Second Leg of Elliptical Sector Calculator

Second Leg of Elliptical Sector Formula:

\[ l_2 = \sqrt{\frac{a_{Sector}^2 \times b_{Sector}^2}{(a_{Sector}^2 \times \sin(\angle Leg(2))^2) + (b_{Sector}^2 \times \cos(\angle Leg(2))^2)}} \]

Semi Major Axis of Elliptical Sector:

Semi Minor Axis of Elliptical Sector:

Second Leg Angle of Elliptical Sector:

rad

Second Leg of Elliptical Sector:

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1. What is the Second Leg of Elliptical Sector?

The Second Leg of Elliptical Sector is the length of the linear edge of the sector which is adjacent to the right most semi major axis of the Elliptical Sector. It represents one of the two straight edges that bound the elliptical sector.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ l_2 = \sqrt{\frac{a_{Sector}^2 \times b_{Sector}^2}{(a_{Sector}^2 \times \sin(\angle Leg(2))^2) + (b_{Sector}^2 \times \cos(\angle Leg(2))^2)}} \]

Where:

\( l_2 \) — Second Leg of Elliptical Sector (m)
\( a_{Sector} \) — Semi Major Axis of Elliptical Sector (m)
\( b_{Sector} \) — Semi Minor Axis of Elliptical Sector (m)
\( \angle Leg(2) \) — Second Leg Angle of Elliptical Sector (rad)

Explanation: The formula calculates the length of the second leg based on the semi-axes of the ellipse and the angle of the leg, using trigonometric functions to account for the elliptical geometry.

3. Importance of Second Leg Calculation

Details: Calculating the second leg is important for determining the complete geometry of elliptical sectors, which is useful in various engineering, architectural, and mathematical applications involving elliptical shapes.

4. Using the Calculator

Tips: Enter semi major axis and semi minor axis in meters, and the second leg angle in radians. All values must be positive numbers.

5. Frequently Asked Questions (FAQ)

Q1: What are the valid ranges for input values?
A: Semi major and semi minor axes must be positive numbers greater than 0. The angle should be between 0 and 2π radians.

Q2: Can this calculator handle degrees instead of radians?
A: No, the calculator requires angle input in radians. Convert degrees to radians by multiplying by π/180.

Q3: What happens if the denominator becomes zero?
A: The result becomes undefined (division by zero), which occurs when the trigonometric terms cancel each other out in a specific way.

Q4: How accurate is the calculation?
A: The calculation uses PHP's built-in trigonometric functions which provide high precision for most practical applications.

Q5: What are some practical applications of this calculation?
A: This calculation is useful in optics, astronomy, architecture, and mechanical engineering where elliptical sectors are encountered.