Arithmetic Mean Method:

To calculate the average depth of rainfall using the arithmetic mean method, we need to find the total rainfall and divide it by the area of the basin.

Step 1: Calculate the total rainfall
To find the total rainfall, we add up the rainfall recorded at each rain gauge.

Total rainfall = 60 + 81 + 73 + 59 + 45 = 318 mm

Step 2: Calculate the area of the basin
The area of a regular pentagon can be calculated using the formula:

Area = (5/4) x s^2 x (cot(π/5))

Given that each side of the pentagon has a length of 20 km, we can substitute this value into the formula to find the area.

Area = (5/4) x (20)^2 x (cot(π/5)) ≈ 688.19 km^2

Step 3: Calculate the average depth of rainfall
The average depth of rainfall is given by the formula:

Average depth = Total rainfall / Area

Substituting the values we calculated in steps 1 and 2:

Average depth = 318 mm / 688.19 km^2 ≈ 0.462 mm/km^2

Therefore, the average depth of rainfall over the basin using the arithmetic mean method is approximately 0.462 mm/km^2.

Theisen Polygon Method:

The Theisen polygon method is a graphical method used to determine the average depth of rainfall over an irregularly shaped area. Here are the steps to calculate the average depth of rainfall using this method:

Step 1: Plot the rain gauges on a graph
Draw a graph representing the basin, with the rain gauges located at the corners of the pentagon.

Step 2: Join the rain gauges with lines
Draw lines connecting the rain gauges in a clockwise direction, starting from any corner.

Step 3: Divide the polygon into triangles
Divide the polygon formed by the lines into triangles. In this case, we will have 3 triangles.

Step 4: Calculate the area of each triangle
Using the formula for the area of a triangle (Area = 1/2 x base x height), calculate the area of each triangle formed by the rain gauges.

Step 5: Calculate the average depth of rainfall for each triangle
Divide the rainfall recorded at each rain gauge by the area of the corresponding triangle to get the average depth of rainfall for each triangle.

Step 6: Calculate the weighted average depth of rainfall
Multiply each average depth of rainfall by the area of the corresponding triangle, and then sum up these values. Finally, divide the sum by the total area of the basin.

By following these steps, we can calculate the average depth of rainfall over the basin using the Theisen polygon method. However, since the specific values for the coordinates of the rain gauges and the angles between the sides of the pentagon are not provided, we cannot provide the exact calculation.