⬜⭕ Square in a Circle Calculator
Calculate dimensions when a square is inscribed in a circle
How to Use This Calculator
Choose What You Know
Select whether you know the square side length, circle radius, or circle diameter.
Enter Your Value
Input the known measurement. Make sure it's a positive number.
Calculate
Click "Calculate" to get all square and circle measurements and their relationships.
Formulas and Relationships
Key Relationship
Square Diagonal = Circle Diameter
a × √2 = 2r = d
If you know square side (a):
- Circle Radius: r = a / √2 = a × (√2 / 2)
- Circle Diameter: d = a × √2
- Square Diagonal: d_square = a × √2
If you know circle radius (r):
- Square Side: a = r × √2 = 2r / √2
- Circle Diameter: d = 2r
- Square Diagonal: d_square = 2r
Area Relationships:
- Square Area = a²
- Circle Area = πr² = π(a²/2) = (π/2) × a²
- Circle Area / Square Area = π/2 ≈ 1.5708
- Square occupies about 63.66% of the circle's area
Example: Square with side = 5 units
Circle Radius = 5 / √2 = 5 × (√2/2) ≈ 3.536 units
Circle Diameter = 5 × √2 ≈ 7.071 units (equals square diagonal)
Square Area = 25 square units
Circle Area = π × (3.536)² ≈ 39.27 square units
About Square in a Circle Calculator
The Square in a Circle Calculator finds all measurements when a square is inscribed in a circle (or a circle circumscribes a square). In this configuration, the square's diagonal equals the circle's diameter.
When to Use This Calculator
- Geometry Problems: Solve square-in-circle problems in mathematics
- Design: Plan square elements within circular spaces
- Architecture: Calculate dimensions for square features in circular areas
- Engineering: Design components with square-in-circle configurations
- Education: Learn geometric relationships between squares and circles
- Verification: Check manual calculations
Why Use Our Calculator?
- ✅ Multiple Input Options: Works with square side, circle radius, or diameter
- ✅ Comprehensive Results: Shows all square and circle measurements
- ✅ Relationship Analysis: Displays how square and circle relate
- ✅ Educational: Shows formulas and calculation steps
- ✅ Accurate: Precise mathematical calculations
- ✅ 100% Free: No registration required
Understanding the Relationship
When a square is inscribed in a circle, all four vertices of the square lie on the circle. The square's diagonal passes through the center and equals the circle's diameter. This creates a specific mathematical relationship.
- The square's diagonal equals the circle's diameter
- Since diagonal = side × √2, we get: 2r = a × √2, so r = a / √2
- The circle's area is always π/2 times the square's area
- The square occupies about 63.66% of the circle's area
Real-World Applications
Architecture: Design square windows or features within circular frames or buildings.
Manufacturing: Calculate dimensions when creating square components that fit within circular constraints.
Design: Plan square elements within circular logos, patterns, or artwork.
Engineering: Design square components that must fit within circular housings or spaces.
Frequently Asked Questions
Why does the square diagonal equal the circle diameter?
When a square is inscribed in a circle, its vertices lie on the circle. The diagonal connects opposite vertices and passes through the center, making it the longest distance across the square and equal to the circle's diameter.
What percentage of the circle does the square occupy?
The square occupies exactly 2/π ≈ 63.66% of the circle's area. This is because Circle Area = πr² and Square Area = 2r², so the ratio is 2/π.
Is this different from a circle in a square?
Yes! A circle in a square (circle inscribed in square) is different. In that case, the square's side equals the circle's diameter, and the circle occupies about 78.54% of the square's area.
Can I calculate this with only the circle area?
Yes! If Circle Area = πr², then r = √(Area/π). Then square side = r × √2 = √(Area/π) × √2 = √(2Area/π).
What's the relationship between square side and circle radius?
Circle Radius = Square Side / √2. This comes from the fact that the square's diagonal (side × √2) equals the circle's diameter (2 × radius).
How do I find the circle radius if I only know the square area?
If Square Area = a², then a = √(Square Area). Since r = a/√2, we get r = √(Square Area) / √2 = √(Square Area / 2).