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

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

• A. 2,900 cm²
• B. 2,338 cm²
• C. 2,540 cm²
• D. 2,678 cm²

Answer: (B)

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: \[ A = \frac{3\sqrt{3}}{2} \times 900 \] \[ A = 1350\sqrt{3} \]

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}

]