Angle properties of polygons
Angles inside regular polygons can usually be calculated by splitting the polygon into triangles around the centre.
Question
Find the interior angle of a regular pentagon.
Split the regular pentagon into triangles.
Each angle at the centre is \(72^\circ\) as the \(360^\circ\) at the centre is split into five parts.
This means that in each triangle we have \(108^\circ\) left for the two angles lying on the side of the pentagon, so each of them is \(54^\circ\).
Each interior angle is two of these.
So the required angle is \(108^\circ\). (You will probably spot that we are halving and then doubling.)
Question
Find the interior angle of a regular hexagon.
Split the regular hexagon into triangles.
Each angle at the centre is \(60^\circ\) as the \(360^\circ\) at the centre is split into six pieces.
This means that in each triangle we have \(120^\circ\) left for the two angles lying on the side of the hexagon, so each of them is \(60^\circ\).
Each interior angle (white) is two of these.
Therefore the answer is \(120^\circ\).
Question
Find the interior angle of a regular octagon.
Split the octagon into small triangles.
Each angle at the centre is \(360^\circ\) divided by eight which is \(45^\circ\).
This means that in each triangle we have \((180-45)^\circ\) is left for the two angles lying on the side of the hexagon, which leaves \(135^\circ\), so each of them is \(67.5^\circ\).
Each interior angle (white) is two of these.
Therefore the answer is \(135^\circ\)
More guides on this topic
- Determine the gradient of a straight line
- Circle geometry
- Calculating the volume of a standard solid
- Applying Pythagoras Theorem
- Using similarity
- Working with two-dimensional vectors
- Working with three-dimensional coordinates
- Using vector components
- Calculating the magnitude of a vector
- An Approximate History of Co-ordinates
