Answer :
Answer:
[tex]A\approx 10.4\text{ Millimeters}^2[/tex]
Step-by-step explanation:
Please consider the attached image.
We have been given that each cell of the honeycomb is a regular hexagon. The side of one of these honeycomb cells is 2 millimeters.
We know that a regular hexagon is made of 6 equilateral triangles.
Let us find apothem of our given hexagon using Pythagoras theorem as:
[tex]a^2=2^2-1^2[/tex]
[tex]a^2=4-1[/tex]
[tex]a^2=3[/tex]
Take positive square root of both sides:
[tex]a=\sqrt{3}[/tex]
Now, we will use formula for the area of the regular polygon to solve our given problem as:
A = ½ap, where,
A = Area of the regular polygon,
a = Apothem,
p = Perimeter of polygon.
Perimeter of our given hexagon would be 6 times 2 millimeters:
[tex]\text{Perimeter of the given hexagon}=6\times 2\text{ millimeters}[/tex]
[tex]\text{Perimeter of the given hexagon}=12\text{ millimeters}[/tex]
[tex]A=\frac{1}{2}\times \sqrt{3}\text{ Millimeters}\times 12\text{ millimeters}[/tex]
[tex]A=6\times \sqrt{3}\text{ Millimeters}^2[/tex]
[tex]A=6\times 1.7320508075688773\text{ Millimeters}^2[/tex]
[tex]A=10.3923048454132638\text{ Millimeters}^2[/tex]
Round to nearest tenth:
[tex]A\approx 10.4\text{ Millimeters}^2[/tex]
Therefore, the area of the given honeycomb is approximately 10.4 millimeters square.
