• Given a polygon with 18 sides, find the sum of the measures of its interior angles. 

    A polygon with 18 sides has an interior angle measure sum of ______°.

     
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    The answer is 180°*16 = 2880°.

    The general formula is Sa=180°*(n-2) for any polygon with n sides.

    This formula may be proved relatively easy for convex polygons. Consider a point P in the interior of the polygon and draw segments from this point to all vertices.

    We obtain...

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    Hello!

    The answer is 180°*16 = 2880°.

    The general formula is Sa=180°*(n-2) for any polygon with n sides.

    This formula may be proved relatively easy for convex polygons. Consider a point P in the interior of the polygon and draw segments from this point to all vertices.

    We obtain n triangles, each has the sum of angles 180°. The sum of angles whose vertex isn't P is the sum of interior angles of the polygon. The sum of angles whose vertex IS P is one complete circle, i.e. 360°.

    So we have 180°*n=Sa+360°, or Sa=180°*(n-2).

    This theorem holds for concave polygons also but the proof is more difficult.

    Approved by eNotes Editorial Team
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    The sum of the measures of the interior angle of any polygon is the number of sides minus 2 multiplied by 180.

    `S=(n-2)*180`

    `S=(18-2)(180)`

    S=16(180)

    S=2880

    A polygon with 18 sides has an interior angle sum of 2880 degrees.

    Approved by eNotes Editorial Team