Question

Find the sum of all the measures of all the exterior angles of a decagon.

Answer 1

Hint: According to the question given in the question we have to determine the sum of all the measures of all the exterior angles of a decagon. So, first of all we have to use the formula to find the each interior angle of a given polynomial which is explained below:

Formula used: Each in interior angle of a polygon $ = \dfrac{{{{360}^\circ}}}{n}........................(a)$
Where n is the number of sides of a given polygon.
After obtaining the interior angle of the decagon with the help of formula (a) given above now we have to obtain the each exterior angle for the polygon with the help of the formula to find the each exterior angle as given below:
Each interior angle of a polygon$ = {180^\circ} - \dfrac{{{{360}^\circ}}}{n}........................(b)$
After obtaining each exterior angle for the given polygon we can obtain the sum of all exterior angles by multiplying each exterior angle with the number of sides of the given polygon decagon.

Complete step-by-step answer:
Step 1: First of all we have to determine the interior angle of the decagon with the help of the formula as mentioned in the solution hint and as we know that the number of sides of a decagon are n = 10. Hence, each interior angle of the decagon:
$
   = \dfrac{{{{360}^\circ}}}{{10}} \\
   = {36^\circ}
 $
Step 2: Now, in this step we have to determine the each exterior angle of the decagon with the help of the formula (B) as mentioned in the solution hint. Hence,
$
   = {180^\circ} - {36^\circ} \\
   = {144^\circ}
 $
Step 3: Now, we have to find the sum of all the exterior angles of the decagon which can be obtained by multiplying the number of sides of the decagon which are n = 10 with the each exterior angle as obtained in the solution step 2. Hence,
$
   = 10 \times {144^\circ} \\
   = {1440^\circ}
 $

Hence, with the help of formula (A) and formula (B) we have obtained the sum of all the measures of all the exterior angles of a decagon is $ = {1440^\circ}$

Note: If we want to determine the number of sides of any given polygon then we just have to use the formula $S = (n - 2) \times {180^\circ}$ where, n is the number of sides and if we know the value of S which is the sum of all the interior angles of a given polygon we can easily determine the number of sides of a given polygon.