UpStudy Free Solution:

To determine which regular polygon has the largest interior angle, we can use the formula for the interior angle of a regular \(n\)-sided polygon:

\[\text { Interior Angle} = \frac { ( n- 2) \times 180^ \circ } { n} \]

Let's calculate the interior angles for each of the given polygons:

1. Pentagon (\(n = 5\)):

\[\text { Interior Angle} = \frac { ( 5- 2) \times 180^ \circ } { 5} = \frac { 3 \times 180^ \circ } { 5} = \frac { 540^ \circ } { 5} = 108^ \circ \]

2. Octagon (\(n = 8\)):

\[\text { Interior Angle} = \frac { ( 8- 2) \times 180^ \circ } { 8} = \frac { 6 \times 180^ \circ } { 8} = \frac { 1080^ \circ } { 8} = 135^ \circ \]

3. Square (\(n = 4\)):

\[\text { Interior Angle} = \frac { ( 4- 2) \times 180^ \circ } { 4} = \frac { 2 \times 180^ \circ } { 4} = \frac { 360^ \circ } { 4} = 90^ \circ \]

4. Hexagon (\(n = 6\)):

\[\text { Interior Angle} = \frac { ( 6- 2) \times 180^ \circ } { 6} = \frac { 4 \times 180^ \circ } { 6} = \frac { 720^ \circ } { 6} = 120^ \circ \]

Comparing the interior angles:

Pentagon: \(108^ \circ \)

Octagon: \(135^ \circ \)

Square: \(90^ \circ \)

Hexagon: \(120^ \circ \)

The regular polygon with the largest interior angle is the octagon with an angle measure of \(135^ \circ \).

So, the correct answer is:

b. octagon

Supplemental Knowledge

A regular polygon is a polygon with all sides and angles equal. The measure of each interior angle of a regular polygon with \(n\) sides can be calculated using the formula:

\[\text { Interior Angle} = \frac { ( n- 2) \times 180^ \circ } { n} \]

where \(n\) is the number of sides.

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