A regular polygon has interior angles that each measure 140^\circ. What is the total numbe… — GRE General Test

A regular polygon has interior angles that each measure $140^\circ$ . What is the total number of diagonals that can be drawn in this polygon?

The total number of diagonals that can be drawn in the polygon is 27.

To find the total number of diagonals in a regular polygon, first determine its number of sides

n

. Using the interior angle formula

\frac{(n-2) \times 180^\circ}{n} = 140^\circ

, we find

n = 9

. Next, applying the diagonal formula

D = \frac{n(n-3)}{2}

for

n = 9

gives

D = \frac{9 \times 6}{2} = 27

. Thus, the value 27 is correct.

Find the number of sides

n

of the regular polygon using the interior angle measure.

The polygon has

n = 9

sides.

Each interior angle of a regular

n

-gon is given by

\frac{(n - 2) \times 180^\circ}{n} = 140^\circ

. Solving for

n

180n - 360 = 140n \implies 40n = 360 \implies n = 9

Calculate the total number of diagonals using the formula

D = \frac{n(n - 3)}{2}

The total number of diagonals is 27.

Substituting

n = 9

into the formula yields

D = \frac{9(9 - 3)}{2} = \frac{9 \times 6}{2} = 27

Interior Angle Measure and Diagonal Formula for Regular Polygons

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