91Ó°ÊÓ

To calculate the apothem of a regular hexagon, you need to understand the formula involved. The apothem is the distance from the center of the hexagon to the middle of any side. For a regular polygon, the apothem can be calculated using the formula: \[ a = \frac{s}{2 \tan(\pi/n)} \]where 'a' is the apothem, 's' is the side length, and 'n' is the number of sides. In the case of a hexagon (n=6), you can substitute the values directly. Since a hexagon has 6 sides and each side is 8 inches,\[ a = \frac{8}{2 \tan(\pi/6)} \]To simplify, you find that \( \tan(\pi/6) \) is approximately 0.577. Therefore,\[ a = \frac{8}{2 \times 0.577} \approx 6.93 \, \text{inches} \]. The apothem is approximately 6.93 inches.

Polygon geometry involves the study of shapes with many sides. A regular hexagon is a polygon with six equal sides and angles. Key properties of a regular hexagon include:

• Each internal angle is 120 degrees.
• All sides are of equal length.
• The shape can be divided into six equilateral triangles.

By dividing it into triangles, we can easily calculate measurements like the circumcircle radius and apothem. This is because properties of triangles, especially equilateral ones, make calculations much simpler. Regular polygons like hexagons have symmetrical properties, making them an essential study topic in geometry.

The circumcircle radius (or simply the radius) of a polygon, especially a regular one, is the distance from the center of the polygon to any of its vertices. For a regular hexagon, this radius is not just any distance but matches the side length. In our problem, with the radius provided as 8 inches, each side of the hexagon is also 8 inches. This key relationship holds due to the symmetrical nature of regular polygons, where the radius remains equal across all vertices. The circumcircle radius helps in simplifying several geometric calculations, like side length and area, making it a fundamental property in polygon geometry.