Angles of polygons: tiling a courtyard floor
Cut polygons into triangles to find their angle sums, then test which regular tiles cover a courtyard floor with no gaps.
Core concept
In short: Fit floor tiles together and find which shapes leave no gaps.
A polygon with n sides has angles adding up to (n − 2) × 180°; regular tiles fit round a point only if their angle divides 360°
Topic: Polygons
A courtyard is to be paved with tiles that are all the same regular polygon. To know whether they fit, we need the angle: The amount of turn between two lines that meet, measured in degrees (°). All key words at each corner of the tile. From one corner, any polygon with n sides can be cut into n − 2 triangles, and the angles of each triangle add up to 180°, so the polygon's angles add up to (n − 2) × 180°. In a regular polygon all angles are equal, so each one is that sum ÷ n. Where tiles meet at a point, their angles must add up to exactly 360°.
What you will learn
- Find the sum of the interior angles of a polygon as (n − 2) × 180°.
- Find each interior and exterior angle of a regular polygon.
- Explain which regular polygons tile a floor.
Before you start
- The angles of a triangle add up to 180°; angles round a point make 360°.
Materials
- Paper polygons cut from card
Equipment
- A protractor and scissors
Procedure
- Tap − or + to change the number of sides.
- In the triangles view, count the triangles and read the angle sum.
- Tap the chip to fit tiles round one point of the floor and look for gaps.
- Record a hexagon's angle sum, an octagon's interior angle, a tile that covers the floor and one that leaves gaps.
How this simulation is simplified
- All the tiles are regular polygons of the same size.