Question

a regular hexagon with center b is shown below.
what is the area of the hexagon?
round your answer to the nearest tenth.
square kilometers

Explanation:

Step1: Recall formula for regular hexagon area

A regular hexagon can be divided into 6 equilateral triangles, each with side length equal to the radius (distance from center to a vertex). The area of an equilateral triangle with side length \( s \) is \( \frac{\sqrt{3}}{4}s^2 \). So the area of the hexagon is \( 6\times\frac{\sqrt{3}}{4}s^2 \).

Here, \( s = 10 \) km (since the distance from center \( B \) to a vertex is 10 km, which is the side length of the equilateral triangles).

Step2: Substitute \( s = 10 \) into the formula

First, calculate the area of one equilateral triangle: \( \frac{\sqrt{3}}{4}\times(10)^2=\frac{\sqrt{3}}{4}\times100 = 25\sqrt{3} \) square kilometers.

Then, multiply by 6 to get the area of the hexagon: \( 6\times25\sqrt{3}=150\sqrt{3} \).

Step3: Calculate the numerical value

\( \sqrt{3}\approx1.732 \), so \( 150\times1.732 = 259.8 \) (rounded to the nearest tenth).

Answer:

259.8