The diagram shows a square, a regular hexagon and part of another regular polygon meeting at point P - OCR - GCSE Maths - Question 18 - 2018 - Paper 1

Let the number of sides of the other regular polygon be ( n ). The interior angle of the other polygon can be expressed using the same formula:

$\text{Interior Angle} = \frac{(n-2) \times 180°}{n}$

From the diagram, it is known that at point P, the angles of the square (90°) and the hexagon (120°) meet. Since the sum of angles around point P is 360°, we have:

$90° + 120° + \text{Interior Angle of Other Polygon} = 360°$

This simplifies to:

$\text{Interior Angle of Other Polygon} = 360° - 90° - 120° = 150°$

Now, substituting this back into the interior angle formula:

$150° = \frac{(n-2) \times 180°}{n}$

Multiplying both sides by ( n ):

$150n = (n-2) \times 180$

Expanding this gives:

$150n = 180n - 360$

Rearranging the terms results in:

$30n = 360$

Dividing both sides by 30 gives:

$n = 12$

Thus, the number of sides of the other regular polygon is 12.