Area of Crescent Calculator
Calculate the area of crescents and lunes formed by two overlapping circles
Crescent Area Calculator
Radius of the first circle
Radius of the second circle
Distance between circle centers
Example Calculation
Moon Crescent Shape
Problem: Calculate the crescent area formed by two overlapping circles
Circle 1 radius: 4 cm
Circle 2 radius: 6 cm
Center distance: 5 cm
Solution
1. Check validity: |r₁ - r₂| < d < r₁ + r₂ → |4-6| < 5 < 4+6 → 2 < 5 < 10 ✓
2. Calculate overlap area using intersection formula
3. Lune 1 area = π × 4² - overlap ≈ 20.96 cm²
4. Lune 2 area = π × 6² - overlap ≈ 83.70 cm²
5. Check crescent: d < r₂ → 5 < 6 ✓ (Lune 1 is a crescent)
Crescent vs Lune
Crescent
A lune that doesn't include the center of the original circle
Lune
Any concave-convex region formed by two circular arcs
Lens
A convex-convex region where both arcs curve outwards
Shape Conditions
Overlapping Circles
|r₁ - r₂| < d < r₁ + r₂
Crescent Formation
d < r (of the other circle)
Maximum Overlap
d approaches 0
Minimal Overlap
d approaches r₁ + r₂
Historical Context
Ancient Egypt used crescents as moon symbols
Ottoman Empire adopted crescent and star
Now symbol of Islam in many countries
Latin "crescere" means "to grow"
Understanding Crescents and Lunes
What is a Crescent?
A crescent is a specific type of lune - a geometric shape formed by two intersecting circular arcs. The defining characteristic of a crescent is that it doesn't include the center of the original circle, which happens when the distance between circle centers is smaller than the radius of the covering circle.
Mathematical Definition
- •Lune: Region bounded by two circular arcs
- •Crescent: Lune where d < r₂ (or d < r₁)
- •Concave-convex shape (one arc curves in, one out)
- •Often resembles moon phases
Calculation Formula
Overlap Area
A = r₁²·arccos((d²+r₁²-r₂²)/(2dr₁)) + r₂²·arccos((d²+r₂²-r₁²)/(2dr₂)) - ½√[(-d+r₁+r₂)(d+r₁-r₂)(d-r₁+r₂)(d+r₁+r₂)]
Lune Areas
Lune₁ = πr₁² - Overlap
Lune₂ = πr₂² - Overlap
Key Parameters
- r₁, r₂: Radii of the two circles
- d: Distance between circle centers
- Validity: |r₁ - r₂| < d < r₁ + r₂
- Crescent condition: d < max(r₁, r₂)
Applications of Crescent Calculations
🌙 Astronomy
- • Moon phase calculations
- • Eclipse coverage areas
- • Planetary transit geometry
- • Solar/lunar limb darkening
🏗️ Engineering
- • Optical lens design
- • Fluid flow analysis
- • Material stress distribution
- • Architectural elements
🎨 Design & Art
- • Logo and symbol design
- • Islamic geometric patterns
- • Flag and emblem creation
- • Decorative elements