Square in Circle Calculator
Calculate inscribed squares, circles, and shapes with equal areas
Calculate Square and Circle Relationships
Linear measurement
Largest Square Inscribed in Circle
Formulas used:
- • Square side: s = r√2 = 0.0000 cm
- • Square area: A = 2r² = 0.0000 cm²
- • Circle area: A = πr² = 0.0000 cm²
Geometric Relationships
Example Calculations
Pizza Box Problem
Problem: What's the largest square pizza box for a 14-inch diameter circular pizza?
Given: Circle diameter = 14 inches, so radius = 7 inches
Solution: Square side = r√2 = 7√2 ≈ 9.90 inches
Answer: A 9.90 × 9.90 inch square box
Pool in Square Room
Problem: Largest circular pool in a 20 ft × 20 ft square room?
Given: Square side = 20 feet
Solution: Circle radius = s/2 = 20/2 = 10 feet
Answer: A 10-foot radius (20-foot diameter) circular pool
Equal Area Challenge
Problem: Square cake with same area as 6-inch radius circular plate?
Given: Circle radius = 6 inches
Solution: Square side = r√π = 6√π ≈ 10.63 inches
Answer: A 10.63 × 10.63 inch square cake
Key Formulas
Square in Circle
Circle in Square
Equal Areas
Geometric Facts
Inscribed square diagonal = circle diameter
Inscribed circle diameter = square side
Inscribed square uses ~63.7% of circle area
Inscribed circle uses ~78.5% of square area
Equal area shapes have different perimeters
Understanding Square and Circle Relationships
Inscribed vs Circumscribed
When a shape is inscribed in another, it fits entirely inside with specific touching points. When a shape is circumscribed around another, it completely surrounds the inner shape.
Mathematical Relationships
- •Inscribed square in circle: maximum square that fits inside
- •Inscribed circle in square: maximum circle that fits inside
- •Equal areas: different shapes with identical area measurements
Practical Applications
Engineering & Design
Optimizing space usage in circular and square containers, architectural design
Manufacturing
Cutting optimal shapes from materials, packaging design, material efficiency
Mathematics
Geometry problems, optimization theory, area and perimeter relationships
Key Constants
- √2 ≈ 1.414 (square in circle ratio)
- √π ≈ 1.772 (equal area ratio)
- π/4 ≈ 0.785 (circle to square area ratio)
- 2/π ≈ 0.637 (square to circle area ratio)