Given a regular pentagon with a radius (distance from center to a vertex) of $11$ meters, what is the length of its apothem (the perpendicular distance from the center to a side)?

• A. $8.5m$
• B. $10.5m$
• C. $11m$
• D. $9.0m$

Answer: (D)

Verified step by step guidance

1. Identify the key elements of the regular pentagon: the radius (distance from the center to a vertex) is given as 11 meters, and we need to find the apothem (the perpendicular distance from the center to a side).
2. Recall that a regular pentagon can be divided into 5 identical isosceles triangles, each with a vertex angle of \(\frac{360^\circ}{5} = 72^\circ\) at the center.
3. Focus on one of these triangles and split it into two right triangles by drawing the apothem. This creates a right triangle with an angle of \(\frac{72^\circ}{2} = 36^\circ\), the hypotenuse as the radius (11 meters), and the adjacent side as the apothem.
4. Use the cosine trigonometric ratio, which relates the adjacent side (apothem) to the hypotenuse (radius): \(\cos(36^\circ) = \frac{\text{apothem}}{11}\).
5. Solve for the apothem by multiplying both sides by 11: \(\text{apothem} = 11 \times \cos(36^\circ)\). This expression gives the length of the apothem.