**Title: Calculation of Magnetic Field Intensity at the Center of a Hexagon**

**Introduction:**
In this question, we are given a regular hexagon inscribed in a circle of radius 1m. The circuit carrying a direct current of 5A is placed along the hexagon. We need to calculate the magnetic field intensity at the center of the hexagon, assuming the medium to be free space.

**Solution:**

**Step 1: Determine the Magnetic Field due to a Straight Current-Carrying Wire**
The magnetic field intensity at the center of the hexagon can be calculated by considering each side of the hexagon as a straight current-carrying wire. The magnetic field at the center of a wire is given by the formula:
B = (μ₀ * I) / (2 * π * r)

Where:
B is the magnetic field intensity
μ₀ is the permeability of free space (4π × 10⁻⁷ T·m/A)
I is the current flowing through the wire
r is the distance from the wire to the point where we want to calculate the magnetic field intensity

**Step 2: Calculate the Distance from the Center of the Hexagon to the Side**
To calculate the magnetic field intensity at the center of the hexagon, we need to determine the distance from the center to any side of the hexagon. The radius of the circle in which the hexagon is inscribed is given as 1m. The distance from the center to any side of the hexagon can be calculated using the formula:
d = r / cos(30°)

Where:
d is the distance from the center to any side of the hexagon
r is the radius of the circle

**Step 3: Calculate the Magnetic Field Intensity at the Center of the Hexagon**
Now that we have the distance from the center to any side of the hexagon, we can calculate the magnetic field intensity at the center by considering each side as a straight current-carrying wire. Since the hexagon has 6 sides, we need to consider the contribution from each side and sum them up.

B_total = B₁ + B₂ + B₃ + B₄ + B₅ + B₆

Where:
B_total is the total magnetic field intensity at the center of the hexagon
B₁, B₂, B₃, B₄, B₅, B₆ are the magnetic field intensities due to each side of the hexagon

**Step 4: Substitute the Values and Calculate B_total**
Substitute the values of I, μ₀, and r into the formula for each side of the hexagon, and then sum up the magnetic field intensities to calculate B_total.

B_total = (μ₀ * I) / (2 * π * d) + (μ₀ * I) / (2 * π * d) + (μ₀ * I) / (2 * π * d) + (μ₀ * I) / (2 * π * d) + (μ₀ * I) / (2 * π * d) + (μ₀ * I) / (2 * π * d)

B_total = (6 * μ₀ * I) / (2 * π * d)

Finally, substitute the value of d into the equation to get the magnetic field intensity at the center of the hexagon.