3D Pythagoras (Edexcel GCSE Maths): Revision Notes

3D Pythagoras

Understanding the 3D Pythagoras theorem

The 3D Pythagoras theorem is simply an extension of the familiar 2D Pythagoras theorem that you already know. Instead of working with just two dimensions, we now work with three dimensions to find the longest diagonal (called the space diagonal) that runs through the interior of a 3D shape.

The fundamental formula for 3D Pythagoras is:

$a^2 + b^2 + c^2 = d^2$

Where $a$, $b$, and $c$ are the three perpendicular edges of a rectangular prism (cuboid), and $d$ is the space diagonal running from one corner to the opposite corner.

How is 3D Pythagoras derived?

The beauty of 3D Pythagoras lies in its simplicity - it's actually just the 2D Pythagoras theorem applied twice in succession. Here's how it works:

Step 1: First, we use the 2D Pythagoras theorem to find the diagonal of the base rectangle. If the base has sides $a$ and $b$, then the diagonal $e$ can be found using: $e^2 = a^2 + b^2$

Step 2: Next, we create a new right-angled triangle using the diagonal $e$ we just found and the height $c$ of the shape. The hypotenuse of this new triangle is our space diagonal $d$: $d^2 = e^2 + c^2$

Step 3: Substituting our expression for $e^2$ into this equation: $d^2 = (a^2 + b^2) + c^2 = a^2 + b^2 + c^2$

Working with cubes

When dealing with cubes, all three dimensions are equal, which makes the calculation slightly simpler. The key advantage is that we only need to know one measurement to find the space diagonal.

Applying 3D Pythagoras to other shapes

The 3D Pythagoras formula isn't limited to just finding diagonals in cuboids. It can be adapted to solve problems involving other 3D shapes, such as pyramids.

Key problem-solving steps

When tackling any 3D Pythagoras problem, it's essential to work systematically. Here's a reliable approach that will help you solve any 3D Pythagoras problem:

Remember!