Area of Section of Lune Calculator

Area of Section of Lune Formula:

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

Radius of Smaller Circle of Lune:

Area of Triangle of Lune:

m²

Radius of Larger Circle of Lune:

Distance of Centers of Circles of Lune:

Area of Section of Lune:

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is the Area of Section of Lune?

The Area of Section of Lune is the total quantity of plane occupied by the section formed between the larger and smaller Lunes in the Lune shape or the portion of intersection of circles in the Lune.

2. How Does the Calculator Work?

The calculator uses the Area of Section of Lune formula:

Where:

\( r_{Smaller} \) — Radius of Smaller Circle of Lune (m)
\( A_{Triangle} \) — Area of Triangle of Lune (m²)
\( r_{Larger} \) — Radius of Larger Circle of Lune (m)
\( d_{Centers} \) — Distance of Centers of Circles of Lune (m)
\( \pi \) — Archimedes' constant (≈3.14159)

Explanation: The equation calculates the area of the section formed between two intersecting circles (lune) by subtracting the area of intersection from the area of the smaller circle.

3. Importance of Area Calculation

Details: Calculating the area of a lune section is important in geometry, architecture, and various engineering applications where circular intersections occur. It helps in determining the exact area of overlapping circular regions.

4. Using the Calculator

Tips: Enter all values in meters and square meters. Ensure the distance between centers is valid for the given radii (must satisfy the triangle inequality: |rLarger - rSmaller| ≤ dCenters ≤ rLarger + rSmaller).

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. The section of lune refers to the area between the two circular arcs.

Q2: What are the constraints for valid input values?
A: The distance between centers must satisfy: |rLarger - rSmaller| ≤ dCenters ≤ rLarger + rSmaller. All values must be positive numbers.

Q3: When is this calculation useful?
A: This calculation is useful in architecture, mechanical engineering, and geometric design where overlapping circular areas need to be precisely measured.

Q4: What if the arccos function receives invalid input?
A: The calculator automatically clamps the input to the valid range [-1, 1] to prevent mathematical errors.

Q5: How accurate is this calculation?
A: The calculation uses PHP's built-in mathematical functions which provide high precision, typically to about 15 decimal places.