Free · Mathematical relations and geometry

Inscribed and Equal-Area Circle–Square Measures

Compare the largest inscribed square or circle and separate shapes having equal area.

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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

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Formula and method

The largest square inside a radius-5 circle has side 5√2 and area 50. The largest circle inside a side-10 square has radius 5.

inscribed square s=√2r; inscribed circle r=s/2; equal area s=√πr

Worked example

Enter these known values and leave the other values blank.

I want to find...
1
Radius of the circle (r)
5 cm
Area of the circle (A)
78.54 cm²
Side of square (s)
7.071 cm
Area of square (Aₛ)
50 cm²

Assumptions and limitations

  • The four modes distinguish containment from equal area. Equal area does not mean one shape fits inside the other.
  • A largest inscribed square has its diagonal equal to the circle diameter. A largest inscribed circle has diameter equal to the square side.
  • Dimensions and areas are positive, and inverse roots select positive dimensions.

Common questions

How can I solve for a missing quantity?

Leave the unknown control empty and enter the quantities needed by the selected relation. Read the assumptions for any branch or domain restrictions.

How should I interpret the worked example?

It is an exact mathematical model with independently checked numbers. Display rounding does not change the values used for subsequent calculations.

References

The calculation equations, inverse formulas, units, and input rules were imported from this source. Bookify provides the interface and equation solver.

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