Each interior angle of a polygon is 179 degrees. How many sides does it have?
So the question tells us the size of each interior angle in a polygon. And we’re asked to determine how many sides it has. There are two possible approaches that we’ll consider for this question. In the first method, we’ll think about the sum of the interior angles.
A key fact about polygons is that the sum of their interior angles can be found by multiplying 180 by 𝑛 minus two, where 𝑛 represents the number of sides in the polygon and is also the number of interior angles in the polygon.
In this polygon, all of the interior angles are the same size. And therefore, each interior angle can be found by dividing the sum by how many there are. So each interior angle is equal to 180 multiplied by 𝑛 minus two over 𝑛. As we know the size of each interior angle in this polygon, we can form an equation which we can then solve in order to find the value of 𝑛.
We have the expression for each interior angle, 180 multiplied by 𝑛 minus two over 𝑛 is equal to 179. Now let’s solve this equation. First, we multiply both sides of the equation by 𝑛. This gives 180 multiplied by 𝑛 minus two is equal to 179𝑛. Next, I’m going to expand the bracket on the left-hand side of the equation. This gives 180𝑛 minus 360 is equal to 179𝑛.
Next, I’ll add 360 to both sides. This gives 180𝑛 is equal to 179𝑛 plus 360. The final step is to subtract 179𝑛 from each side. This solves the equation and tells us that 𝑛 is equal to 360. So that’s our first method: considering the sum of the interior angles and hence an expression for the size of each interior angle.
Our second method is going to focus on exterior angles. Remember, the relationship between the interior and exterior angles of any polygon is that they sum to 180 degrees because they sit on a straight line together. Therefore, in this polygon each exterior angle can be found by subtracting the interior angle 179 degrees from 180 degrees. And so each exterior angle is equal to one degree.
Now a key fact about the exterior angles in regular polygons, which this polygon is as all of its interior angles are equal, is that they can be calculated by dividing 360 by the number of sides or the number of angles in the polygon, 360 over 𝑛. So we can form an equation using this expression and the fact that we know the exterior angle is one degree.
We have the simple equation 360 over 𝑛 is equal to one. This equation can be solved by multiplying both sides by 𝑛. This gives 360 is equal to 𝑛. So both methods lead us to the same conclusion, which is it the polygon has 360 sides.