Question

How many degrees will the skaters turn if they go once around a regular hexagon? A regular octagon? A regular polygon with n sides?

Answer 1

Hint: Here in this question, we have to find the degree, when the skaters go around for the different shapes. We have formula \[(n - 2) \times {180^ \circ }\] , by using this we determine the degree the skater goes around. The “n” represents the number of sides. It depends on the exterior and interior angle.

Complete step-by-step answer:
The skaters while skating go around the surface. The surface depends on the number of sides containing the surface. For any surfaces we have interior angles and the exterior angles. as we know that the sum of interior angles of the polygon surface is \[(n - 2){180^ \circ }\] and the sum of exterior angles of the polygon surface is
\[{180^ \circ }(n - 2)\] .
To find the degree that the skater goes around we use the formula
\[(n - 2) \times {180^ \circ }\] where n is the number of sides.
Now consider the regular hexagon, the hexagon is one of the polygons and hexagon has six sides therefore n is 6.
Therefore by using the formula
\[(n - 2) \times {180^ \circ }\] and substituting the value of n as 6 we get
 \[(n - 2) \times {180^ \circ } \\
\Rightarrow (6 - 2){180^ \circ } \\
\Rightarrow 4 \times {180^ \circ } \;
= {720^ \circ }\]
So, the correct answer is “\[ {720^ \circ }\] ”.

Therefore when the skater goes around the regular hexagon he covers \[{720^ \circ }\] .
Now consider the regular octagon, the octagon is one of the polygons and octagon has eight sides therefore n is 8.
Therefore by using the formula
\[(n - 2) \times {180^ \circ }\] and substituting the value of n as 8 we get
\[(n - 2) \times {180^ \circ } \\
\Rightarrow (8 - 2){180^ \circ } \\
\Rightarrow 6 \times {180^ \circ } \;
= {1080^ \circ }\]
So, the correct answer is “\[ {1080^ \circ }\] ”.

Therefore when the skater goes around the regular hexagon he covers \[{1080^ \circ }\] .
Now consider the regular polygon with n sides and polygon has n sides therefore n will be n.
Therefore by using the formula \[(n - 2) \times {180^ \circ }\] and substituting the value of n as n we get
 \[(n - 2) \times {180^ \circ } \\
\Rightarrow (n - 2){180^ \circ }\]
Therefore when the skater goes around the regular polygon with n sides he covers
\[(n - 2){180^ \circ }\] .
Hence we have determined the solution.
So, the correct answer is “\[(n - 2){180^ \circ }\] ”.

Note: In the polygon we have interior angles and the exterior angles. the interior angles which are lies or present inside the polygon surface. The exterior angles which are lies or present outside the polygon surfaces. The word poly means number and gon means sides.