Angles that fill the space around a point

The idea

If some angles add up to 360°, their vertices can be joined at a single point so that the angles neither overlap nor leave any gap around the point.

A gap around a point

Several angles are joined at one point O. For the angles to leave no gap and no overlap, what must they add up to?

They must add up to:

• 360°
• 180°
• 90°

Fill the gap

Six angles are already joined at O. A gap is left. Try the pieces from the tray.

Count up to 360°

The six angles add up to 40° + 60° + 50° + 30° + 40° + 90° = 310°. A full turn is 360°, so the gap is 360° − 310° = 50°. Only the 50° piece fits.

If some angles add up to 360°, they can be joined at one point with no gap and no overlap.

• Piece | Total around O | Result
• 20° | 330° | gap of 30° left
• 50° | 360° | fits exactly
• 70° | 380° | 20° overlap

Notes

If some angles add up to 360°, they can be joined at a point with no gap and no overlap.

Check yourself

Angles of 100°, 80° and 70° are joined at a point. Another angle is added so that nothing is left over and nothing overlaps. How many degrees is the new angle?

Answer: 110

100° + 80° + 70° = 250°, and 360° − 250° = 110°.

Which angles can be joined at a point with no gap and no overlap?

Five equal angles are joined at a point and fill all the space around it. How many degrees is each angle?

Answer: 72

360° ÷ 5 = 72°.

• 100°, 100° and 100°. They add up to 300°, so a 60° gap is left.
• 120°, 120° and 130°. They add up to 370°: a 10° overlap.
• 150°, 150° and 150°. They add up to 450°, which is more than a full turn.
• 90°, 90°, 90° and 90° — correct. Yes! 90° × 4 = 360°.