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If an exterior angle of a regular polygon is 30 degrees, then find the number of diagonals of the polygon.
44
54
64
74
54
Since the exterior angle is 30, n=360/30=12. Immediately calculate D=12×(9/2)=6×9=54.
Exterior angle of a regular polygon is 30 degrees.
n = \frac{360°{\text{Exterior Angle}}, \quad D = \frac{n(n-3)}{2}
Since the exterior angle is 30, n=360/30=12. Immediately calculate D=12×(9/2)=6×9=54.
Students often calculate the number of sides n correctly as 12 but then use the wrong formula for diagonals, such as n(n−1)/2, which counts sides as diagonals.
Calculate number of sides
The sum of exterior angles of any convex polygon is 360°. For a regular polygon with exterior angle θ=30°, the number of sides n is calculated using n=360°/θ.
n = \frac{360°{30° = 12
Apply diagonal formula
The formula for the number of diagonals D in a polygon with n sides is D=2n(n−3).
D=212(12−3)
Final Calculation
Substitute n=12 into the formula and simplify.
D=212×9=6×9=54
B is correct because the polygon has 12 sides, and applying the diagonal formula n(n−3)/2 yields 54.
This formula for diagonals is derived from selecting 2 vertices out of n (nC2) and subtracting the n sides that are not diagonals, i.e., n(n−1)/2−n=n(n−3)/2.