Question

find the area of a regular hexagon with apothem length 9 cm. if necessary, write your answer in simplified radical form. area = cm²

Explanation:

Step1: Find side - length from apothem

In a regular hexagon, the apothem forms a 30 - 60 - 90 triangle with half of the side - length. If the apothem \(a = 9\) cm, and in a 30 - 60 - 90 triangle, the ratio of the sides is \(1:\sqrt{3}:2\). Let the half - side length be \(x\). We know that \(\tan30^{\circ}=\frac{x}{a}\), so \(x = 3\sqrt{3}\) cm. Then the side - length \(s=6\sqrt{3}\) cm.

Step2: Calculate perimeter

The perimeter \(P\) of a regular hexagon with side - length \(s\) is \(P = 6s\). Substituting \(s = 6\sqrt{3}\) cm, we get \(P=36\sqrt{3}\) cm.

Step3: Calculate area

The area formula of a regular polygon is \(A=\frac{1}{2}aP\), where \(a\) is the apothem and \(P\) is the perimeter. Substituting \(a = 9\) cm and \(P = 36\sqrt{3}\) cm, we have \(A=\frac{1}{2}\times9\times36\sqrt{3}=162\sqrt{3}\text{ cm}^2\).

Answer:

\(162\sqrt{3}\)