Answer :
To determine the number of sides in a polygon when the sum of its interior angles is known, we can use a specific formula based on the properties of polygons.
The formula to find the sum of the interior angles of a polygon is:
[tex]\[ \text{Sum of interior angles} = (n - 2) \times 180^\circ \][/tex]
where [tex]\( n \)[/tex] is the number of sides of the polygon.
Given that the sum of the interior angles is [tex]\( 3,420^\circ \)[/tex], we can set up the following equation:
[tex]\[ 3,420 = (n - 2) \times 180 \][/tex]
Now, we need to solve for [tex]\( n \)[/tex]:
1. Divide both sides of the equation by 180 to isolate [tex]\( n - 2 \)[/tex]:
[tex]\[ \frac{3,420}{180} = n - 2 \][/tex]
2. Simplify the left-hand side:
[tex]\[ 19 = n - 2 \][/tex]
3. Add 2 to both sides to solve for [tex]\( n \)[/tex]:
[tex]\[ 19 + 2 = n \][/tex]
[tex]\[ n = 21 \][/tex]
Therefore, the polygon has 21 sides.
The formula to find the sum of the interior angles of a polygon is:
[tex]\[ \text{Sum of interior angles} = (n - 2) \times 180^\circ \][/tex]
where [tex]\( n \)[/tex] is the number of sides of the polygon.
Given that the sum of the interior angles is [tex]\( 3,420^\circ \)[/tex], we can set up the following equation:
[tex]\[ 3,420 = (n - 2) \times 180 \][/tex]
Now, we need to solve for [tex]\( n \)[/tex]:
1. Divide both sides of the equation by 180 to isolate [tex]\( n - 2 \)[/tex]:
[tex]\[ \frac{3,420}{180} = n - 2 \][/tex]
2. Simplify the left-hand side:
[tex]\[ 19 = n - 2 \][/tex]
3. Add 2 to both sides to solve for [tex]\( n \)[/tex]:
[tex]\[ 19 + 2 = n \][/tex]
[tex]\[ n = 21 \][/tex]
Therefore, the polygon has 21 sides.
