Question

One of the most amazing things that happens in nature is the ability of bees to make a honeycomb. Each cell of the honeycomb is a regular hexagon. If the side of one of these honeycomb cells is 2 millimeters, then what is its area? Round your answer to the nearest tenth. Use the equation A = ½ap for the area of the regular polygon.

Answer 1

`A\approx 10.4\text{ Millimeters}^2`

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:

`a^2=2^2-1^2`

`a^2=4-1`

`a^2=3`

Take positive square root of both sides:

`a=√(3)`

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:

`\text{Perimeter of the given hexagon}=6* 2\text{ millimeters}`

`\text{Perimeter of the given hexagon}=12\text{ millimeters}`

`A=(1)/(2)* √(3)\text{ Millimeters}* 12\text{ millimeters}`

`A=6* √(3)\text{ Millimeters}^2`

`A=6* 1.7320508075688773\text{ Millimeters}^2`

`A=10.3923048454132638\text{ Millimeters}^2`

Round to nearest tenth:

`A\approx 10.4\text{ Millimeters}^2`

Therefore, the area of the given honeycomb is approximately 10.4 millimeters square.