Question

the sum of all the interior angles of a regular polygon is 4 times the sum of its exterior angles​

Answer 1

Answer:

the interior angle = 4 times the exterior angle.

the interior angle and the exterior angle are supplementary.

the interior angle plus the exterior angle = 180 degrees.

since the interior angle = 4 times the exterior angle, then you can replace the interior angle with the equivalent value of 4 times the exterior angle to get:

4 times the exterior angle plus the exterior angle = 180 degrees.

combine like terms to get 5 times the exterior angle = 180 degrees.

divide both sides of this equation by 5 to get:

the exterior angle = 180 / 5 = 36 degrees.  

you get interior angle = 4 * exterior angle = 4 * 36 = 144 degrees.

144 + 36 = 180 degrees so they are supplementary as required.  

the exterior angle of the regular polygon is equal to 360 / number of sides.  

solve for number of sides to get number of sides = 360 / exterior angle = 360 / 36 = 10.  

the interior angle of the regular polygon is equal to (180 * (n-2)) / n

this formula becomes 144 = (180 * (n-2)) / n.

you want to solve this equation for n.

multiply both sides of this equaiton by n to get 144 * n = 180 * (n-2)

simplify to get 144 * n = 180 * n - 360

add 360 to both sides of this equation and subtract 144 * n from both sides of this equaiton to get 360 = 180 * n - 144 * n.

simplify to get 360 = 36 * n

divide both sides of this equation by 36 to get 360 / 36 = n

solve for n to get n = 360 / 36 = 10.  

you get 10 sides to the regular polygon either way.  

something else to note:

the exterior angle is equal to the central angle.

the number of sides is equal to 360 / the central angle or 360 / the exterior angle.

either formula will do.  

finding the number of sides using the interior angle is messier than finding the number of sides using the exterior angle.

if you are given the interior angle, just subtract that from 180 to get the exterior angle and then use the number of sides = 360 / exterior angle to find the number of sides.

it's much easier.  

the number of sides is equal to 10.  

the sum of the interior angles is equal to 10 * 144 = 1440 degrees.  

this can also be found using the sum of the interior angles of a regular polygon which is:  

sum of interior angles = (180 * (n-2)) = 180 * 8 = 1440.

Step-by-step explanation:

Answer 2

Every polygon has total 360 degrees in exterior when added.

So,4 x 360=1440 degrees.

The number of sides=

Let the number of sides be X

(X-2)x180=1440

(X-2)=1440/180=8

X=8+2=10

10 sides.