# Pythagoras' theorem in 3D · Grade 8

> Pythagoras' theorem in 3D, Grade 8: Will the stick fit in the box? and more. Explanation, worked examples and free practice following the Israeli curriculum.

In Pythagoras' theorem in 3D (Grade 8) you practise 3 skills: The diagonal of a face and of a cuboid · Will the stick fit in the box? · The height of a pyramid and the diagonal cross section.

Inside a solid there are right triangles, and in each of them you can use Pythagoras' theorem.

The diagonal of a face is the hypotenuse of a triangle whose two shorter sides are edges of the face.

The diagonal of a cuboid is the hypotenuse of a triangle: one shorter side is the diagonal of the base, and the other is the height. In short: the diagonal squared is equal to the length squared plus the width squared plus the height squared. The longest stick that fits in a cuboid is as long as its diagonal.

In a square pyramid, the height, half the side of the base and the height of a side face make a right triangle.

**Tip:** Look for the right triangle and find its hypotenuse: the side opposite the right angle.

## What you learn

### The diagonal of a face and of a cuboid

- Example: A cuboid measures 6 cm by 8 cm by 12 cm. How long is the diagonal of its bottom face, which measures 6 by 8? (A cuboid, with a dashed diagonal on its bottom face marked with a question mark.)
- Answer: 10 cm
- Solution: 6² + 8² = 36 + 64 = 100 √100 = 10 The diagonal is 10 cm long.

### Will the stick fit in the box?

- Example: A cuboid measures 9 cm by 11 cm by 10 cm. Will a straight stick 12 cm long fit inside the cuboid when it is closed? (A cuboid, with its diagonal drawn as a dashed line.)
- Options: Yes · No
- Answer: Yes
- Solution: The diagonal, squared: 9² + 11² + 10² = 81 + 121 + 100 = 302 The stick, squared: 12² = 144 So 144 < 302 The stick is shorter than the diagonal, so it fits along the diagonal. The answer is yes.

### The height of a pyramid and the diagonal cross section

- Example: In a square pyramid, each side of the base is 16 cm long, and the height of each side face is 10 cm. What is the height of the pyramid? (A square pyramid. Its height, from the apex to the centre of the base, is marked with a question mark.)
- Answer: 6 cm
- Solution: Half the side of the base is 8 cm. The height, squared: 10² − 8² = 100 − 64 = 36 √36 = 6 The height of the pyramid is 6 cm.

The exercises are made by rules and the answers are worked out in code. The topic follows the Israeli Ministry of Education's grade 8 transition maths curriculum (2026–27).

[Practise on the site](https://kixplo.com/il/en/) · [All Grade 8 topics](https://kixplo.com/il/en/math/grade-8/)