A regular polygon has n sides - OCR - GCSE Maths - Question 11 - 2018 - Paper 6

Let the exterior angle be represented as ( x ). Then, the interior angle can be expressed as ( 5x ).

The formula for the sum of the exterior angles of any polygon is always ( 360^\circ ), so each exterior angle can be calculated as:

[ x = \frac{360}{n} ]

Substituting this value into the equation for the interior angle gives:

[ 5x = 5 \times \frac{360}{n} = \frac{1800}{n} ]

The relationship between the interior and exterior angles is also given by:

[ \text{Interior angle} = 180 - x ]

Thus, substituting for the exterior angle we have:

[ 5 \times \frac{360}{n} = 180 - \frac{360}{n} ]

Now, simplifying this equation:

[ 5 \times \frac{360}{n} + \frac{360}{n} = 180 ]

This simplifies to:

[ \frac{1800}{n} + \frac{360}{n} = 180 ] [ \frac{2160}{n} = 180 ]

Next, multiplying both sides by ( n ) gives:

[ 2160 = 180n ]

Dividing both sides by 180 results in:

[ n = \frac{2160}{180} = 12 ]