A regular polygon has an interior angle equal to 150 degrees. Work out the number of sides that the polygon has.
If a polygon is regular, each of its interior angles are the same size. This also means that each of its exterior angles will be equal. In order to calculate the number of sides of this polygon, we need to use two formulas. Firstly, the exterior angle plus the interior angle will always add up to 180 degrees. This is because the interior and exterior angles are on a straight line. And we know that angles on a straight line sum to 180 degrees.
We’re told in this question that the interior angle is equal to 150 degrees. If we let the exterior angle be the letter 𝑥, we have the equation 𝑥 plus 150 is equal to 180. Subtracting 150 from both sides of the equation gives us 𝑥 equals 30 as 180 minus 150 is equal to 30. This means that each exterior angle of the regular polygon is equal to 30 degrees.
We were asked to calculate the number of sides of the polygon. In order to do this, we need to use a second formula. To calculate the number of sides of any regular polygon, we divide 360 by the size of each exterior angle. In this case, we need to divide 360 by 30 as each exterior angle was 30 degrees. We can divide the top and bottom of this fraction by 10. 360 divided by 10 is 36 and 30 divided by 10 is equal to three. The number of sides is equal to 36 divided by three which is 12 as 12 multiplied by three is 36.
We can, therefore, conclude that a regular polygon with interior angle of 150 degrees has 12 sides.