How to apply Pythagoras' theorem to 3D shapes in Sec 2

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Pythagoras theorem states that in a 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. This is often written as a² + b² = c², where c is the hypotenuse.
In 3D shapes, you often need to find the length of a diagonal within the shape. You can use Pythagoras theorem by first finding a diagonal on one face of the shape, and then using that diagonal as one side of a new right-angled triangle to find the 3D diagonal.
A space diagonal is a line segment connecting two vertices of a 3D shape that are not on the same face. It essentially cuts through the interior of the shape.
First, find the diagonal of one of the rectangular faces using Pythagoras theorem. Then, use this diagonal and the height of the cuboid as the two shorter sides of another right-angled triangle. Apply Pythagoras theorem again to find the length of the space diagonal (the hypotenuse of this new triangle).
Imagine a cuboid with length 4cm, width 3cm, and height 12cm. To find the space diagonal, first find the diagonal of the base: √(4² + 3²) = 5cm. Then, use this 5cm diagonal and the height (12cm) to find the space diagonal: √(5² + 12²) = 13cm.
Cuboids, cubes, pyramids, and cones are common examples. In pyramids and cones, you might need to find the slant height or the height of the shape.
If you know the side length of the square base and the height of the pyramid, you can find the slant height (the height of a triangular face) by forming a right-angled triangle with half the base side length, the pyramids height, and the slant height as the hypotenuse.
Mastering this concept helps build a strong foundation for more advanced geometry and trigonometry topics in higher secondary levels, crucial for streaming into Express and accessing pure sciences or Additional Mathematics.
A common mistake is forgetting to apply the theorem twice – first to find a 2D diagonal on a face, and then again to find the 3D diagonal. Students also sometimes struggle to visualize the right-angled triangles within the 3D shape.
Practice visualizing the right-angled triangles within the 3D shapes. Draw diagrams, and clearly label the sides you know and the side you need to find. Work through a variety of problems, starting with simpler cuboids and progressing to more complex shapes.

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