Questions: Use the Pythagorean theorem to help you determine the area of a regular hexagon with sides of length 1 inch.

Solution Steps

To find the area of a regular hexagon with side length 1 inch, we can divide the hexagon into 6 equilateral triangles. The area of one equilateral triangle can be calculated using the formula: \((\sqrt{3}/4) \times \text{side}^2\). Multiply the area of one triangle by 6 to get the total area of the hexagon.

The area \( A \) of one equilateral triangle with side length \( s \) is given by the formula:

\[ A = \frac{\sqrt{3}}{4} s^2 \]

Substituting \( s = 1 \):

\[ A = \frac{\sqrt{3}}{4} (1)^2 = \frac{\sqrt{3}}{4} \]

Calculating this gives:

\[ A \approx 0.4330 \]

The total area \( A_{hex} \) of the regular hexagon is the sum of the areas of the 6 equilateral triangles:

\[ A_{hex} = 6 \times A = 6 \times \frac{\sqrt{3}}{4} = \frac{6\sqrt{3}}{4} = \frac{3\sqrt{3}}{2} \]

Calculating this gives:

\[ A_{hex} \approx 2.5981 \]

Final Answer