Area of Large Lune Calculator

Area of Large Lune Formula:

\[ A_{Large} = \pi(r_{Larger}^2 - r_{Smaller}^2) + 2A_{Triangle} + r_{Smaller}^2 \cdot \arccos\left(\frac{r_{Larger}^2 - r_{Smaller}^2 - d_{Centers}^2}{2r_{Smaller}d_{Centers}}\right) - r_{Larger}^2 \cdot \arccos\left(\frac{r_{Larger}^2 + d_{Centers}^2 - r_{Smaller}^2}{2r_{Larger}d_{Centers}}\right) \]

Radius of Larger Circle (r_Larger):

Radius of Smaller Circle (r_Smaller):

Distance Between Centers (d_Centers):

Area of Triangle (A_Triangle):

m²

Area of Large Lune (A_Large):

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1. What is the Area of Large Lune?

The Area of Large Lune is the total quantity of plane occupied by the larger lune portion in the Lune shape, which is created by two intersecting circles with different radii.

2. How Does the Calculator Work?

The calculator uses the Area of Large Lune formula:

Where:

\( r_{Larger} \) — Radius of the larger circle (m)
\( r_{Smaller} \) — Radius of the smaller circle (m)
\( d_{Centers} \) — Distance between centers of the circles (m)
\( A_{Triangle} \) — Area of the triangle formed by the centers and intersection point (m²)
\( \pi \) — Mathematical constant pi (approximately 3.14159)
\( \arccos \) — Inverse cosine function

Explanation: The formula calculates the area of the larger lune by combining circular segments and triangular areas formed by the intersecting circles.

3. Importance of Area Calculation

Details: Calculating the area of a lune is important in geometry, architecture, and design applications where curved shapes intersect. It helps in determining material requirements and understanding spatial relationships.

4. Using the Calculator

Tips: Enter all measurements in meters. Ensure the distance between centers is valid for the given radii (must satisfy the triangle inequality). All values must be positive numbers.

5. Frequently Asked Questions (FAQ)

Q1: What is a lune in geometry?
A: A lune is a crescent-shaped figure formed by two circular arcs, typically created by the intersection of two circles.

Q2: What are the constraints for valid inputs?
A: The distance between centers must be greater than the absolute difference of the radii and less than the sum of the radii for the circles to intersect.

Q3: How is the triangle area determined?
A: The triangle area is formed by connecting the centers of the two circles and one of their intersection points, creating a triangle with sides rLarger, rSmaller, and dCenters.

Q4: Can this calculator handle very small values?
A: Yes, the calculator can handle decimal values with up to 4 decimal places precision.

Q5: What if the circles don't intersect?
A: The formula assumes intersecting circles. If circles don't intersect, the lune area calculation is not applicable.