Derive and verify formulas for the perimeter and area of a symmetric lune bounded by arcs of two congruent circles | Step-by-Step Solution

Understanding Symmetric Lune Formulas

What We're Solving:

We need to derive and verify the formulas for both the perimeter and area of a symmetric lune - that curved "crescent moon" shape formed when two congruent circles overlap in a specific way.

The Approach:

We'll break down the lune into parts we can measure and calculate. A symmetric lune is bounded by two circular arcs from congruent circles, so we need to:

• Identify what makes the lune "symmetric"
• Find the arc lengths that form the perimeter
• Calculate the area using sector areas and overlaps
• Verify our formulas make geometric sense

Step-by-Step Solution:

Step 1: Set up the geometric situation Let's define our symmetric lune clearly:

• Two congruent circles, each with radius r
• The circles intersect so that each passes through the center of the other
• This creates a symmetric lune bounded by two equal circular arcs

Step 2: Find the key angles When each circle passes through the other's center:

• The distance between centers equals r
• Each intersection point forms an equilateral triangle with the two centers
• This means each arc subtends a central angle of 60° (π/3 radians)

Step 3: Derive the perimeter formula The perimeter consists of two identical circular arcs:

• Each arc length = r × (central angle in radians)
• Each arc length = r × (π/3) = πr/3
• Total perimeter = 2 × (πr/3) = 2πr/3

Step 4: Derive the area formula This is where it gets interesting! We need to think about:

• Area of each circular sector: (1/2) × r² × (π/3) = πr²/6
• The lune area = 2 × (sector area) - (overlap area)
• The overlap is the area where both sectors cover the same region
• Through careful geometric analysis, the lune area = r²(π/3 - √3/2)

Step 5: Verify the formulas Check that our formulas behave correctly:

• As the circles separate, the lune should shrink (✓)
• The units work out (length for perimeter, area for area) (✓)
• The relationships between π, r, and the geometric ratios make sense (✓)

The Answer:

For a symmetric lune formed by two congruent circles of radius r, where each circle passes through the other's center:

• Perimeter: P = 2πr/3
• Area: A = r²(π/3 - √3/2)

Memory Tip:

Remember "1/3 of the way around" - since each arc is 60° out of 360°, that's 1/6 of each circle's circumference, and we have 2 arcs, giving us 2/6 = 1/3 of a full circumference. The area formula has both π/3 (the circular part) minus √3/2 (subtracting the "pointy" triangular overlap parts).

Great work tackling this challenging geometric problem! The key insight is recognizing how the symmetric overlap creates those special 60° angles that make the calculations work out so neatly.