Question

If each interior angle of a regular polygon is 144°. The polygon has: * a. 24 sides b. 10 sides c. 12 sides d. 20 sides​

Answer 1

$\large\underline{\text{Required Knowledge}}$

$\red{\bigstar}$Angle sum of a triangle.

The angles of a triangle sum to $180^{\circ}$.

$\red{\bigstar}$The number of diagonals.

In a regular polygon of $n$ sides, we can draw $(n-3)$ diagonals from a point since we cannot connect adjacent sides or itself to get a diagonal. Also, inside is $(n-2)$ triangles.

$\red{\bigstar}$Property of regular polygons.

Every single angle is equal in regular polygons.

$\large\underline{\text{Solution}}$

Since there are $(n-2)$ triangles, the angle sum of all the triangles is $180^{\circ}\times(n-2)$.

Then since the regular polygon has $n$ equal angles, one angle will be $\dfrac{180^{\circ}\times(n-2)}{n}$.

Now we get the equation.

$\implies\dfrac{180^{\circ}\times(n-2)}{n}=144^{\circ}$

$\implies180^{\circ}\times n-360^{\circ}=144^{\circ}\times n$

$\implies(180^{\circ}-144^{\circ})\times n=360^{\circ}$

$\implies36^{\circ}\times n=360^{\circ}$

$\implies n=10$

Hence, the shape is a regular 10-sided polygon.

Answer 2

Step-by-step explanation:

Given :

• Each interior angle = 144°

Formula used :

• Each interior angle = 180(n - 2)/n (n = no. of sides)

Calculation :

• Each interior angle = 144°

• ⇒ 180(n - 2) = 144n

• ⇒ n = 10

The no. of side of polygon is 10

• Each interior angle = 144°

• The each exterior angle = 180 - 144 = 36°

• The number of sides = 360/36 = 10

The no. of side of polygon is 10.