📚 Pythagoras’ Theorem: Finding Missing Sides | 勾股定理：求缺失边长
In KS3 mathematics, one of the most powerful tools you will learn is Pythagoras’ Theorem. This relationship between the sides of a right-angled triangle allows you to calculate a missing length when the other two sides are known. It is named after the ancient Greek mathematician Pythagoras and has been used for thousands of years in construction, navigation and design. Whether you are working out the diagonal of a rectangle, the height of a ladder against a wall, or the shortest distance between two points on a coordinate grid, Pythagoras’ Theorem is the key. In this article, we will go through the theorem step by step, practise finding both the hypotenuse and the shorter sides, and explore common tricks and mistakes.
在 KS3 数学中，你将学到的最强大的工具之一就是勾股定理。直角三角形三条边之间的这种关系可以让你在已知其他两边长度时计算出缺失的边长。这个定理以古希腊数学家毕达哥拉斯的名字命名，几千年来一直被用于建筑、导航和设计领域。无论你是要计算长方形的对角线长度、靠在墙上的梯子的高度，还是坐标网格上两点之间的最短距离，勾股定理都是关键。在本文中，我们将逐步学习这个定理，练习如何求出斜边和直角边的长度，并探讨常见的解题技巧和易犯错误。
1. What is Pythagoras’ Theorem? | 什么是勾股定理？
Pythagoras’ Theorem states that 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 other two sides. This only works for right-angled triangles. If the triangle does not have a 90° angle, the rule cannot be applied directly. The theorem provides an exact relationship, not an approximation, which makes it extremely useful for precise calculations in geometry and real-life problems.
勾股定理指出，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长度的平方之和。这个定理仅适用于直角三角形。如果三角形没有 90° 角，就不能直接应用这个规则。该定理给出的是一个精确的关系，而不是近似值，因此它在几何和现实问题中的精确计算方面非常有用。
2. Identifying the Hypotenuse | 识别斜边
The hypotenuse is always the longest side of a right-angled triangle and is directly opposite the right angle. It does not matter how the triangle is rotated; the side that faces the 90° corner is the hypotenuse. Many mistakes occur when students label the wrong side as c in the formula a² + b² = c². A simple check is to compare the lengths: if you have measured or labelled the sides, the hypotenuse will be larger than each of the other two sides. Never assume the sloping side is the hypotenuse without checking the angle first.
斜边总是直角三角形中最长的边，并且正对着直角。不管三角形如何旋转，正对 90° 角的那条边就是斜边。当学生错误地将另一条边标为公式 a² + b² = c² 中的 c 时，常常会出现错误。一个简单的检查方法是比较各边的长度：如果你已经测量或标注了各边，斜边一定会比另外两条边都长。在没有先确认直角的情况下，千万不要仅凭倾斜方向就认定某条边是斜边。
3. The Formula and How to Use It | 公式及其用法
The standard formula is written as:
a² + b² = c²
where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides (the legs). When you are given any two sides of a right-angled triangle, you can substitute them into the equation and solve for the missing side. If you are finding the hypotenuse, you square the two known legs, add the results, and then take the square root. If you are finding a shorter side, you subtract the square of the known leg from the square of the hypotenuse, then take the square root. Write the formula down at the start of every Pythagoras question to avoid confusion.
标准公式写作：
a² + b² = c²
其中 c 代表斜边的长度，a 和 b 代表另外两条直角边的长度。当你已知直角三角形的任意两条边时，就可以将它们代入方程，求出缺失的边长。如果求的是斜边，你需要先将两条已知直角边分别平方，相加之后再开平方。如果求的是一条直角边，你就要用斜边的平方减去已知直角边的平方，然后再开平方。做任何一道勾股定理题目时，都先写出公式，以免混淆。
4. Worked Example: Finding the Hypotenuse | 例题演示：求斜边长度
Problem: A right-angled triangle has legs of lengths 6 cm and 8 cm. Find the length of the hypotenuse.
Solution: Let a = 6, b = 8, and c be the hypotenuse. Using the formula:
6² + 8² = c²
36 + 64 = c²
100 = c²
c = √100 = 10 cm
Always include the units in your final answer and check that the hypotenuse is indeed longer than both 6 cm and 8 cm. This quick sense-check will catch many careless errors.
题目：一个直角三角形的两条直角边长度分别为 6 cm 和 8 cm，求斜边的长度。
解答：设 a = 6, b = 8, 斜边为 c。代入公式：
6² + 8² = c²
36 + 64 = c²
100 = c²
c = √100 = 10 cm
最后别忘了写上单位，并检查斜边是否确实比 6 cm 和 8 cm 都长。这个快速的合理性检查可以帮你找出许多粗心导致的错误。
5. Finding a Shorter Side | 求一条直角边的长度
When the hypotenuse and one leg are known, you need to rearrange the formula. The rearranged form for a missing leg a is:
a² = c² – b²
Example: A right-angled triangle has a hypotenuse of 13 m and one leg of 5 m. Find the missing leg.
a² = 13² – 5²
a² = 169 – 25 = 144
a = √144 = 12 m
Make sure you always subtract before taking the square root; some students take the square root of each number individually, which is incorrect. The subtraction must be done inside the square, not outside.
当已知斜边和一条直角边时，你需要对公式进行变形。求直角边 a 的变形公式为：
a² = c² – b²
例题：一个直角三角形的斜边长为 13 m，一条直角边长为 5 m，求另一条直角边。
a² = 13² – 5²
a² = 169 – 25 = 144
a = √144 = 12 m
一定要先相减再开平方；有些学生会先对每个数分别开平方，这是不正确的。减法必须在平方之后、开平方之前进行，而不能在外面。
6. Rearranging the Formula Safely | 安全地变形公式
It is easy to get confused when rearranging a² + b² = c². A reliable method is to write down the version of the formula that isolates the unknown before substituting numbers. For the hypotenuse, use c = √(a² + b²). For a leg, say b, use b = √(c² – a²). Notice the square root covers the entire expression inside the brackets. On a calculator, you can use the square root button after computing the value inside. Always show your working steps clearly so you can trace back any mistake.
在对 a² + b² = c² 进行变形时很容易搞混。一个可靠的方法是，在代入数值之前先写出将未知数单独分离出来的公式形式。求斜边时，用 c = √(a² + b²)。求直角边 b 时，则用 b = √(c² – a²)。要注意平方根号覆盖了括号内的整个表达式。在计算器上，你可以先算出括号内的值，再按平方根键。始终清晰地展示你的解题步骤，这样你就可以追溯任何错误。
7. Estimation and Checking | 估算与检验
Before calculating the exact answer, make an estimate by rounding the given numbers to the nearest whole number that is easy to square. For example, if a = 5.8 and b = 9.1, round to a = 6 and b = 9. Then c ≈ √(36 + 81) = √117 ≈ 10.8. The exact answer will be close to this. If your calculator gives a wildly different result, you have probably made an input error. Always ask yourself: does my answer make sense in the context of the triangle? The hypotenuse must be the longest side, and each leg must be shorter than the hypotenuse.
在算出精确答案之前，你可以先估算一下，把给定的数值四舍五入到容易平方的整数。例如，如果 a = 5.8, b = 9.1，可以大致取 a = 6, b = 9。那么 c ≈ √(36 + 81) = √117 ≈ 10.8。精确的答案应该与此接近。如果计算器得出了相差甚远的结果，你很可能是在输入时犯了错误。要时常问自己：这个答案在具体的三角形中合理吗？斜边必须是最长的边，而每条直角边都必须比斜边短。
8. Applying Pythagoras’ Theorem to Word Problems | 勾股定理在实际问题中的应用
Many KS3 problems are presented in words rather than with a diagram. You will need to sketch a right-angled triangle yourself and label the sides with the given information. Classic examples include a ladder leaning against a wall (ladder = hypotenuse, wall and ground = legs), a ship sailing east then north (the direct distance back to the start is the hypotenuse), and the diagonal of a rectangular sports field. Always identify which lengths are the legs and which is the hypotenuse, then apply the formula. Drawing a clear diagram is half the battle.
许多 KS3 题目是以文字形式呈现的，而没有给出图形。你需要自己画出一个直角三角形，并用题目给出的信息标出各边。经典例子包括：一架梯子靠在墙上（梯子为斜边，墙和地面为直角边），一艘船先向东再向北航行（返回起点的直线距离就是斜边），以及一个长方形运动场的对角线长度。一定要先确认哪些长度是直角边、哪条是斜边，然后再应用公式。画出一个清晰的示意图就等于成功了一半。
9. The Converse of Pythagoras’ Theorem | 勾股定理的逆定理
Sometimes you are given three side lengths and need to determine if the triangle is right-angled. This is where the converse of the theorem is used: if the square of the longest side equals the sum of the squares of the other two sides, then the triangle is right-angled. For example, a triangle with sides 9, 12 and 15: 15² = 225 and 9² + 12² = 81 + 144 = 225, so it is a right-angled triangle. This is very useful when checking whether a corner is exactly 90°, for instance in carpentry or tiling.
有时题目会给出三条边长，让你判断该三角形是否为直角三角形。这时就要用到逆定理：如果最长边的平方等于另外两条边的平方和，那么这个三角形就是直角三角形。例如，一个三角形的边长分别是 9, 12, 15：15² = 225，而 9² + 12² = 81 + 144 = 225，所以它是一个直角三角形。这在检查某个角落是否恰好成 90° 时非常有用，比如在木工或铺瓷砖的场合。
10. Common Mistakes and How to Avoid Them | 常见错误及如何避免
(1) Forgetting to take the square root at the end. After finding c² = 25, some students write c = 25 instead of c = 5. Always remember that you need the square root to get back to the side length.
(1) 计算出 c² = 25 后忘记开平方。有些学生会写成 c = 25，而正确答案是 c = 5。要时刻记住，你需要开平方才能得到边长。
(2) Adding when you should subtract for a shorter side. If you are given the hypotenuse and one leg, you must subtract, not add the squares. Writing the rearranged formula first helps enormously.
(2) 求直角边时，本应相减却做成了相加。如果已知斜边和一条直角边，你必须用斜边的平方减去已知直角边的平方，而不是相加。先写下变形后的公式会有很大帮助。
(3) Mixing up which side is the hypotenuse. Even if the triangle is drawn with a sloping side, check for the right-angle symbol. The hypotenuse is the side exactly opposite that 90° angle box.
(3) 弄错斜边。即使三角形画有一条倾斜的边，也要查看直角符号。斜边是正好与那个 90° 角框相对的那条边。
(4) Not squaring the whole length or rounding too early. Use the exact values until the final square root step to ensure accuracy.
(4) 没有对整条边长进行平方，或者过早地四舍五入。在最后开平方之前要使用精确值，以保证结果的准确性。
11. Practice Tips for Success | 取得成功的练习技巧
To become confident with Pythagoras’ Theorem, practise a mix of straightforward calculations and word problems. Create your own triangles with simple whole-number sides (Pythagorean triples) such as 3-4-5, 5-12-13, 8-15-17 and their multiples. Check your answers by plugging them back into the original a² + b² to see if it matches c². When revising, try to solve each problem without a calculator for the squaring and square root of perfect squares, then use a calculator for decimals. More importantly, always show your working; this not only helps you gain method marks in exams but also makes it easier to spot errors.
要想熟练掌握勾股定理，你需要混合练习直接计算题和文字应用题。可以自己构造一些具有简单整数边长的三角形（勾股数），例如 3-4-5, 5-12-13, 8-15-17 以及它们的倍数。通过将答案代回原来的 a² + b² 来检验是否等于 c²，也是一种好方法。在复习时，对于完全平方数的平方和开平方运算，可以尝试不借助计算器完成，然后再用计算器处理小数。更重要的是，一定要写出解题过程；这不仅有助于你在考试中获得过程分，还能让你更容易发现错误。
12. Summary | 总结
Pythagoras’ Theorem is a fundamental tool for solving right-angled triangle problems. Remember: it works only for right-angled triangles; the hypotenuse is the longest side opposite the right angle; the formula is a² + b² = c²; for the hypotenuse, calculate c = √(a² + b²); for a leg, use a = √(c² – b²); always include units and check your answer makes sense. With consistent practice, finding missing sides will become second nature and will prepare you for more advanced topics such as trigonometry and vectors in later years.
勾股定理是解决直角三角形问题的基本工具。请记住：它只适用于直角三角形；斜边是正对直角的最长边；公式为 a² + b² = c²；求斜边用 c = √(a² + b²)；求直角边用 a = √(c² – b²)；答案一定要带单位，并检验其合理性。通过持续不断的练习，求缺失边长将会变得像本能一样自然，同时也为你今后学习三角函数、向量等更深入的内容做好准备。
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