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Pythagoras' Theorem: Finding Missing Sides | 毕达哥拉斯定理：求缺失边长

2026-08-21 10:53 · 本文部分内容由AI辅助创作

📚 Pythagoras' Theorem: Finding Missing Sides | 毕达哥拉斯定理：求缺失边长

Pythagoras' theorem is one of the most important ideas in KS3 geometry. It connects the three sides of a right-angled triangle and allows you to calculate an unknown side when the other two sides are known. This article will guide you through the formula, worked examples, common errors and practice questions so you can use the theorem confidently in Cambridge Checkpoint and beyond.

毕达哥拉斯定理是 KS3 几何中最重要的思想之一。它将直角三角形的三条边联系起来，当已知另外两条边时，可以计算未知边。本文将引导你掌握公式、范例、常见错误和练习题，使你在剑桥 Checkpoint 考试及以后的学习中自信地运用这一定理。

1. What is Pythagoras' Theorem? | 什么是毕达哥拉斯定理？

Pythagoras' theorem describes a special relationship between the sides of a right-angled triangle. It works only when one angle is exactly 90°. The theorem states that the square of the longest side is equal to the sum of the squares of the other two sides. In a right-angled triangle, the longest side is called the hypotenuse, and it is always opposite the right angle.

毕达哥拉斯定理描述了直角三角形三条边之间的特殊关系。它只适用于其中一个角恰好为 90° 的情况。定理指出，最长边的平方等于另外两条边的平方和。在直角三角形中，最长的边称为斜边，它总是在直角的对面。

2. The Formula a² + b² = c² | 公式 a² + b² = c²

letters a and b represent the two shorter sides, which meet at the right angle. You can swap a and b, but c must always be the hypotenuse. This formula is the foundation for solving missing-side problems.

a² + b² = c²

该定理通常写作 a² + b² = c²，其中 c 是斜边。字母 a 和 b 表示两条较短的直角边，它们在直角处相交。你可以交换 a 和 b，但 c 必须始终是斜边。这个公式是解决求缺失边长问题的基础。

a² + b² = c²

3. Identifying the Hypotenuse | 识别斜边

Before using the formula, you must identify the hypotenuse. Look for the right angle first. The side directly across from the right angle is the hypotenuse. It is longer than either of the other two sides. Never label the hypotenuse as a or b, because the formula depends on c being the longest side.

在使用公式之前，你必须先识别斜边。首先找到直角。正对直角的边就是斜边。它比另外任何一条边都长。不要把斜边标记为 a 或 b，因为公式依赖于 c 是最长边。

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4. Finding the Hypotenuse | 求斜边

of the sum. For example, if a = 3 cm and b = 4 cm, then c² = 3² + 4² = 9 + 16 = 25, so c = √25 = 5 cm. This method works for any right-angled triangle when both shorter sides are known.

c = √a² + b²

要求斜边，先将两条直角边分别平方，把结果相加，然后取和的平方根。例如，如果 a = 3 厘米，b = 4 厘米，那么 c² = 3² + 4² = 9 + 16 = 25，因此 c = √25 = 5 厘米。当两条直角边已知时，这种方法适用于任何直角三角形。

c = √a² + b²

5. Finding a Shorter Side | 求较短的直角边

. Square c, subtract the square of b, and take the square root. This gives the missing shorter side. Be careful not to add the squares in this case.

a = √c² − b²

要求较短的直角边，需要变形公式。如果你已知斜边 c 和一条直角边 b，那么 a² = c² − b²。将 c 平方，减去 b 的平方，然后取平方根。这样就得到缺失的直角边。注意在这种情况下不要将平方相加。

a = √c² − b²

6. Checking Your Answer | 检查答案

Always check your answer by comparing the three sides. The hypotenuse should be the longest. Your answer should be positive and reasonable. Estimate before calculating: if the two shorter sides are around 6 and 8, the hypotenuse should be a little more than 8 but less than 14. If your answer falls outside this range, look for a mistake.

始终通过比较三条边来检查答案。斜边应该是最长的。答案应为正数且合理。计算前先估算：如果两条直角边约为 6 和 8，斜边应该略大于 8 但小于 14。如果你的答案超出这个范围，请检查是否有错误。

7. Common Mistakes | 常见错误

Common mistakes include using the theorem on a triangle that is not right-angled, misidentifying the hypotenuse, and forgetting to take the square root. Another error is adding instead of subtracting when finding a shorter side. Always double-check the right angle and the side you are solving for.