Regular polygons are fascinating geometric shapes with equal sides and angles. To fully understand their properties and measurements, it’s crucial to know how to calculate the apothem. The apothem, denoted as ‘a,’ is defined as the perpendicular distance from the center of the polygon to any of its sides. In this blog post, we will guide you through the process of calculating the apothem of regular polygons step by step.

Step 1: Identify the Necessary Information

Before diving into the actual calculation, you need to ensure you have all the required information. To calculate the apothem, you need to know the length of any side (s) and the number of sides (n) in the regular polygon.

Step 2: Calculate the Interior Angle

The interior angle (I) of a regular polygon can be determined using the formula:

I = (n – 2) × 180° / n

In this formula, ‘n’ represents the number of sides in the polygon. By substituting the value of ‘n’, you can find the interior angle.

Step 3: Calculate the Central Angle

The central angle (C) of a regular polygon is formed by two radii drawn to adjacent vertices. It can be calculated using the formula:

C = 360° / n

By substituting the value of ‘n’ into the formula, you can calculate the central angle.

Step 4: Calculate the Apothem

Finally, armed with the values of the interior angle (I) and the central angle (C), you can calculate the apothem (a) using the formula:

a = (s/2) × tan(I/2)

In this formula, ‘s’ represents the length of any side of the regular polygon, and ‘tan’ refers to the tangent function. By substituting the values of ‘s,’ ‘I,’ and performing the necessary calculations, you can obtain the value of the apothem.

Example Calculation:

Let’s say we have a regular hexagon with a side length of 5 cm. We can calculate the apothem following the steps mentioned above:

  • Step 1: We have the side length as 5 cm, and since it’s a hexagon, n = 6.
  • Step 2: Applying the formula, I = (6 – 2) × 180° / 6 = 120°.
  • Step 3: Using the formula, C = 360° / 6 = 60°.
  • Step 4: Substituting the values into the formula, a = (5/2) × tan(120°/2).
  • Calculating further, a = (5/2) × tan(60°) ≈ 4.33 cm.

Thus, the apothem of the given regular hexagon is approximately 4.33 cm.

Calculating the apothem of regular polygons is essential to understand their dimensions and characteristics. By following the steps outlined in this blog post, you can easily determine the apothem using basic information about the polygon. So, the next time you encounter a regular polygon, you’ll be confident in calculating its apothem.

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