The formula 180^(∘)(n-2)n describes the measure of an interior angle in a regular n-gon. We can also use the fact that the exterior angles of a polygon always sum to 360^(∘).
8 sides. See solution for different methods.
Practice makes perfect
Method 1
If we know the number of sides in a polygon n, the sum of its angles can be calculated using the following formula.
180^(∘)(n-2)
Furthermore, if a polygon is regular all interior angles are congruent. Therefore, if we divide the formula by the number of sides the quotient must equal the measure of an interior angle. From the exercise, we know that this measure is 135^(∘). With this information, we can write the following equation.
180^(∘)(n-2)/n=135^(∘)
Let's solve this equation for n.
The exterior angles of an n-gon always sum to 360^(∘). Since the exterior angles of a regular polygon are congruent, we can write an equation describing the measure of our polygons exterior angles, m∠ θ.
m∠ θ=360^(∘)/n
The exterior and corresponding interior angle in a polygon form a linear pair. With this information, we can figure out the measure of an exterior angle in the polygon.
135^(∘)+m∠ θ = 180^(∘) ⇔ m∠ θ = 45^(∘)
When we know the measure of the exterior angle, we can use this to determine the number of sides.
