Free · Physics models
Spherical Indentation Area and Mean Pressure
Calculate the actual spherical indentation contact area and mean load pressure, including the conventional kgf/mm² hardness-number display.
- Formula & worked example
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Calculator inputs
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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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Formula and method
A 1,000 N load, 10 mm ball and 2 mm indentation give contact area about 3.174 mm² and mean pressure about 315.10 MPa.
A = πD[D−√(D²−d²)]/2; mean pressure = F/A. Selecting kgf/mm² gives the conventional Brinell-number scale.
Worked example
Enter these known values and leave the other values blank.
- Applied load (P)
- 1000 N
- Diameter of indenter (D)
- 10 mm
- Diameter of indentation (d)
- 2 mm
- Spherical indentation surface area
- 3.1737 mm²
- Mean indentation pressure / hardness scale
- 315.1 MPa
Assumptions and limitations
- The area is the spherical cap surface area of the ideal indentation, not the projected circular area.
- The output in MPa is mean indentation pressure; the conventional Brinell numerical scale is the corresponding load in kgf divided by area in mm². Select kgf/mm² for that scale.
- This calculation does not certify compliance with a hardness test standard. Test load, dwell time, specimen thickness and permitted indentation ratios require the applicable test procedure.
- Ball and indentation diameters, load, area and pressure are positive; indentation diameter cannot exceed ball diameter.
- Enter known quantities and leave the unknown blank. Editable quantities support inverse solutions and locks; inconsistent locked quantities are reported. Unit selectors convert the quantity independently of its display.
- Reviewed hash-pinned model correction: The area is the spherical cap surface area of the ideal indentation, not the projected circular area. The output in MPa is mean indentation pressure; the conventional Brinell numerical scale is the corresponding load in kgf divided by area in mm². Select kgf/mm² for that scale. This calculation does not certify compliance with a hardness test standard. Test load, dwell time, specimen thickness and permitted indentation ratios require the applicable test procedure. Ball and indentation diameters, load, area and pressure are positive; indentation diameter cannot exceed ball diameter. Enter known quantities and leave the unknown blank. Editable quantities support inverse solutions and locks; inconsistent locked quantities are reported. Unit selectors convert the quantity independently of its display.
Common questions
Is the displayed area the projected circle?
No. It is the spherical cap contact surface area, which differs from the projected indentation area.
How do I display the usual hardness-number scale?
Select kgf/mm² for the mean-pressure output. MPa displays the corresponding physical pressure instead.
How do I use the inverse calculation?
Clear the quantity you want to find and enter the remaining known values in their selected units. The model can only solve branches with enough independent information.
References
Correct the spherical indentation surface area to pi D(D-sqrt(D²-d²))/2 and mean indentation pressure to F/A. The pressure selector in kgf/mm² gives the conventional hardness number; MPa gives the physical pressure.