Square & Circle Geometry Calculator

Solve for properties of a square in a circle or a circle in a square.

Select the geometric configuration and enter a known value to calculate all related properties, including areas, lengths, and perimeters.

Examples

Load an example to see how the calculator works.

Square in Circle: Given Circle Radius

squareInCircle_circleRadius

If a circle has a radius of 10, find the properties of the square inscribed within it.

Configuration: squareInCircle

Property: circleRadius

Value: 10

Square in Circle: Given Square Side

squareInCircle_squareSide

If an inscribed square has a side length of 14.142, find the properties of the circumscribing circle.

Configuration: squareInCircle

Property: squareSide

Value: 14.142

Circle in Square: Given Circle Area

circleInSquare_circleArea

If an inscribed circle has an area of 78.54, find the properties of the surrounding square.

Configuration: circleInSquare

Property: circleArea

Value: 78.54

Circle in Square: Given Square Perimeter

circleInSquare_squarePerimeter

If a square has a perimeter of 40, find the properties of the circle inscribed within it.

Configuration: circleInSquare

Property: squarePerimeter

Value: 40

Other Titles
Understanding Square-Circle Geometry: A Comprehensive Guide
An in-depth exploration of the relationship between squares and circles, covering formulas, applications, and core concepts.

What is the Relationship Between a Square and a Circle?

  • Defining 'Inscribed' and 'Circumscribed'
  • Key Geometric Properties
  • The Foundational Formulas
The relationship between a square and a circle is a fundamental concept in geometry, typically explored in two main configurations: a square inscribed in a circle, and a circle inscribed in a square.
Square Inscribed in a Circle
A square is 'inscribed' in a circle when all four of its vertices lie on the circle's circumference. The key property here is that the diagonal of the square is equal to the diameter of the circle. This connection is the basis for all calculations.
Circle Inscribed in a Square
A circle is 'inscribed' in a square when it touches all four sides of the square from the inside. In this case, the diameter of the circle is equal to the side length of the square.

Key Relationships

  • Square in Circle: Square Diagonal = Circle Diameter
  • Circle in Square: Circle Diameter = Square Side

Step-by-Step Guide to Using the Calculator

  • Selecting the Correct Configuration
  • Entering Your Known Value and Property
  • Interpreting the Comprehensive Results
Our calculator simplifies these geometric problems into a few easy steps.
1. Choose Configuration
Start by selecting whether you have a 'Square Inscribed in a Circle' or a 'Circle Inscribed in a Square' from the first dropdown menu. This choice determines the formulas used.
2. Provide Known Information
In the 'Known Value' field, enter the numeric value of the dimension you have. Then, in the 'Known Property' dropdown, specify what that value represents (e.g., Circle Radius, Square Area).
3. Analyze the Results
Click 'Calculate' to see a full breakdown of all related properties for both the square and the circle, including lengths, perimeters, areas, and the ratio between them.

Calculation Steps

  • Step 1: Select 'Square in Circle'.
  • Step 2: Enter '10' for Known Value, select 'Circle Radius' as Known Property.
  • Step 3: View results for square side, area, etc.

Mathematical Derivations and Formulas

  • Formulas for a Square Inscribed in a Circle
  • Formulas for a Circle Inscribed in a Square
  • How Area and Perimeter are Calculated
All calculations are derived from the foundational geometric relationships between the two shapes.
Case 1: Square Inscribed in a Circle
Let r be the circle radius and s be the square's side. The square's diagonal d_s is s * sqrt(2). This diagonal is also the circle's diameter 2r. So, 2r = s * sqrt(2). From this, we derive: s = r * sqrt(2) and r = s / sqrt(2).
Case 2: Circle Inscribed in a Square
Here, the circle's diameter 2r is equal to the square's side length s. The relationship is simpler: s = 2r and r = s / 2.
Area and Perimeter Formulas
Circle: Area = πr², Circumference = 2πr. Square: Area = s², Perimeter = 4s.

Core Formulas

  • Square in Circle: side = radius * √2
  • Circle in Square: side = radius * 2

Real-World Applications

  • Engineering and Manufacturing
  • Architecture and Design
  • Optimization Problems
These geometric principles are not just academic; they have numerous practical uses.
Engineering
Engineers use these calculations to determine the maximum size of a square component that can be milled from a round piece of stock, or to find the right-sized circular pipe to fit a square shaft, minimizing waste and ensuring structural integrity.
Architecture
Architects might use this to design a square room within a circular building, or to fit a circular fountain into a square courtyard, ensuring the most efficient and aesthetically pleasing use of space.

Practical Uses

  • Cutting a square beam from a round log.
  • Placing a circular swimming pool in a square backyard.

Common Misconceptions and Key Distinctions

  • Inscribed vs. Circumscribed: A Critical Difference
  • Area vs. Perimeter Calculations
  • The Importance of the Correct Configuration
A common point of confusion is mixing up the two main configurations, which leads to incorrect formulas.
Inscribed is 'Inside'
Remember that 'inscribed' means the inner shape's points are on the outer shape's boundary. A 'square in a circle' has its corners on the circle. A 'circle in a square' has its edge touching the square's sides.
Don't Confuse the Formulas
The formula s = 2r is for a circle inside a square. The formula s = r * sqrt(2) is for a square inside a circle. Using the wrong one is the most frequent mistake. Always confirm your configuration before calculating.

Avoiding Errors

  • Mistake: Using side = 2 * radius for a square inside a circle.
  • Correct: For a square in a circle, its diagonal equals the circle's diameter.