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Quarter-Disk Chord, Arc, Boundary and Square Remainder

It is the area of the radius-by-radius square outside the quarter disk, r²−πr²/4. It is not the area bounded by the chord and arc.

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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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Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.

Formula and method

Example in initial displayed units: Radius (r) 4 gives Chord (c) 5.656854, Curved quarter arc L 6.283185, Full boundary perimeter 14.28319, Quarter area (A1) 12.56637, Square area outside quarter disk 3.433629.

c = r√2; L = πr/2; P = L+2r; A₁ = πr²/4; A₂ = r²−A₁

Worked example

Enter these known values and leave the other values blank.

Radius (r)
4 cm
Chord (c)
5.66 cm
Curved quarter arc L
6.28 cm
Full boundary perimeter
14.28 cm
Quarter area (A1)
12.57 cm²
Square area outside quarter disk
3.43 cm²

Assumptions and limitations

  • A quarter disk occupies part of an r-by-r square. External area is the square portion outside that disk, not the circular segment between chord and arc. All measures are positive. Radius, chord, arc or quarter area provides a starting measure. For perimeter or square remainder, supply another compatible linked measure because those captured equations require more than one known value.
  • Clear unknown measurements before solving backward. Use compatible units and independent values. Reference pictures are original schematic drawings. Independently rounded measurements can fail strict compatibility at a boundary.
  • Where available, the soccer-field area option is a fixed 7,140 m² comparison (105 m by 68 m). Actual playing fields vary; it is not a survey or building unit.

Common questions

What does external area mean?

It is the area of the radius-by-radius square outside the quarter disk, r²−πr²/4. It is not the area bounded by the chord and arc.

How do I use this calculator with different units?

Select the unit beside each measure. Length, squared length and volume use different dimensions; changing a unit preserves the physical value. Compound length controls represent feet/inches or meters/centimeters together.

References

The international acre is exactly 43560 square international feet; Bookify replaces the rounded captured area factor.

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