What is the area of a regular hexagon inscribed in a circle with a radius of 30 centimeters, rounded to the nearest whole number?

To find the area of a regular hexagon inscribed in a circle (circumscribed by the circle), we can use the formula that relates the area of the hexagon to the radius of the circle.

A regular hexagon can be thought of as being made up of 6 equilateral triangles. Each triangle has a vertex at the center of the circle and its base as one side of the hexagon. The formula for the area ( A ) of a regular hexagon can also be derived using the radius ( r ) of the circumscribing circle:

[

A = \frac{3\sqrt{3}}{2} r^2

]

In this case, the radius of the circle is given as 30 cm. Plugging this radius into the formula gives:

[

A = \frac{3\sqrt{3}}{2} (30^2)

]

Calculating ( 30^2 ):

[

30^2 = 900

]

Now substituting back we get:

[

A = \frac{3\sqrt{3}}{2} \times 900

]

[

A = 1350\sqrt{3}

]