Introduction to Pythagoras’ Theorem | 毕达哥拉斯定理简介
Pythagoras’ Theorem is one of the most famous and useful results in mathematics. Named after the ancient Greek mathematician Pythagoras (c. 570–495 BC), this theorem describes the fundamental relationship between the three sides of a right-angled triangle. For students studying the Cambridge Lower Secondary (KS3) Mathematics curriculum, mastering Pythagoras’ Theorem is an essential stepping stone toward IGCSE and beyond.
毕达哥拉斯定理是数学中最著名、最有用的结论之一。该定理以古希腊数学家毕达哥拉斯（约公元前570–495年）命名，描述了直角三角形三条边之间的基本关系。对于学习剑桥初中（KS3）数学课程的学生来说，掌握毕达哥拉斯定理是通向IGCSE及更高阶段的关键基石。
The Statement of the Theorem | 定理的陈述
In any right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (called the legs or catheti). Written as a formula:
在任何直角三角形中，斜边（直角所对的边）长度的平方等于另外两条边（称为直角边）长度的平方和。用公式表示为：
a² + b² = c²
where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides. This simple yet powerful equation allows us to calculate any side of a right-angled triangle if we know the other two.
其中 c 代表斜边的长度，a 和 b 代表另外两条边的长度。这个简单而强大的方程式使我们能够在已知另外两条边的情况下计算直角三角形的任意一条边。
Understanding the Geometry Behind the Theorem | 理解定理背后的几何意义
The theorem can be visualised geometrically: if you draw a square on each side of a right-angled triangle, the area of the square drawn on the hypotenuse equals the sum of the areas of the squares drawn on the other two sides. For a 3-4-5 triangle, this means a 3×3 square (area 9) plus a 4×4 square (area 16) together equal a 5×5 square (area 25).
这个定理可以用几何方式直观展示：如果在直角三角形的每条边上各画一个正方形，斜边上的正方形面积等于另外两条边上正方形面积之和。以3-4-5三角形为例，3×3的正方形（面积9）加上4×4的正方形（面积16）等于5×5的正方形（面积25）。
This visual proof is one of the most elegant demonstrations in all of mathematics and has been independently discovered by cultures around the world, including ancient Chinese, Indian, and Babylonian mathematicians.
这种视觉证明是整个数学中最优雅的演示之一，被世界各地的文化独立发现，包括古代中国、印度和巴比伦的数学家。
Finding the Hypotenuse | 求斜边长度
When you know the lengths of both legs (a and b), finding the hypotenuse (c) is straightforward. Simply square both legs, add them together, and take the square root:
当你知道两条直角边的长度（a 和 b）时，求斜边（c）非常简单。只需将两条直角边分别平方，相加，然后取平方根：
c = √(a² + b²)
Example 1: A right-angled triangle has legs of 6 cm and 8 cm. Find the hypotenuse.
示例 1：一个直角三角形的直角边分别为 6 厘米和 8 厘米。求斜边长度。
c² = 6² + 8² = 36 + 64 = 100
c = √100 = 10 cm
Example 2: A ladder leans against a wall. The foot of the ladder is 3 metres from the wall, and the top reaches 4 metres up the wall. How long is the ladder?
示例 2：一架梯子靠在墙上。梯脚离墙 3 米，梯顶到达墙上 4 米高。梯子有多长？
Ladder length² = 3² + 4² = 9 + 16 = 25
Ladder length = √25 = 5 metres
梯子长度² = 3² + 4² = 9 + 16 = 25
梯子长度 = √25 = 5 米
Finding a Shorter Side | 求直角边长度
When you know the hypotenuse and one leg, you can find the missing leg by rearranging the formula:
当你已知斜边和一条直角边时，可以通过重新排列公式来求另一条直角边：
a = √(c² − b²) or b = √(c² − a²)
Example 3: A right-angled triangle has a hypotenuse of 13 cm and one leg of 5 cm. Find the other leg.
示例 3：一个直角三角形的斜边为 13 厘米，一条直角边为 5 厘米。求另一条直角边。
b² = 13² − 5² = 169 − 25 = 144
b = √144 = 12 cm
Notice that 5-12-13 is another Pythagorean triple, just like 3-4-5.
注意，5-12-13也是一组勾股数，就像3-4-5一样。
Pythagorean Triples | 勾股数（毕达哥拉斯三元组）
A Pythagorean triple consists of three positive integers a, b, and c that satisfy a² + b² = c². These special sets of numbers are invaluable for quick mental calculations and appear frequently in exam questions. The most common triples are:
勾股数（毕达哥拉斯三元组）由三个正整数 a、b 和 c 组成，满足 a² + b² = c²。这些特殊的数字组对于快速心算非常有用，并且在考试题目中频繁出现。最常见的勾股数有：
| a | b | c |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 7 | 24 | 25 |
| 8 | 15 | 17 |
| 9 | 40 | 41 |
Multiples of these triples also work: for example, 6-8-10 (2 × 3-4-5) and 10-24-26 (2 × 5-12-13).
这些三元组的倍数也同样成立：例如 6-8-10（3-4-5的两倍）和 10-24-26（5-12-13的两倍）。
Real-World Applications | 实际应用
Pythagoras’ Theorem is not just an abstract mathematical concept — it has countless practical applications in everyday life and various professions:
毕达哥拉斯定理不仅仅是一个抽象的数学概念——它在日常生活和各种职业中有着无数的实际应用：
1. Construction and Architecture | 建筑与施工：Builders use the 3-4-5 rule to ensure walls are perpendicular. By measuring 3 units along one wall, 4 units along the other, and checking that the diagonal is exactly 5 units, they can confirm a perfect right angle.
建筑工人使用 3-4-5 法则来确保墙壁垂直。沿着一面墙量出 3 个单位，沿着另一面墙量出 4 个单位，检查对角线是否刚好为 5 个单位，就可以确认完美的直角。
2. Navigation | 导航：Ships and aircraft use Pythagoras’ Theorem to calculate the shortest distance between two points when traveling at an angle to the grid lines (latitude and longitude).
船舶和飞机使用毕达哥拉斯定理来计算与网格线（经纬度）成一定角度时两点之间的最短距离。
3. Computer Graphics | 计算机图形学：The distance between any two pixels on a screen is calculated using Pythagoras’ Theorem. This is fundamental to rendering, collision detection in games, and GPS systems.
屏幕上任意两个像素之间的距离使用毕达哥拉斯定理计算。这是渲染、游戏中的碰撞检测和 GPS 系统的基础。
4. Sports | 体育：In football, a player running diagonally across the pitch covers a distance that can be calculated using Pythagoras’ Theorem. Coaches use this to analyse player movement and positioning.
在足球中，球员沿对角线跑过球场所覆盖的距离可以使用毕达哥拉斯定理计算。教练用它来分析球员的移动和站位。
5. Astronomy | 天文学：Astronomers use the theorem to calculate distances to stars and planets using parallax measurements.
天文学家使用该定理通过视差测量来计算恒星和行星的距离。
The Converse of Pythagoras’ Theorem | 毕达哥拉斯定理的逆定理
The converse of Pythagoras’ Theorem is equally important: if the square of the longest side of a triangle equals the sum of the squares of the other two sides, then the triangle is right-angled. This provides a powerful method for determining whether a triangle contains a right angle without measuring angles directly:
毕达哥拉斯定理的逆定理同样重要：如果一个三角形最长边的平方等于另外两条边的平方和，那么这个三角形是直角三角形。这提供了一种强大的方法，可以在不直接测量角度的情况下确定一个三角形是否包含直角：
Example 4: Is a triangle with sides 9 cm, 12 cm, and 15 cm right-angled?
示例 4：边长为 9 厘米、12 厘米和 15 厘米的三角形是直角三角形吗？
Check: 9² + 12² = 81 + 144 = 225
15² = 225
Since 9² + 12² = 15², the triangle IS right-angled. (This is 3 × the 3-4-5 triple.)
检查：9² + 12² = 81 + 144 = 225
15² = 225
因为 9² + 12² = 15²，所以这个三角形是直角三角形。（这是 3-4-5 勾股数的 3 倍。）
Example 5: Is a triangle with sides 7 cm, 10 cm, and 12 cm right-angled?
示例 5：边长为 7 厘米、10 厘米和 12 厘米的三角形是直角三角形吗？
Check: 7² + 10² = 49 + 100 = 149
12² = 144
Since 149 ≠ 144, this triangle is NOT right-angled.
检查：7² + 10² = 49 + 100 = 149
12² = 144
因为 149 ≠ 144，所以这个三角形不是直角三角形。
Applying Pythagoras in 3D | 在三维空间中应用毕达哥拉斯定理
For more advanced KS3 students, Pythagoras’ Theorem extends naturally into three dimensions. The length of the space diagonal of a rectangular box (cuboid) can be found by applying the theorem twice:
对于更高水平的 KS3 学生，毕达哥拉斯定理自然地延伸到三维空间。长方体的空间对角线长度可以通过两次应用该定理来求得：
d = √(l² + w² + h²)
where l, w, and h are the length, width, and height of the cuboid. This is effectively Pythagoras’ Theorem in 3D — the square of the space diagonal equals the sum of the squares of the three dimensions.
其中 l、w 和 h 分别是长方体的长、宽和高。这实际上是三维中的毕达哥拉斯定理——空间对角线的平方等于三个维度的平方和。
Example 6: Find the length of the longest diagonal of a box measuring 4 cm × 3 cm × 12 cm.
示例 6：求一个尺寸为 4 厘米 × 3 厘米 × 12 厘米的盒子中最长对角线的长度。
d² = 4² + 3² + 12² = 16 + 9 + 144 = 169
d = √169 = 13 cm
Common Mistakes and How to Avoid Them | 常见错误及如何避免
Mistake 1: Forgetting to take the square root. Students often calculate a² + b² and stop there, forgetting that this gives c², not c. Always remember the final square root step.
错误 1：忘记开平方根。学生常常计算出 a² + b² 后就停止了，忘记这得到的是 c² 而不是 c。请务必记住最后一步开平方根。
Mistake 2: Confusing which side is the hypotenuse. The hypotenuse is always the longest side and always opposite the right angle. Double-check before substituting into the formula.
错误 2：混淆哪条边是斜边。斜边始终是最长的边，始终对着直角。在代入公式前要仔细确认。
Mistake 3: Applying the theorem to non-right-angled triangles. Pythagoras’ Theorem ONLY works for right-angled triangles. If the triangle does not contain a 90° angle, you must use other methods such as the sine rule or cosine rule (covered at IGCSE).
错误 3：将定理应用于非直角三角形。毕达哥拉斯定理仅适用于直角三角形。如果三角形不包含 90° 角，则必须使用其他方法，如正弦定理或余弦定理（在 IGCSE 中学习）。
Mistake 4: Incorrect subtraction when finding a shorter side. When finding a leg, you must subtract the known leg’s square from the hypotenuse’s square (c² − a²), not the other way around. The hypotenuse is always the largest number.
错误 4：求直角边时减法顺序错误。求直角边时，必须用斜边的平方减去已知直角边的平方（c² − a²），而不是反过来。斜边始终是最大的数。
Practice Questions | 练习题
Test your understanding with these practice problems. Try to solve them before checking the answers:
用以下练习题检验你的理解。在查看答案之前先尝试自己解答：
Q1: A right-angled triangle has legs of 9 cm and 12 cm. Find the hypotenuse.
问题 1：一个直角三角形的直角边分别为 9 厘米和 12 厘米。求斜边长度。
Q2: The hypotenuse of a right-angled triangle is 17 cm. One leg is 8 cm. Find the other leg.
问题 2：一个直角三角形的斜边为 17 厘米。一条直角边为 8 厘米。求另一条直角边。
Q3: A ship sails 30 km east and then 40 km north. How far is it from its starting point?
问题 3：一艘船向东航行 30 公里，然后向北航行 40 公里。它离起点有多远？
Q4: Is a triangle with sides 20 cm, 21 cm, and 29 cm right-angled?
问题 4：边长为 20 厘米、21 厘米和 29 厘米的三角形是直角三角形吗？
Q5: A rectangular room is 8 m long and 6 m wide. What is the diagonal distance from one corner to the opposite corner?
问题 5：一个长方形房间长 8 米，宽 6 米。从一个角到对角线的距离是多少？
Answers | 答案
A1: c² = 9² + 12² = 81 + 144 = 225, c = 15 cm
A2: b² = 17² − 8² = 289 − 64 = 225, b = 15 cm
A3: d² = 30² + 40² = 900 + 1600 = 2500, d = 50 km
A4: 20² + 21² = 400 + 441 = 841; 29² = 841; YES, it is right-angled
A5: d² = 8² + 6² = 64 + 36 = 100, d = 10 m
Historical Note | 历史注记
Although named after Pythagoras, evidence suggests that the relationship between the sides of a right-angled triangle was known to Babylonian mathematicians over 1,000 years before Pythagoras was born. The Babylonian clay tablet known as Plimpton 322 (dating to around 1800 BC) contains a table of Pythagorean triples. In China, the theorem appears in the ancient mathematical text Zhoubi Suanjing (周髀算经), where it is known as the Gougu Theorem (勾股定理). The Indian mathematician Baudhayana also described the theorem in his Sulba Sutras (c. 800 BC). This fascinating piece of mathematical history shows how fundamental truths transcend cultures and eras.
虽然以毕达哥拉斯命名，但证据表明，直角三角形边之间的关系在毕达哥拉斯出生前 1000 多年就已经被巴比伦数学家所知。被称为普林顿 322（约公元前 1800 年）的巴比伦泥板上就包含了一张勾股数表。在中国，该定理出现在古代数学著作《周髀算经》中，被称为勾股定理。印度数学家 Baudhayana 也在他的 Sulba Sutras（约公元前 800 年）中描述了这个定理。这段迷人的数学历史表明，基本真理超越了文化和时代。
Summary | 总结
Pythagoras’ Theorem (a² + b² = c²) is a cornerstone of geometry that every KS3 Cambridge Mathematics student should master. It enables you to find missing sides in right-angled triangles, determine whether a triangle is right-angled (the converse), and solve a wide range of practical problems. The key skills to develop are: recognising when the theorem applies, correctly identifying the hypotenuse, substituting values accurately, and remembering to take the square root at the end. With regular practice using real-world problems and exam-style questions, you will build confidence and fluency with this essential mathematical tool.
毕达哥拉斯定理（a² + b² = c²）是几何学的基石，每位 KS3 剑桥数学学生都应该掌握。它使你能够求出直角三角形中缺失的边长，判断一个三角形是否为直角三角形（逆定理），以及解决各种各样的实际问题。需要培养的关键技能是：识别定理何时适用，正确识别斜边，准确代入数值，并记住最后开平方根。通过定期练习实际问题和考试风格的题目，你将建立对这一基本数学工具的信心和熟练度。
This article is part of the Cambridge Lower Secondary (KS3) Mathematics series at aleveler.com. For more practice questions, worked examples, and exam preparation resources across all Cambridge IGCSE and A-Level subjects, explore our Past Papers Hub.
本文是 aleveler.com 剑桥初中（KS3）数学系列的一部分。如需更多练习题、例题解析和跨所有剑桥 IGCSE 及 A-Level 科目的备考资源，请访问我们的试卷中心。
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