Investigating polygons
A polygon is a two dimensional shape enclosed by three or more straight lines.
A short video showing how to prove the sum of the angles in a n-sided polygon is 180° × (n-2). For example a hexagon has 6 sides, so (n-2) is 4, and the internal angles add up to 180° × 4 = 720°.
How to show angles in a n-sided polygon add to 180° × (n-2)
Image caption, Click to see a step-by-step slideshow.
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The exterior angle of a polygon
The exterior angle at a vertex (corner) of a shape is made by extending a side.
Sum of exterior angles
The exterior angles of a polygon add up to \(360^\circ\).
Imagine walking round the outside of the polygon.
By the time you get back to where you started you have completed one full turn.
So all the corners you turned must add to \(360^\circ\).
In this diagram the exterior angles have been given different colours.
You can see how they can be put together to make a full circle.
The interior angle of a polygon
An interior angle is the angle inside the polygon at a vertex. The interior and exterior angles together lie on a straight line.
For each vertex of a polygon: \(interior~angle + exterior~angle = 180 ^\circ\)
Angles of a regular polygon
You have already seen that the sum of the exterior angles is \(360^\circ\) and that the interior and the exterior angles add up to \(180^\circ\).
A regular polygon is a polygon whose interior angles are all equal.
Use this information to find the exterior and interior angles of a regular polygon.
Question
Find the exterior and interior angle of a regular pentagon.
Answer
A pentagon has \({5}\) sides and \({5}\) exterior angles.
The exterior angles add up to \(360^\circ\).
\(one~exterior~angle = 360^\circ \div 5 = 72^\circ\)
\(interior~angle = 180 - exterior~angle = 180 - 72 = 108^\circ\)
Test section
Question 1
What is a polygon?
Answer
A polygon is a 2D shape with straight sides.
Question 2
What type of polygon is this shape?
Answer
A shape with five sides is called a pentagon.
Question 3
What is the sum of the interior angles in a pentagon?
Answer
The sum of the interior angles in a pentagon is \({540}^\circ~({180}^\circ\times{3})\).
Question 4
What is the sum of the interior angles in a decagon?
Answer
The sum of the interior angles in a decagon is \({1440}^\circ~({180}^\circ\times{8})\).
Question 5
What is the sum of the exterior angles of a quadrilateral?
Answer
The exterior angles of any polygon add up to \({360}^\circ\).
Question 6
What is the sum of the exterior angles of an octagon?
Answer
The exterior angles of any polygon add up to \({360}^\circ\).
Question 7
Five of the six exterior angles of an irregular hexagon measure \({15}^\circ\), \({30}^\circ\), \({45}^\circ\), \({110}^\circ\) and \({120}^\circ\).
What is the size of the sixth exterior angle?
Answer
The exterior angles of any polygon add up to \({360}^\circ\), so \({360}^\circ-{120}^\circ-{110}^\circ-{45}^\circ-{30}^\circ-{15}^\circ={40}^\circ\).
Question 8
Four of the five interior angles of an irregular pentagon measure \({155}^\circ\), \({60}^\circ\), \({75}^\circ\) and \({100}^\circ\).
What is the size of the fifth interior angle?
Answer
The interior angles of a pentagon add up to \({540}^\circ~({180}^\circ\times{3})\), so \({540}^\circ-{155}^\circ-{60}^\circ-{75}^\circ-{100}={150}^\circ\).
Question 9
What is the size of an interior angle of a regular pentagon?
Answer
The interior angles of a pentagon add up to \({540}^\circ~({180}^\circ\times{3})\).
So in a regular pentagon, an interior angle is \({540}^\circ\div{5}={108}^\circ\).
Question 10
What is the size of an interior angle of a regular hexagon?
Answer
The interior angles of a hexagon add up to \({720}^\circ~({180}^\circ\times{4})\).
So in a regular hexagon, an interior angle is \({720}^\circ\div{6}={120}^\circ\).
More on 2D shapes
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