Question

name: andrew carter 12/10/25
the following review sheet is for the benefit of them whomst fill it out. the recommendation is that those who do so write clearly and show all necessary steps so that they might make sense of it when the time comes.
formula for calculating the interior angle sum of a polygon with n number of sides;
identify the interior angle sum of the following polygons:
three polygon images
what is the interior angle sum of a 54 sided polygon?
how many sides does a polygon with an interior angle sum of 2340° have?
define regular polygon:
formula for calculating the number of sides n of a regular polygon with an interior angle a°:

Explanation:

Step1: Recall the formula for interior angle sum

The formula for the interior angle sum \( S \) of a polygon with \( n \) sides is \( S=(n - 2)\times180^{\circ} \).

Step2: Calculate for 54 - sided polygon

For a 54 - sided polygon, \( n = 54 \). Substitute into the formula:
\( S=(54 - 2)\times180^{\circ}=(52)\times180^{\circ}=9360^{\circ} \)

Step3: Find the number of sides for sum \( 2340^{\circ} \)

We know \( S=(n - 2)\times180^{\circ}=2340^{\circ} \).
First, divide both sides by \( 180^{\circ} \): \( n - 2=\frac{2340^{\circ}}{180^{\circ}} = 13 \)
Then, add 2 to both sides: \( n=13 + 2=15 \)

Step4: Define Regular Polygon

A regular polygon is a polygon that is equiangular (all interior angles are equal in measure) and equilateral (all sides have the same length).

Step5: Derive formula for \( n \) with interior angle \( a^{\circ} \)

In a regular polygon, each interior angle \( a^{\circ}=\frac{(n - 2)\times180^{\circ}}{n} \)
Multiply both sides by \( n \): \( a^{\circ}\times n=(n - 2)\times180^{\circ} \)
Expand the right - hand side: \( a^{\circ}n = 180^{\circ}n-360^{\circ} \)
Bring terms with \( n \) to one side: \( 180^{\circ}n-a^{\circ}n = 360^{\circ} \)
Factor out \( n \): \( n(180^{\circ}-a^{\circ})=360^{\circ} \)
So, \( n=\frac{360^{\circ}}{180^{\circ}-a^{\circ}} \)

Answer:

s:

• Interior Angle Sum of 54 - sided polygon: \( 9360^{\circ} \)
• Number of sides for sum \( 2340^{\circ} \): 15
• Definition of Regular Polygon: A polygon that is equiangular and equilateral.
• Formula for \( n \) with interior angle \( a^{\circ} \): \( n=\frac{360}{180 - a} \) (where angles are in degrees)