Two regular pentagons and a regular decagon, all with the same side length, can completely surround a point. An equilateral triangle, a regular octagon, and a regular n-gon, all with the same side length, also completely surround a point. Find n.
+23246 The formula to find the number of degrees in each vertex angle of a regular polygon is: Degrees = (n - 2) · 180 / n
Each angle of an equilateral triangle is 60°.
In a regular octagon, n = 8 and Degrees = (n - 2) · 180 / n ---> Degrees = (8 - 2) · 180 / 8 ---> Degrees = 135°.
Since there are 360° around a point, the number of degrees in each vertex of the regular n-gon is:
360° - 60° - 135° = 165°.
To find the value of n: 165° = (n - 2) · 180 / n
---> 165 · n = (n - 2) · 180
---> 165n = 180n - 360
---> -15n = -360
---> n = 24
