Answer :
To find the value of [tex]\( n \)[/tex] (the number of sides) for a polygon whose sum of interior angles is [tex]\( 1440^\circ \)[/tex], you can follow these steps:
1. Recall the formula for the sum of interior angles of an [tex]\( n \)[/tex]-sided polygon:
The sum of the interior angles of an [tex]\( n \)[/tex]-sided polygon is given by the formula:
[tex]\[ \text{Sum of interior angles} = (n - 2) \times 180^\circ \][/tex]
2. Set up the equation:
Given that the sum of the interior angles is [tex]\( 1440^\circ \)[/tex], we can set up the equation using the formula:
[tex]\[ (n - 2) \times 180 = 1440 \][/tex]
3. Solve for [tex]\( n \)[/tex]:
Divide both sides of the equation by [tex]\( 180 \)[/tex] to isolate [tex]\( (n - 2) \)[/tex]:
[tex]\[ n - 2 = \frac{1440}{180} \][/tex]
4. Calculate the value inside the equation:
[tex]\[ n - 2 = 8 \][/tex]
5. Add 2 to both sides to solve for [tex]\( n \)[/tex]:
[tex]\[ n = 8 + 2 \][/tex]
[tex]\[ n = 10 \][/tex]
6. Identify the polygon:
A polygon with [tex]\( 10 \)[/tex] sides is called a decagon.
Therefore, the value of [tex]\( n \)[/tex] is [tex]\( 10 \)[/tex], which means that the polygon is a decagon.
1. Recall the formula for the sum of interior angles of an [tex]\( n \)[/tex]-sided polygon:
The sum of the interior angles of an [tex]\( n \)[/tex]-sided polygon is given by the formula:
[tex]\[ \text{Sum of interior angles} = (n - 2) \times 180^\circ \][/tex]
2. Set up the equation:
Given that the sum of the interior angles is [tex]\( 1440^\circ \)[/tex], we can set up the equation using the formula:
[tex]\[ (n - 2) \times 180 = 1440 \][/tex]
3. Solve for [tex]\( n \)[/tex]:
Divide both sides of the equation by [tex]\( 180 \)[/tex] to isolate [tex]\( (n - 2) \)[/tex]:
[tex]\[ n - 2 = \frac{1440}{180} \][/tex]
4. Calculate the value inside the equation:
[tex]\[ n - 2 = 8 \][/tex]
5. Add 2 to both sides to solve for [tex]\( n \)[/tex]:
[tex]\[ n = 8 + 2 \][/tex]
[tex]\[ n = 10 \][/tex]
6. Identify the polygon:
A polygon with [tex]\( 10 \)[/tex] sides is called a decagon.
Therefore, the value of [tex]\( n \)[/tex] is [tex]\( 10 \)[/tex], which means that the polygon is a decagon.
