A regular polygon with 12 sides (dodecagon) is inscribed in a square of area 24 square units as shown in the figure where four of the vertices are mid points of the sides of the square . The area of the dodecagon in square units is
Answers
Given that the the polygon is inscribed inside a square, we can get the sides of the square.
Area = 24 square units.
Side = √24 = 4.899 units
Given that the four vertices of the polygon are the midpoint of the sides, the polygon is divided into four right angled triangle whose height and base are equal.
The height of each triangle is half the sides of the square.
Therefore :
h = b = 4.899/2
= 2.449
Area of one right angled triangle :
0.5 × 2.449 × 2.449 = 2.9988 square units.
We have four right angled triangles in the polygon hence the area of the polygon is :
2.9988 × 4 = 11.995 square units
Provided that, the the polygon is inscribed inside a square, we can calculate sides of the square as below,
Area of the Square = 24 square units.
Each side of the given Square= √24 = 4.8989794855 units
And the four vertices of the polygon are at the midpoint of the Square's each sides, the polygon is divided into 12 Isosceles triangle whose two sides are equal.
So, the polygon has 12 Isosceles triangle with a side of √24/2 ( equal side). and a included angle with 30 degree because 360 / 12= 30.
Now, area of each triangle is = (√6 * √6 * sin 30) /2
= 1.5
Therefore, Area of total polygon is = 12*1.5
= 18