Answer:
[tex]A=512\sqrt{3}\ cm^2[/tex]
Step-by-step explanation:
we know that
The regular hexagon can be divided into six equilateral triangles
Remember that an equilateral triangle has three equal sides and three equal interior angles (the measure of each interior angle is 60 degrees)
The length side of the triangle is equal to the length side of the regular hexagon
step 1
Find the length side of triangle
Let
b ----> the length side of triangle
see the attached figure to better understand the problem
In the right triangle OAN
Applying the Pythagorean Theorem
[tex]OA^2=AN^2+ON^2[/tex]
we have
[tex]OA=b\ cm[/tex]
[tex]AN=\frac{b}{2}\ cm[/tex]
[tex]ON=16\ cm[/tex] ----> the apothem
substitute
[tex]b^2=(\frac{b}{2})^2+16^2[/tex]
[tex]b^2-\frac{b^2}{4}=256[/tex]
[tex]\frac{3b^2}{4}=256[/tex]
[tex]b=\sqrt{\frac{1,024}{3}}\ cm[/tex]
simplify
[tex]b=\frac{32\sqrt{3}}{3}\ cm[/tex]
step 2
Find the area of the regular hexagon
Find the area of one triangle and multiply by 6
[tex]A=6[\frac{1}{2}(b)(h)][/tex]
we have'
[tex]b=\frac{32\sqrt{3}}{3}\ cm[/tex]
[tex]h=16\ cm[/tex]
substitute
[tex]A=6[\frac{1}{2}(\frac{32\sqrt{3}}{3})(16)][/tex]
[tex]A=512\sqrt{3}\ cm^2[/tex]
