In this activity, we will look at using Pythagoras' theorem on 3D shapes.

Let's remind ourselves of the theory with an example:

**a² + b² = c²**

**To find the hypotenuse of a triangle:**

**1. Square one side:** 2² = 4²

**2. Square the other side:** 6² = 36²

**3. Add your answers together:** 36 + 4 = 40²

**4. Square root your answer:** √40 = 6.32 (to 2 decimal places)

__Pythagoras in 3D__

Now let's consider using this in 3D shapes instead of 2D.

Calculating a length is **just the same** as when working with a 2D triangle; the trick is being able to **spot a right-angled triangle within the shape**.

Let's look at this cuboid as an example:

We can find any diagonal line across any face of a cube or cuboid, as the **diagonal** forms the **hypotenuse of a right-angled triangle**.

For example:

Triangle ADH is a right-angled triangle, and we can use **a² + b² = c² **to calculate length AH (the hypotenuse).

Here is an overhead view of the base with our line drawn:

We can then take this into 3 dimensions by drawing **another diagonal through the shape**:

Here is a cross-section of the shape we have created.

We now have a new triangle, AGH, which extends from the bottom corner of the cuboid into the opposite top corner. Significantly,** it is still a right-angled triangle**, so we can continue to use **a² + b² = c² **to calculate the hypotenuse, AG. By doing this, we are using Pythagoras in 3D.

__Example__

Calculate length **AG** in this cube.

The first thing we can do is work out the diagonal across the base, **AH**:

a^{2} + b^{2} = c^{2}

3^{2} + 3^{2} = AH^{2}

AH^{2} = 18

AH = √18

As we will soon be squaring this answer, **it is useful to leave the number in root form at this stage**, rather than working with a decimal.

We can then use this length to work out the hypotenuse of triangle AGH:

a^{2} + b^{2} = c^{2}

(√18)^{2} + 3^{2} = AG^{2 }

AG^{2} = 18 + 9 = 27

AG = √27 = 5.2 (1 d.p.)

Now that you have seen these examples, have a go at 10 questions to see if you can apply Pythagoras to 3D shapes.