Answered

The sum of the interior angles, $s$, in an $n$-sided polygon can be determined using the formula $s=180(n-2)$, where $n$ is the number of sides.



Using this formula, how many sides does a polygon have if the sum of the interior angles is 1,260°? Round to the nearest whole number.


A6 sides
B7 sides
C8 sides
D9 sides

Verified Answer :

  • Use the formula \(s = 180(n - 2)\) with the given sum of interior angles \(s = 1260^{\circ}\).
  • Divide the total sum by 180 to find the number of triangles that compose the polygon, which is \(n - 2 = 7\).
  • Add 2 to this value to determine the number of sides, resulting in \(n = 9\).
  • Conclude that the polygon has \(\boxed{9}\) sides.

Explanation

  1. Analyze the problem and given information.

The problem asks to determine the number of sides \(n\) of a polygon given that the sum of its interior angles is \(1{,}260^{\circ}\). We are provided with the formula for the sum of interior angles, \(s = 180(n - 2)\), where \(n\) is the number of sides. Our goal is to solve for \(n\) given \(s = 1260\).

  1. Solve for n using the given formula.

To find the number of sides \(n\), we substitute the given sum \(s = 1260^{\circ}\) into the formula \(s = 180(n - 2)\):
\[1260 = 180(n - 2)\]
Next, we isolate the expression \((n - 2)\) by dividing both sides of the equation by \(180\):
\[\frac{1260}{180} = n - 2\]
Calculating the quotient:
\[7 = n - 2\]
Finally, solve for \(n\) by adding \(2\) to both sides:
\[n = 7 + 2\]
\[n = 9\]
Thus, the polygon has \(9\) sides.

Examples

Understanding the relationship between the number of sides and the sum of interior angles is fundamental in geometry. For example, architects and structural engineers often use this formula to design polygonal structures, such as a nonagonal (9-sided) room or a complex floor tile layout, ensuring the sum of the corner angles fits correctly in their construction plans. If a floor plan requires a room with interior angles summing up to \(1{,}260^{\circ}\), they would know exactly that a nonagon is the correct shape to implement.

Related Questions