Pythagoras’ Theorem | 勾股定理

📚 Pythagoras’ Theorem | 勾股定理

Pythagoras’ theorem is one of the most famous results in geometry. It describes a special relationship between the three sides of a right-angled triangle. For KS3 learners following the Cambridge curriculum, mastering this theorem means you can calculate unknown side lengths, check whether a triangle has a right angle, and solve a wide range of practical problems. This article provides a complete revision guide, worked examples, and useful tips to help you feel confident when using Pythagoras’ theorem.

勾股定理是几何学中最著名的结论之一。它描述了直角三角形三条边之间的一种特殊关系。对于学习剑桥课程的 KS3 学生来说，掌握这一定理意味着你可以计算未知的边长、判断一个三角形是否含有直角，并解决大量实际问题。本文将提供完整的复习指南、详细例题和实用技巧，帮助你自信地运用勾股定理。

1. Introduction to Right Triangles | 直角三角形入门

A right-angled triangle, or right triangle, is a triangle that has one angle exactly equal to 90°. The side opposite the right angle is always the longest side, and we call it the hypotenuse. The other two sides are usually referred to as the ‘legs’ or simply the shorter sides. In Cambridge KS3 mathematics, you will often see these labelled as a, b and c, where c is the hypotenuse.

直角三角形是有一个角恰好等于 90° 的三角形。直角所对的边总是最长的那条边，我们称其为斜边。另外两条边通常被称为“直角边”，或者简单地称为较短的边。在剑桥 KS3 数学中，你经常会看到它们被标注为 a、b 和 c，其中 c 表示斜边。

Before you apply the theorem, you must be able to identify the hypotenuse. A handy tip is to look for the side that does not touch the right angle. In diagrams, the right angle is often marked with a small square. The side directly opposite that square is the hypotenuse. Remember that the hypotenuse only exists in a right triangle, so always confirm the 90° angle first.

在应用定理之前，你必须能够识别斜边。一个简单的诀窍是找到不与直角接触的那条边。在示意图中，直角通常用一个带小方框标记。正对着这个小方框的边就是斜边。请记住，斜边只存在于直角三角形中，因此一定要先确认图中确实有一个 90° 角。

---

2. Statement of the Theorem | 定理的陈述

Pythagoras’ theorem states: in any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. If we label the hypotenuse as c and the other two sides as a and b, the rule can be written as a² + b² = c². This simple algebraic relationship is the foundation for all the work on right triangles at KS3 level.

勾股定理可以表述为：在任何直角三角形中，斜边的平方等于另外两条边的平方之和。如果我们把斜边记为 c，另外两条边记为 a 和 b，那么这一定则就可以写成 a² + b² = c²。这个简洁的代数关系是 KS3 阶段所有直角三角形问题的基础。

It is important to understand what ‘squaring’ means. Squaring a number means multiplying it by itself. For example, if a = 3, then a² = 3 × 3 = 9. The theorem connects geometry and algebra, so you will often need to substitute values and then solve an equation to find the missing side. Always write the formula first, then substitute the numbers carefully.

理解“平方”的含义很重要。求一个数的平方就是将这个数自身相乘。例如，如果 a = 3，那么 a² = 3 × 3 = 9。勾股定理将几何与代数联系起来，因此你通常需要代入数值，然后通过解方程来求出缺失的边长。解题时一定要先写出公式，然后再仔细地代入数值。

---

3. Understanding the Hypotenuse | 理解斜边

The hypotenuse is a key concept. It is the side of greatest length and sits opposite the right angle. No matter how the triangle is oriented, the hypotenuse remains the longest side. Confusion often happens when the right triangle is rotated so that the hypotenuse is not sloping. In such cases, simply trace the right angle and move to the opposite side – that is your hypotenuse.

斜边是一个核心概念。它是直角三角形中最长的边，并且坐在直角的对面。无论三角形如何摆放，斜边始终是最长的边。当直角三角形经过旋转、使得斜边看起来不再倾斜时，学生常常会感到困惑。遇到这种情况，你只需从直角出发、走到它的对边——那里就是斜边。

In the standard formula a² + b² = c², the letter c is reserved exclusively for the hypotenuse. Never place a shorter side as c by mistake. A common error is to label the sides randomly. To avoid this, underline the hypotenuse first and label it c. Then assign a and b to the remaining two sides in any order – addition is commutative, so it makes no difference which leg is a or b.

在标准公式 a² + b² = c² 中，字母 c 专门留给斜边。切勿错误地将某条直角边当成 c。一个常见错误是随意地标注边。为避免此类错误，请先标出斜边并把它记为 c，然后再把剩下的两条边任意地记为 a 和 b——加法具有交换律，所以哪条直角边是 a、哪条是 b 并不影响结果。

---

4. Formula and Notation | 公式与符号

The standard Pythagoras equation is written as:

a² + b² = c²

When you want to find the hypotenuse, the formula can be rearranged to:

c = √(a² + b²)

The standard Pythagoras equation is written as:

a² + b² = c²

When you want to find the hypotenuse, the formula can be rearranged to:

c = √(a² + b²)

If you are missing one of the shorter sides, you need to subtract. The rearranged versions are:

a² = c² − b² or b² = c² − a²

Then take the square root to find the side length.

如果你想要求出直角边，就需要进行减法运算。重排后的公式为：

a² = c² − b² 或 b² = c² − a²

然后再开平方即可求得边长。

At KS3, you will often leave your answer as a square root if it is not a perfect square, or round it to a required number of decimal places. The surd form is sometimes acceptable, but do check your question instructions. Your calculator is essential for evaluating square roots of non-square numbers.

在 KS3 阶段，如果计算结果不是完全平方数，你通常可以保留根号形式，或者按照题目要求四舍五入到指定的小数位数。保留不尽根式有时是可以接受的，但请务必看清题意。计算非平方数的平方根时，你的计算器是必不可少的。

---

5. Worked Example 1: Finding the Hypotenuse | 示例 1：求斜边

Question: A right triangle has shorter sides of length 6 cm and 8 cm. Find the length of the hypotenuse.

问题：一个直角三角形的两条直角边分别为 6 cm 和 8 cm。求斜边的长度。

Step 1 – Write the formula: a² + b² = c². Let a = 6 and b = 8.
Step 2 – Substitute: 6² + 8² = c² → 36 + 64 = c² → 100 = c².
Step 3 – Take the square root: c = √100 = 10 cm.
Always include the unit in your final answer.

步骤 1——写出公式：a² + b² = c²。设 a = 6，b = 8。
步骤 2——代入数值：6² + 8² = c² → 36 + 64 = c² → 100 = c²。
步骤 3——开平方：c = √100 = 10 cm。
最终答案一定要标明单位。

Notice that 6, 8 and 10 form a well-known Pythagorean triple. Recognising these triples can save you time in exams. However, if you do not spot the triple, simply follow the substitution procedure and you will obtain the correct answer safely. This is a classic KS3 question that tests both squaring and square root skills.

注意，6、8 和 10 组成了一组著名的勾股数。能够认出这些勾股数组可以在考试中节省时间。但是，如果你没有发现这组勾股数，只需按照代入步骤进行计算，同样可以稳妥地得到正确答案。这是一道经典的 KS3 题目，同时考查了平方和开平方的技能。

---

6. Worked Example 2: Finding a Shorter Side | 示例 2：求直角边

Question: The hypotenuse of a right triangle is 13 cm. One of the shorter sides is 5 cm. Work out the length of the remaining side.

问题：一个直角三角形的斜边为 13 cm，其中一条直角边为 5 cm。请计算剩下一条边的长度。

Step 1 – Write a² + b² = c². Let c = 13 and the known leg a = 5.
Step 2 – Rearrange: b² = c² − a².
Step 3 – Substitute: b² = 13² − 5² = 169 − 25 = 144.
Step 4 – Take the square root: b = √144 = 12 cm.

步骤 1——写出 a² + b² = c²。设 c = 13，已知直角边 a = 5。
步骤 2——变形：b² = c² − a²。
步骤 3——代入：b² = 13² − 5² = 169 − 25 = 144。
步骤 4——开平方：b = √144 = 12 cm。

A very common mistake here is adding the squares instead of subtracting. Because the hypotenuse is the longest side, its square is always larger than either of the shorter squares. Therefore, you must subtract the square of the known shorter side from the square of the hypotenuse. Checking that your answer is less than the hypotenuse is a quick sense-check.

这里一个极为常见的错误是将平方相加而不是相减。由于斜边是最长的边，其平方值总是大于任何一条直角边的平方值。因此，你必须用斜边的平方减去已知直角边的平方。快速检查一下你的答案是否小于斜边的长度，就可以完成一个有效的合理性检验。

---

7. Pythagorean Triples | 勾股数

Pythagorean triples are sets of three whole numbers that fit the rule a² + b² = c² exactly. The most common triples you will meet at KS3 include (3, 4, 5), (5, 12, 13), (6, 8, 10), (9, 12, 15) and (8, 15, 17). Notice that (6, 8, 10) is simply a multiple of the (3, 4, 5) family. Multiples of any triple are also valid triples.

勾股数是能够完美满足 a² + b² = c² 的三组整数。在 KS3 阶段你将会遇到的最常见的数组有 (3, 4, 5)、(5, 12, 13)、(6, 8, 10)、(9, 12, 15) 以及 (8, 15, 17)。请注意，(6, 8, 10) 只是 (3, 4, 5) 这一组勾股数的整数倍。任何勾股数组的倍数同样是有效的勾股数。

Memorising a few basic triples can make problem-solving much faster. For instance, if you spot a triangle with sides 9 and 12, you can predict the hypotenuse is 15 without needing to square and add. However, never force a triple when the numbers do not match. Always be prepared to use the full algebraic method.

熟记几组基本的勾股数可以极大地提高解题速度。例如，如果你看到一个三角形的两条边分别是 9 和 12，就可以直接推断出斜边是 15，而无需经过平方再相加的过程。不过，当题目给出的数字无法匹配时，千万不要强行套用勾股数。一定要做好使用完整代数方法的准备。

---

8. Applying Pythagoras in Real Life | 勾股定理在实际生活中的应用

Pythagoras’ theorem is not confined to textbook triangles. It is used to find heights of ladders leaning against walls, distances across rectangular fields, diagonal lengths of TV screens, and even the shortest path between two points on a map. In every case, you need to identify or visualise a right triangle within the situation.

勾股定理并不只局限于课本上的三角形。它被用来求解斜靠在墙上的梯子的高度、矩形场地的对角线距离、电视屏幕的对角线长度，甚至地图上两点之间的最短路径。在任何一种情境下，你都需要在现实情境中找到或者想象出一个直角三角形。

Example: A ladder of length 5 m leans against a vertical wall. The foot of the ladder is 2 m away from the base of the wall. How high up the wall does the ladder reach? Here the ladder is the hypotenuse c = 5, and the distance on the ground is a = 2. The height up the wall b is missing. Using b² = 5² − 2² = 25 − 4 = 21, so b = √21 ≈ 4.58 m (to 3 s.f.). This is a classic Cambridge Checkpoint style question.

示例：一架 5 米长的梯子斜靠在一面竖直的墙上。梯子脚距离墙脚 2 米。问梯子能达到墙上的多高位置？在这里，梯子就是斜边 c = 5，地面上的距离为 a = 2。墙上到达的高度 b 是需要求解的未知量。计算 b² = 5² − 2² = 25 − 4 = 21，所以 b = √21 ≈ 4.58 米（保留三位有效数字）。这是一道典型的剑桥 Checkpoint 风格的题目。

---

9. Problem Solving with Multiple Steps | 多步骤问题解决

Higher-achieving KS3 students need to use Pythagoras’ theorem as part of a chain of reasoning. For instance, you might have to find the area of an isosceles triangle by first splitting it into two right triangles. The altitude becomes one leg, half the base becomes the other, and the equal side is the hypotenuse. You use Pythagoras to find the height, then apply the area formula.

程度较高的 KS3 学生需要将勾股定理当作一系列推理链条中的一环来使用。例如，你可能需要先通过分割等腰三角形得到两个直角三角形，再计算等腰三角形的面积。此时，底边上的高是一条直角边，底边的一半是另一条直角边，等腰三角形的腰则是斜边。你先用勾股定理求出高的长度，然后再使用面积公式。

Another common multi-step problem involves finding the distance between two points on a coordinate grid. Draw a right triangle using the horizontal and vertical distances between the points. The length of the hypotenuse is the straight-line distance. This directly prepares you for the distance formula in later years.

另一种常见的多步骤问题涉及求坐标网格上两点之间的距离。你可以利用两点之间的水平距离和垂直距离画出一个直角三角形，斜边的长度就是两点间的直线距离。这为你将来学习距离公式做好了直接铺垫。

---

10. Converse of Pythagoras’ Theorem | 勾股定理的逆定理

The converse of Pythagoras’ theorem states: if the sides of a triangle satisfy a² + b² = c², where c is the longest side, then the triangle is right-angled. You can use this to test whether an angle is exactly 90°. This is particularly useful in construction problems and when analysing given side lengths in a Cambridge exam.

勾股定理的逆定理指出：如果一个三角形的三条边满足 a² + b² = c²，其中 c 是最长的边，那么这个三角形一定是直角三角形。你可以利用这一点来检验某个角是否正好是 90°。这在结构问题以及分析剑桥考试所给的三边长度时特别有用。

Example: A triangle has sides 7 cm, 24 cm and 25 cm. Is it right-angled? Check: 7² + 24² = 49 + 576 = 625, and 25² = 625. The sum of the squares of the two shorter sides equals the square of the longest side, so the triangle does contain a right angle. The right angle lies opposite the side of length 25 cm.

示例：一个三角形三条边的长度分别为 7 cm、24 cm 和 25 cm。它是不是直角三角形？检验：7² + 24² = 49 + 576 = 625，而 25² = 625。因为较短两边的平方和等于最长边的平方，所以这个三角形确实含有一个直角，并且直角就在 25 cm 那条边的对面。

---

11. Common Mistakes to Avoid | 常见错误避免

One frequent error is forgetting to identify the hypotenuse correctly and putting the longest side in the wrong position. Always write down ‘c is the hypotenuse’ before you start. Another mistake is adding when you should subtract. This typically happens when students blindly use a² + b² = c² without thinking about which side is missing.

一个常见的错误是没能正确识别斜边，把最长的边放在了错误的位置上。开始计算之前，务必先写下“c 是斜边”。另一个错误是应当用减法时却用了加法。这往往是在学生不假思索地套用 a² + b² = c² ，却没想过究竟缺少了哪条边时所发生的情况。

Many learners forget to take the square root at the end. They find c² = 100 and then give 100 as the final answer. You must remember that the theorem gives the square of the side, so the last step is always to take the positive square root. Lengths are positive, so ignore the negative root.

很多同学到最后会忘记开平方。他们算出了 c² = 100，然后把 100 当做最终答案。必须记住，定理给出的是边长的平方，所以最后一步永远都是取正平方根。因为边长是正数，所以要舍去负的平方根。

Rounding errors can also accumulate. Work with the full calculator display until the very end, then round your answer as required. In Cambridge KS3, you may be asked to give your answer correct to 3 significant figures or to 2 decimal places, depending on the context.

舍入误差也可能会累积。在最终答案之前，应当一直使用计算器显示的完整数值进行计算，然后再按照要求取近似值。在剑桥 KS3 考试中，视情况不同，你可能会被要求将答案保留三位有效数字或者两位小数。

---

12. Practice Questions Summary | 练习总结

To master Pythagoras’ theorem, consistent practice is essential. Try a mix of straightforward calculations, problems where the right triangle is hidden within a larger shape, and real-world contexts. With each question, follow a disciplined approach: draw a sketch, label the sides, write the formula, substitute, solve and check your answer makes sense.

要掌握勾股定理，持之以恒的练习是必不可少的。你可以混合练习直接计算的题目、直角三角形隐藏在一个更大图形内的问题，以及真实情境的应用题。每做一道题，都要遵循有条不紊的步骤：画出草图、标注各边、写出公式、代入数值、解方程，并检查答案是否合理。

Below is a short table of sample questions to test yourself. Cover the answers and try them first.

下面是一个简短的自我检测样题表。你可以先把答案遮住，自己先试着做一下。

Question 问题	Working hints 解题提示	Answer 答案
Find hypotenuse: legs 9 and 12	9² + 12² = 81 + 144 = 225	15
Find missing leg: c = 17, leg = 8	b² = 17² − 8² = 289 − 64 = 225	15
Is triangle with sides 10, 24, 26 right-angled?	Check 10²+24²=100+576=676, 26²=676	Yes
Ladder 6.5 m, base 2.5 m from wall: height?	h² = 6.5² − 2.5² = 42.25 − 6.25 = 36	6 m

Regular timed practice will improve your accuracy and speed. Keep a revision card with the two forms of the formula: c² = a² + b² and a² = c² − b². With these tools, you will be well prepared for the Pythagoras questions in your Cambridge KS3 assessments.

定期进行限时练习可以提高你的准确度和解题速度。你可以制作一张复习卡片，写下公式的两种形式：c² = a² + b² 和 a² = c² − b²。有了这些工具，你就能为剑桥 KS3 考试中的勾股定理题目做好充分准备。

---

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）