# Tilings · Year 10

> Tilings, Year 10: The angle sum of a polygon and more. Explanation, worked examples and free practice for the national curriculum in England.

In Tilings (Year 10) you practise 3 skills: The angle sum of a polygon · Angles of a regular polygon · Which polygons tile the plane.

A tiling covers the plane with shapes, with no gaps and no overlaps. Round every vertex of a tiling the angles fill a full turn, 360°.

The diagonals from one vertex split a polygon into triangles, two fewer than it has sides. So the angles of a polygon add up to the number of sides minus 2, times 180°.

In a regular polygon all the sides are equal and all the angles are equal. A regular polygon tiles the plane alone only when its angle goes into 360° a whole number of times: the equilateral triangle, the square and the regular hexagon.

Every triangle and every quadrilateral tiles the plane. To find how many tiles are needed, put the lengths in the same unit and work out how many tiles fit along each side.

**Tip:** Before you work anything out, check every length is in the same unit: one metre is 100 cm.

## What you learn

### The angle sum of a polygon

- Example: (A polygon with the diagonals from one vertex dashed) What is the sum of the interior angles of a polygon with 4 sides?
- Answer: 360 °
- Solution: The diagonals from one vertex make 2 triangles, so the sum is (4 − 2) × 180° = 360°

### Angles of a regular polygon

- Example: (A square) What is the size of each interior angle of a square?
- Answer: 90 °
- Solution: The angle sum: (4 − 2) × 180° = 360° . Each angle: 360° ÷ 4 = 90°

### Which polygons tile the plane

- Example: Each angle of a square is 90°. Can the plane be tiled with copies of it alone, with no gaps and no overlaps?
- Options: No · Yes
- Answer: Yes
- Solution: The quotient 360 ÷ 90 = 4 is a whole number, so it tiles.

The exercises are made by rules and the answers are worked out in code. The topic follows the national curriculum in England: mathematics (key stages 3 and 4).

[Practise on the site](https://kixplo.com/uk/en/) · [All Year 10 topics](https://kixplo.com/uk/en/maths/year-10/)