Question

How many diagonals are there of a polygon having 12 sides?
A. 12
B. 24
C. 36
D. 54

Answer 1

Hint: Here, we are given that there is a polygon having 12 sides. We need to find out the number of diagonals in it. We will use the formula \[\dfrac{1}{2} \times n \times (n - 3)\] to find and get the final output.

Complete step by step answer:
Given that, a polygon has 12 sides.We know that a 12-sided polygon is called a dodecagon. Dodecagons can be regular, in which all interior angles and sides are equal in measure. Also, they can be irregular, with different angles and sides of different measurements.As we also know, each interior angle of a regular dodecagon is equal to 150 degrees.

Now, we will find the diagonals of the polygon as below: We know that, the number of distinct diagonals that can be drawn in a dodecagon from all its vertices. We will use the formula to get the number of diagonals of the polygon having n sides, where n is the number of sides = 12 (given).
\[\dfrac{1}{2} \times n \times (n - 3)\]
Substituting the value of n = 12, we will get,
\[\dfrac{1}{2} \times 12 \times (12 - 3)\]
On evaluating this, we will get,
\[6 \times (9)\]
Removing the brackets, we will get,
\[6 \times 9 = 54\]

Hence, there are 54 diagonals in a polygon having 12 sides.

Note: We can calculate each interior angle by using the formula: \[\dfrac{{180n - 360}}{n}\] where n is the number of sides of the polygon. And the sum of the interior angles can be calculated using the formula: \[(n - 2) \times {180^ \circ }\]. A dodecagon is a polygon with 12 sides, 12 angles and 12 vertices. Dodecagons can be of different types depending upon the measure of their sides, angles, and many such properties. They are 4 types as below:
- Regular
- Irregular
- Concave
- Convex