Question

If the difference between the exterior angle of a 'n' sided regular polygon and an (n + 1) sided regular polygon is 12°, find the value of n.

Answer 1

We know that sum of exterior angles of a polygon = 360°

Each exterior angle of a regular polygon of 360°

n sides = `360^circ/"n"`

and exterior angle of the regular polygon of 

(n + 1) sides = `360^circ/("n" + 1)`

`therefore 360^circ/"n" - 360^circ/("n" + 1) = 12`

`=> 360 [1/"n" - 1/("n" + 1)] = 12`

`=> 360 [("n" + 1 - "n")/("n"("n" + 1))] = 12`

`=> (30 xx 1)/("n"^2 + "n") = 12`

`=> 12 ("n"^2 + "n") = 360^circ`

⇒ n² + n = 36    (Dividing by 12)

⇒ n² + n − 30 = 0

⇒ n² + n − 30 = 0

⇒ n² + 6n − 5n - 30 = 0  ...{∵ −30 = 6 × (−5), 1 = 6 − 5}

⇒ n(n + 6) − 5(n + 6) = 0

⇒ (n + 6)(n + 5) = 0

Either n + 6 = 0, then n = −6 which is not possible being negative

orn - 5 = 0 then n = 5

Hence n = 5.

Answer 2

Step 1: Exterior Angle of a Regular Polygon

The formula for the exterior angle of a regular polygon with n sides is:

Exterior angle = `(360°)/n`

For an n-sided polygon, the exterior angle is `(360°)/n`

For an (n + 1)-sided polygon, the exterior angle is `(360°)/(n+1)`

Step 2: Difference Between the Exterior Angles

According to the problem, the difference between the exterior angle of an n-sided polygon and an (n + 1)-sided polygon is 12^∘. Therefore:

`360/n - 360/(n+1) = 12`

Step 3: Solve for n

Factor out 360 on the left-hand side: `360 (1/n - 1/(n+1)) =12`

Divide both sides by 360:

`1/n - 1/(n+1) = 12/360`

Simplify `12/360: 1/n - 1/(n+1) = 1/30`

Step 4: Combine the Terms on the Left

`1/n - 1/(n+1) = ((n+1) - n)/(n(n+1))`

`1/n - 1/(n+1) = 1/(n(n+1))`

`1/(n(n+1)) = 1/30`

Step 5: Solve for n

Cross-multiply to eliminate the denominators:

n(n + 1) = 30

n² + n − 30 = 0

Step 6: Solve the Quadratic Equation

`n = (-b+- sqrt(b^2 - 4ac))/(2a)`

a = 1, b = 1, and c = −30

Substitute the values:

`n = (-1 +- sqrt(1^2 - 4(1)(-30)))/(2(1))`

`n = (-1 +- sqrt(1+120))/(2)`

`n = (-1 +- sqrt(121))/(2)`

`n = (-1 +- 11)/(2)`

Now, solve for the two possible values of n:

`n = (-1+11)/2 = 10/2 = 5`

`n = (-1-11)/2 = (-12)/2 = -6` (not valid as n must be positive).

The value of n is 5.