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Angles and Polygons

Angles and Polygons. By Mr Robertson. In and Out. Exterior angle. In this pentagon there are: 5 Interior angles 5 exterior angles 5 sides. Interior angle. Exterior Angles. The sum of the exterior angles = 360

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Angles and Polygons

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  1. Angles and Polygons By Mr Robertson

  2. In and Out Exterior angle • In this pentagon there are: • 5 Interior angles • 5 exterior angles • 5 sides Interior angle

  3. Exterior Angles The sum of the exterior angles = 360 Another way to express this is to say that the number of exterior angles × the size of each exterior angle = 360 And since the number of sides is the same as the number of angles we can write: number of sides x exterior angle = 360 se = 360

  4. Exterior Angles And if we know the size of the exterior angle we can find the number of sides: number of sides= 360 ÷ exterior angle s = 360 e

  5. Exterior Angles We can rearrange this so if we know the number of sides we can find the exterior angle: exterior angle= 360 ÷ number of sides e = 360 s

  6. In and Out Look at the diagram and you will see that the exterior and the interior angle combine to make a straight line. This means that: interior angle = 180 – exterior angle exterior angle = 180 – interior angle i = 180 – ee = 180 – i

  7. Summing Up Sometimes you have to calculate the sum of the interior angles. This is going to be equal to the number of sides × the size of the interior angle. This can be written like this: Sum of interior angles= si But the interior angle is just 180 – the exterior angle, so we can write it like this instead: si = s(180 – e) But the exterior angle is just 360 ÷ the number of sides, so we can write it like this instead: si = s(180 – 360 ÷ s) And if we multiply out this bracketed expression and simplify we get: si = 180s – 360 So it turns out we can work out the sum of the interior angles, just by knowing the sides!

  8. In and Out • So it turns out that we can work out everything else just by knowing one of: • Number of sides • Size of exterior angle • Size of interior angle • Sum of interior angles • Let’s see an example

  9. Example • The sum of the interior angles of a regular polygon is 9000°. How many sides has it got? • Start with:si = 9000 • Then replace i with 180 – e to get:s(180 – e) = 9000 • Then replace e with 360/s to get:s(180 – 360/s) = 9000 • Multiply out the bracket and solve:180s – 360 = 9000 180s = 9360 s = 9360 ÷ 180 s = 52 • So the polygon has 52 sides interior angle = 180 – exterior angle exterior angle = 360 ÷ number of sides

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