Free · Circle calculators ⭕

Quarter-Disk Area and Remaining Square Corner

Its endpoints are one radius apart horizontally and vertically. The Pythagorean theorem gives √(r²+r²)=r√2 for an exact 90-degree quarter disk.

Formula & worked example
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How to use this calculator

Enter the known values in the units shown. Results update as you type.
Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
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 Quarter area (A1) 12.56637, Square area outside quarter disk 3.433629, Curved quarter arc L 6.283185, Full boundary perimeter 14.28319, Chord (c) 5.656854.

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

Worked example

Enter these known values and leave the other values blank.

Radius (r): 4 cm

Quarter area (A1): 12.57 cm²
Square area outside quarter disk: 3.434 cm²
Curved quarter arc L: 6.28 cm
Full boundary perimeter: 14.28 cm
Chord (c): 5.66 cm

Assumptions and limitations

The disk has an exact right-angle central sweep. Spare area is the r-by-r square outside that quarter disk. Perimeter includes the arc and both radii. The chord uses exact √2 geometry rather than the rounded 1.570796-radian angle in the capture. Radius, chord, arc or quarter area starts a solve; perimeter/remainder needs another linked measure. All public measures are positive.
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

Why is the chord r times square root of two?

Its endpoints are one radius apart horizontally and vertically. The Pythagorean theorem gives √(r²+r²)=r√2 for an exact 90-degree quarter disk.

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

Use the exact quarter-circle chord r√2 and its positive inverse instead of the rounded captured 1.570796-radian right angle. The international acre is exactly 43560 square international feet; Bookify replaces the rounded captured area factor.

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