3D Pythagoras and Trigonometry | 三维勾股定理与三角学

📚 3D Pythagoras and Trigonometry | 三维勾股定理与三角学

This revision guide covers the IGCSE Mathematics topic of applying Pythagoras’ theorem and trigonometric ratios to three-dimensional shapes. You will learn to calculate lengths, angles and diagonals in cuboids, prisms and pyramids, and to explain each step clearly in an exam.

本复习指南涵盖 IGCSE 数学中将勾股定理和三角比应用于三维图形的内容。你将学会计算长方体、棱柱和棱锥中的长度、角度和对角线，并能在考试中清晰地解释每一步。

1. Pythagoras’ Theorem in 2D Recap | 二维勾股定理回顾

In any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides: a² + b² = c², where c is the longest side opposite the right angle.

在任何直角三角形中，斜边的平方等于另外两条边的平方和：a² + b² = c²，其中 c 是直角所对的最长边。

a² + b² = c²

Always label the sides carefully and check that the triangle is right-angled before applying the theorem.

在应用该定理之前，务必仔细标注各边，并确认三角形是直角三角形。

For example, in a triangle with shorter sides 3 and 4, the hypotenuse is √(3² + 4²) = √25 = 5. This 3-4-5 triangle appears often in IGCSE questions.

例如，在一个较短边为 3 和 4 的三角形中，斜边为 √(3² + 4²) = √25 = 5。这个 3-4-5 三角形经常出现在 IGCSE 题目中。

2. Trigonometric Ratios in Right-Angled Triangles | 直角三角形中的三角比

The three trigonometric ratios are sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, tan θ = opposite / adjacent.

三个三角比分别为：sin θ = 对边 / 斜边，cos θ = 邻边 / 斜边，tan θ = 对边 / 邻边。

Use SOH CAH TOA to remember the ratios. Choose the ratio that uses two known sides and the unknown side or angle.

可用 SOH CAH TOA 来记忆这些比值。选择包含两条已知边和未知边或角的那个比值。

Remember: the hypotenuse is always the side opposite the right angle, not simply the longest-looking side in a diagram.

记住：斜边始终是直角所对的边，而不仅仅是图中看起来最长的边。

3. Identifying Right-Angled Triangles in 3D | 在三维图形中识别直角三角形

In 3D problems, right-angled triangles are often hidden in the faces, cross-sections or internal diagonals of a solid.

在三维问题中，直角三角形通常隐藏在立体图形的面、截面或内部对角线中。

Draw a separate 2D sketch of the triangle you are using and label all known lengths. This helps avoid errors with the three-dimensional diagram.

单独画出所使用的三角形的二维草图，并标出所有已知长度。这有助于避免在三维图中出错。

Two helpful constructions are the base diagonal and the vertical height. These two lengths usually form the legs of the right-angled triangle you need.

两个有用的构造是底面对角线和垂直高度。这两个长度通常构成所需直角三角形的两条直角边。

When an angle is required, check whether it lies in a vertical plane or in a sloping face, because the adjacent and opposite sides will differ.

当需要求角度时，检查它是在竖直平面内还是在斜侧面内，因为邻边和对边会不同。

4. Space Diagonal of a Cuboid | 长方体的空间对角线

For a cuboid with length l, width w and height h, the space diagonal d can be found using a double application of Pythagoras’ theorem: first find the diagonal of the base, then use that with the height.

对于长为 l、宽为 w、高为 h 的长方体，空间对角线 d 可通过两次应用勾股定理求得：先求底面对角线，再与高一起使用。

base diagonal = √(l² + w²)

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