Where it is defined
Defined for every point except the origin, where x² + y² is 0. The page plots -5 to 5 and caps the height at 4.
surface example
3D Inverse Square Surface Graph
A radial surface that blows up at the origin and levels off at a flat top of height 4.
z = min(4, 1 / (x^2 + y^2))Quick facts
- Formula: z = min(4, 1 / (x^2 + y^2))
- Category: surface
- Tags: surface, calculus, z=f(x,y)
What this graph represents
Height depends only on distance r from the origin and grows without bound as r approaches 0. The formula shown caps it at 4 so one infinite spike does not flatten the rest of the picture.
Where it appears in calculus
Read it next to radial-decay 1/(1+r²): that one is finite everywhere, this one is not.
How to use this graph
- Rotate the surface by dragging with your mouse or finger
- Use the Embed button to copy an
<iframe>for your LMS or lesson page - Click PNG to export a classroom-ready image
- Click Formula or LaTeX to copy the equation for your notes
Embed this graph
Use the Embed button in the calculator to copy a ready iframe for blogs, LMS pages, and lesson notes.
Open embed pageExplain this graph
A six-part reading guide to Inverse Square: where the formula is defined, what to notice, what it models, how to read the camera angle, what students usually get wrong, and something to try.
What to notice
- Height depends only on distance from the origin.
- The top is a flat disc of radius 0.5, the set where 1/r² would exceed 4.
- Outside that disc the surface falls off much faster than the Gaussian or 1/(1+r²).
Where it shows up
- Point-charge and point-mass potentials, which blow up at the source
- Why inverse-square laws look harmless far away and impossible up close
How to read the picture
- The edge of the flat top is where 1/r² = 4; drag the camera low and line it up with the grid to see it.
- Plot radial-decay 1/(1+r²) beside this one: same family, no singularity.
Common misconceptions
- The function has no value at the origin; the cap is a plotting choice, not a height the function takes.
- Doubling the distance divides the height by four, not by two.
Try it
At what radius does 1/r² equal 4? Compare your answer with the radius of the flat top on screen.
Teacher prompt
Question: What does the function do as you approach the origin?
Answer: It grows without bound. The flat top at z = 4 belongs to the capped formula on this page, not to 1/r² itself.
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