Area Of Small Lune Calculator

Formula Used:

\[ A_{Small} = (2 \times A_{Triangle}) + (r_{Smaller}^2 \times \arccos(\frac{r_{Larger}^2 - r_{Smaller}^2 - d_{Centers}^2}{2 \times r_{Smaller} \times d_{Centers}})) - (r_{Larger}^2 \times \arccos(\frac{r_{Larger}^2 + d_{Centers}^2 - r_{Smaller}^2}{2 \times r_{Larger} \times d_{Centers}})) \]

Area of Triangle of Lune:

m²

Radius of Smaller Circle of Lune:

Radius of Larger Circle of Lune:

Distance of Centers of Circles of Lune:

Area of Small Lune:

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1. What is the Area of Small Lune?

The Area of Small Lune is the total quantity of plane occupied by the smaller lune portion in the Lune shape. A lune is a crescent-shaped figure formed by two circular arcs.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

\( A_{Triangle} \) — Area of Triangle of Lune (m²)
\( r_{Smaller} \) — Radius of Smaller Circle of Lune (m)
\( r_{Larger} \) — Radius of Larger Circle of Lune (m)
\( d_{Centers} \) — Distance of Centers of Circles of Lune (m)

Explanation: The formula calculates the area of the smaller lune by combining the area of the triangle formed by the circle centers and intersection point with circular segment areas.

3. Importance of Area Calculation

Details: Calculating the area of a lune is important in geometry, architecture, and design applications where crescent-shaped areas need to be precisely measured.

4. Using the Calculator

Tips: Enter all values in meters and square meters. Ensure all values are positive and the distance between centers is valid for the given radii.

5. Frequently Asked Questions (FAQ)

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

Q2: When is this calculation applicable?
A: This calculation is used in geometric problems, architectural design, and any application involving crescent-shaped areas.

Q3: What are the constraints for valid input values?
A: The distance between centers must be less than the sum of the two radii and greater than the absolute difference of the radii for the circles to intersect.

Q4: How accurate is this calculation?
A: The calculation is mathematically precise when the input values represent a valid geometric configuration of intersecting circles.

Q5: Can this formula be used for any size circles?
A: Yes, the formula works for any circle sizes as long as they intersect and form a lune shape.