# Problem Description
The problem involves calculating the area of a triangle formed by the positions of the hour and minute hands on a clock after a given time has elapsed. The initial positions of the hour and minute hands are given, along with the lengths of the arrows representing these hands. We need to determine the area of the triangle formed by the initial position of the hour hand, the final position of the minute hand, and the center of the clock after a specified duration in minutes.

# Visual Facts
1. The image shows two pairs of clocks, each pair representing an initial time and a time after a certain duration has passed.
2. Each clock has:
   - An orange arrow representing the hour hand.
   - A green arrow representing the minute hand.
   - A blue triangle formed by the initial position of the hour hand, the final position of the minute hand, and the center of the clock.
3. The first pair of clocks:
   - Initial time: 3:00
   - Time after 40 minutes: 3:40
4. The second pair of clocks:
   - Initial time: 4:20
   - Time after 70 minutes: 5:30

# Visual Patterns
1. **Hand Movement Patterns**:
   - The hour hand moves 30 degrees per hour (360 degrees in 12 hours).
   - The minute hand moves 6 degrees per minute (360 degrees in 60 minutes).
   - The hour hand also moves 0.5 degrees per minute (since it moves 30 degrees in 60 minutes).

2. **Geometric Relationships**:
   - The blue triangle is formed by the initial position of the hour hand, the final position of the minute hand, and the center of the clock.
   - The angle between the initial and final positions of the hands can be calculated using the degrees moved by each hand.

3. **Calculation Pattern**:
   - To find the area of the triangle, we need to calculate the angle between the initial and final positions of the hands.
   - The area of the triangle can be calculated using the formula for the area of a triangle given two sides and the included angle:
     Area = 1/2 * a * b * sin(θ)
     where a and b are the lengths of the hour and minute hands, and θ is the angle between them.

4. **Angle Calculation**:
   - Initial hour hand position:  θ_h = h * 30 + m * 0.5  degrees
   - Final minute hand position:  θ_m = (m + k) * 6  degrees
   - The angle between the initial hour hand position and the final minute hand position is:
     θ = |θ_m - θ_h|


# Solution
import math

def solution(h: int, m: int, a: int, b: int, k: int) -> float:
    """
    Given an initial time (in 12-hour format) and the lengths of the orange and green arrows, calculate the area of the blue region after k minutes

    Parameters:
        h (int): Hour.
        m (int): Minute.
        a (int): Length of the orange arrow.
        b (int): Length of the green arrow.
        k (int): Duration passed (in minutes).

    Returns:
        float: Area of the blue region (rounded to two decimal places).
    """
    # Calculate the new time for the minute hand
    total_minutes = h * 60 + m + k
    new_m = total_minutes % 60

    # Calculate the angles
    minute_angle = new_m * 6  # 6 degrees per minute
    hour_angle = (h % 12) * 30 + m * 0.5  # 30 degrees per hour + 0.5 degrees per minute

    # Convert angles to radians
    minute_angle_rad = math.radians(minute_angle)
    hour_angle_rad = math.radians(hour_angle)

    # Convert to Cartesian coordinates
    minute_x = b * math.cos(minute_angle_rad)
    minute_y = b * math.sin(minute_angle_rad)
    hour_x = a * math.cos(hour_angle_rad)
    hour_y = a * math.sin(hour_angle_rad)

    # Calculate the area of the triangle using the cross product formula
    area = 0.5 * abs(hour_x * minute_y - hour_y * minute_x)

    return round(area, 2)


# Test Script
def test():
    test_cases = [
        (3, 0, 3, 4, 30, 6.0),
        (3, 0, 4, 3, 180, 6.0),
        (3, 0, 2, 7, 40, 3.5),
        (4, 20, 6, 8, 70, 18.39),
        (6, 0, 6, 8, 150, 0),
        (8, 22, 5, 9, 137, 6.58),
        (7, 12, 4, 6, 29, 6.0),
        (1, 56, 2, 7, 173, 5.8),
        (1, 56, 3, 4, 104, 0.21),
        (0, 56, 3, 4, 84, 6.0),
    ]

    for h, m, a, b, k, expected_output in test_cases:
        try:
            output = solution(h, m, a, b, k)
        except Exception as e:
            error_message = (
                f"An exception occurred while running the test case:\n"
                f"`h`: {h}\n"
                f"`m`: {m}\n"
                f"`a`: {a}\n"
                f"`b`: {b}\n"
                f"`k`: {k}\n\n"
                f"Exception: '{str(e)}'"
            )
            assert False, error_message

        if output != expected_output:
            error_message = (
                f"The following Test Case failed:\n"
                f"`h`: {h}\n"
                f"`m`: {m}\n"
                f"`a`: {a}\n"
                f"`b`: {b}\n"
                f"`k`: {k}\n\n"
                f"Actual Output: {output}\n"
                f"Expected Output: {expected_output}"
            )
            assert False, error_message

test()