One arc fixes four numbers
Set a transition tangent to the ground and put its centre one radius up, and the shape is completely determined by two inputs. The height you want fixes the angle swept, because the cosine of that angle is the radius minus the height, all over the radius. The angle then fixes the arc length, which is radius times angle in radians, and the horizontal run comes out of Pythagoras as the square root of the radius squared minus the quantity radius minus height, squared.
The defaults work out like this. A six foot radius reaching four feet gives a cosine of two over six, which is 0.3333, so the swept angle is 70.53 degrees. The run is the square root of 36 minus 4, which is the square root of 32, or 5.66 feet. The arc is six times 1.2310 radians, which is 7.39 feet. And the tangent at the top is 70.53 degrees from horizontal — steep, but 19.5 degrees short of vertical.
That last figure is the one people misjudge from a drawing. A transition only becomes vertical at the top when the height equals the radius exactly; below that it is always leaning. Above that it would lean back over the rider, which is a different shape entirely, so the page refuses to produce a cut list and says why.
Where the plywood actually goes
The riding surface is the obvious consumer of sheet goods and it is not the big one. With the defaults the surface is 7.39 feet of arc across 8 feet of width, which is 59.1 square feet — two courses of one sheet each per layer, four sheets for two layers. Reasonable.
Now the stringers. Each one has a bounding box of the run by the height: 67.9 inches by 48 inches. On a 4 by 8 sheet that is one per sheet, because two of them will not fit side by side along the 96 inch dimension and the transposed orientation does not fit at all. Seven stringers at 16 inch spacing across 8 feet of width therefore consumes seven sheets — nearly twice what the riding surface takes, and the offcut from each is a large piece of scrap with a curve bitten out of it.
This is the single biggest cost lever on a small ramp and it is invisible until the sheets are being counted. Widening the stringer spacing cuts the count directly. So does splitting the profile into two jointed pieces that nest better, at the cost of a joint that then has to be handled. The arithmetic here is worth running twice with different spacings before anything is bought.
Why the sheet has to bend the short way
Plywood bends far more easily across the grain of its face plies, and in practice on a ramp that means the four foot dimension of a sheet runs up the curve and the eight foot dimension runs across the width. The course count above is calculated on that basis: arc length divided by the short dimension of the sheet, rounded up.
It also explains the layers field. Two or three thin layers laid one over another will each take the radius and can have their joints staggered so no seam runs through the full thickness. A single sheet thick enough to be stiff on its own will not follow a six foot radius; it will sit flat between crosspieces and facet, and every crosspiece becomes a ridge you can feel. The layer count is a construction decision and there is no right answer here — the page just multiplies by whatever you enter.
Crosspieces around the arc, not up the face
The crosspiece spacing field is measured along the curve, and that is not the same as spacing measured up the vertical face. The difference is small at the bottom of the transition where the surface is nearly horizontal and large at the top where it is nearly vertical, and laying out by vertical rise puts the pieces further apart exactly where the surface is steepest and the sheet is trying hardest to facet.
Marking out along the arc is done on the stringer itself, which is why the first stringer is a template rather than a part: it gets marked, cut, checked, and only then used to trace the rest. The crosspiece count here — arc length divided by spacing, rounded up, plus one for the end — assumes exactly that layout.
What the page is not doing
There is no member size here, no fixing schedule, no bracing, no anchoring and no verdict on any of it. The numbers are counts and lengths off the two dimensions you typed. Whether the stringers are stiff enough at that spacing, what the crosspieces are, how the surface is fixed to them, how the whole thing is held down and what it carries are questions for whoever is building it, and for an engineer if it is going somewhere other people will use.
The hazard is worth saying plainly: a ramp is a hard, unforgiving surface with a fall onto it from height, and the injuries that actually happen are to heads and wrists. That is not something a cut list addresses either way.
For flat sheet work the plywood sheet calculator handles the counts; for the platform behind the coping the deck framing calculator does joists and beams; and for a straight ramp with a fixed slope and landings, which is a completely different problem, the ramp run calculator.
Questions people ask
How do I work out the run of a quarter pipe transition?
The horizontal run is the square root of the radius squared minus the quantity radius minus height, squared. A six foot radius reaching four feet gives the square root of 36 minus 4, which is 5.66 feet. The arc length along the riding surface is longer than that — 7.39 feet on the same numbers — because it is a curve rather than a straight line.
Can the height of a quarter pipe be more than its radius?
Not for a transition that starts tangent to the ground. Such an arc reaches vertical when it has climbed exactly one radius, so a height equal to the radius is a full quarter circle. Ask for more and the surface leans back over the rider, which is a different shape with a different name. Raise the radius, or cap the transition at the radius and put the extra elevation into a flat wall or a deck above it.
How many sheets of plywood does a quarter pipe take?
More for the stringers than for the riding surface, usually. On the defaults here the surface takes four sheets across two layers, and seven stringers take seven more, because a stringer with a 68 by 48 inch bounding box only nests one to a 4 by 8 sheet. Changing the stringer spacing moves that number far more than anything you do to the surface.
Which way should the plywood run on the curve?
The four foot dimension of the sheet runs up the arc and the eight foot dimension runs across the width, because plywood bends much more readily across the face grain. The course count on this page assumes that. Multiple thin layers with staggered joints are used instead of one thick sheet, because a sheet stiff enough to stand alone will not take the radius and will facet between the crosspieces.
Does this tell me what size the framing should be?
No. Everything on the page is a count or a length taken from the radius, height, width and spacings you entered. Member sizes, fixings, bracing, how the ramp is held down and what it will carry are not derived here and are not implied by the counts. If the ramp is being built for other people to use, those are questions for an engineer.