. For example, if the base is the side opposite the obtuse angle, you must extend the base line beyond the triangle and draw the perpendicular to that extended line. The length of this extended segment is still the height.

然而，在钝角三角形中，从锐角顶点引出的高可能落在三角形外部。例如，如果底边是钝角所对的边，你必须延长底边所在的直线，并向该延长线作垂线。这条垂线段的长度仍然是高。

• The height must form a 90° angle with the base — never use a slanted side as the height.

  高必须与底边成 90° 角——绝不要把斜边当作高来使用。

• Draw a dashed line for the height and mark the right-angle symbol in your diagram.

  用虚线画出高，并在图中标出直角符号。

• Any side can be chosen as the base; select the one that makes the height easiest to determine.

  任何一条边都可选作底边；选择使高最容易确定的那条边。

In exam questions, the perpendicular height is not always drawn for you. Reading the question carefully for the word ‘perpendicular’ will guide you to the correct height.

在考试题中，垂直高并不总是直接画出来的。仔细阅读题目中的 “perpendicular”（垂直）一词，可以引导你找到正确的高。

3. Right-Angled Triangles | 直角三角形

A right-angled triangle is the simplest case. If the two legs (the perpendicular sides) have lengths a and b, then the area is ½ × a × b. The hypotenuse is never used in the basic area formula unless a corresponding height to the hypotenuse is given.

直角三角形是最简单的情形。如果两条直角边（互相垂直的边）的长度分别为 a 和 b，则面积为 ½ × a × b。在基本面积公式中，斜边绝不直接使用，除非给出了斜边上的高。

Example 1: A right-angled triangle has legs of 5 cm and 12 cm. Area = ½ × 5 × 12 = 30 cm².

例1：一个直角三角形的直角边分别为 5 cm 和 12 cm。面积 = ½ × 5 × 12 = 30 cm²。

Example 2: A right-angled triangle has an area of 24 m² and one leg of 8 m. The other leg is found by solving ½ × 8 × h = 24, giving h = 6 m.

例2：一个直角三角形的面积为 24 m²，一条直角边为 8 m。解方程 ½ × 8 × h = 24，可得另一条直角边 h = 6 m。

This second example shows that the area formula can be rearranged to find a missing side, a very common exam skill. If you know the area and one perpendicular dimension, divide twice the area by that dimension.

第二个例子说明，面积公式可以变形来求缺失的边长，这是一个非常常见的考试技能。如果已知面积和一条垂直方向的尺寸，可用面积的两倍除以该尺寸。

4. Area Using Trigonometry: ½ ab sin C | 使用三角学的面积公式

When the perpendicular height is not given but two sides and the included angle are known, the IGCSE Edexcel syllabus requires you to use the formula: Area = ½ × a × b × sin C. Here, a and b are the lengths of two sides, and C is the angle enclosed between them.

当没有给出垂直高，但已知两条边及它们的夹角时，IGCSE Edexcel 考纲要求你使用公式：面积 = ½ × a × b × sin C。其中 a 和 b

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