
A regular polygon has angles of size 150 degrees each. How many sides does the polygon have?
Answer
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Hint: We will look at the definition of a regular polygon. We will see some examples of regular polygons and also the properties of a regular polygon. We will find the measure of the exterior angle using the given measure of the interior angle. We will use the property of the exterior angles of a regular polygon to find the number of sides of the regular polygon.
Complete step by step answer:
A regular polygon is defined as a polygon that has all the sides of equal length and all the angles of the same measure. An equilateral triangle is a regular polygon that has three sides. A square is a regular polygon that has four sides.
In a regular polygon, the sum of an interior angle and its exterior angle is . We will use this property to find the measure of the exterior angle of the regular polygon given in the question. Its interior angle has measure . Therefore, the exterior angle has measure .
Regular polygons have the property that the sum of all its exterior angles is . Using this property, we can see that
Substituting the respective values, we get the following,
Therefore, the number of sides of the regular polygon with interior angle is 12. This regular polygon looks like the following,
Note: The sum of the interior angles of this regular polygon will be . It is useful to know the relation between the interior angles, the exterior angles and the number of sides of a regular polygon. There is a formula to find the sum of interior angles of a polygon using the number of sides. This formula uses the fact that we can divide a regular polygon into triangles.
Complete step by step answer:
A regular polygon is defined as a polygon that has all the sides of equal length and all the angles of the same measure. An equilateral triangle is a regular polygon that has three sides. A square is a regular polygon that has four sides.
In a regular polygon, the sum of an interior angle and its exterior angle is
Regular polygons have the property that the sum of all its exterior angles is
Substituting the respective values, we get the following,
Therefore, the number of sides of the regular polygon with interior angle

Note: The sum of the interior angles of this regular polygon will be
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