Sine for the reach, cosine for the height
Hang a seat from a pin and everything about where it can be is fixed by the length of the chain and the angle it makes with vertical. Horizontal distance from the pin is the length times the sine of the angle. Height gained above the rest position is the length times one minus the cosine. Those two functions behave nothing alike over the range a swing actually uses, and that is the single most useful thing on this page.
Take the defaults: a 6.5 foot pendulum. At 15 degrees the seat reaches 1.68 feet forward and rises 0.22 feet. At 30 degrees it reaches 3.25 feet — near enough double — but the rise is 0.87 feet, which is almost four times as much. At 45 degrees the reach is 4.60 feet and the rise 1.90 feet. Reach is close to linear across that range; height is not, because cosine is flat near zero and falls away fast after that.
Energy tracks the height, so it tracks the fast-growing one. That is why the speed column climbs the way it does, and why a swing that appears to be going only slightly higher than it was is a materially different thing. The speed shown is the frictionless ideal for a pendulum released from rest — a floor. A rider pumping adds energy every cycle and beats it.
The tangent is the whole point
The number that matters for anyone standing near a swing is not on this page and cannot be. A child leaving a swing does not continue round the arc; they leave along the tangent to it and then travel as a projectile. Near the top of the swing that tangent is close to horizontal, so the departure carries forward well past the front of the arc, and the higher the swing the further and the faster.
This is the reason the protective area in front of and behind a swing is much deeper than the swept path and much deeper than the area at the sides, and it is why the rectangular-footprint model that works for a climbing frame badly understates the ground a swing needs. The actual dimension is published — it comes from the equipment manufacturer for that specific swing and from the current national playground safety guidance — and it is a safety figure rather than a geometric one. This page gives you the seat travel and stops. Treat the seat travel as a lower bound on the area involved and never as the area itself.
What the allowance field is for
The seat is not the widest thing on the swing. A rider sitting on it has legs out in front, and on the forward stroke the feet are the leading edge by a couple of feet. The allowance field lets you add your own measurement of that, and it is deliberately your measurement: it depends entirely on who is using the swing and how, and there is no honest default for it.
The option to add it front and back is there because a rider swinging backwards has their back and head as the leading edge, and because things that get put behind a swing — a fence, a shed wall, a tree trunk, a second frame — are struck by something. Adding the allowance on both sides is the conservative reading of a symmetric arc.
Reading the seat height at the top
The seat height at the end of the swing is worth checking against anything overhead or to the side. With the defaults the seat sits 1.50 feet above the surface at rest and 3.40 feet at 45 degrees, so the seat has climbed nearly two feet and the rider on it more than that. Low branches, a pergola beam, a washing line, a light fitting or the underside of a deck all live in that space and none of them are visible from a plan view.
The rest height itself comes out as the pin height minus the pendulum length, which is why a negative figure there is a sign that the two inputs disagree rather than a real result. If you are setting the chain length rather than measuring it, the swing beam and hanger layout calculator works the subtraction from the beam down.
Where this fits with everything else
The surface under the arc is the highest-wear spot on a play area, and the playground surfacing calculator is direct about the fact that installed depth and effective depth are different numbers by the end of a season. For whether the structure and the zone your instructions specify actually land inside the yard you have, the play area yard fit calculator does the rectangle arithmetic. If the swing is going anywhere near a tree, the tree spacing calculator covers mature spread, which is the dimension that catches people out when the branch that was clear at planting is not clear five years later.
Questions people ask
How far does a swing seat travel forward?
The seat reaches the pendulum length times the sine of the swept angle, measured horizontally from the pin. A 6.5 foot chain-and-seat at 45 degrees reaches 4.60 feet forward, so 9.19 feet front to back in total. That is the seat by itself, with no rider allowance and no departure distance, and it is not a use zone.
Why is the safe area in front of a swing bigger than the arc?
Because a rider leaving a swing travels along the tangent to the arc rather than continuing round it, and then flies as a projectile. From near the top of the swing that tangent is close to horizontal and the landing point is well beyond the front of the arc. The dimension that accounts for this is published by the equipment manufacturer and set out in the national playground safety guidance. This page deliberately gives the seat travel only.
How much higher does the seat get if I swing further?
Much faster than the reach grows. Rise is the pendulum length times one minus the cosine of the angle, and cosine is flat near vertical then drops quickly. On a 6.5 foot pendulum, going from 15 to 30 degrees roughly doubles the horizontal reach but multiplies the height gained by nearly four. Energy tracks the height, which is why the speed at the bottom of the arc rises so steeply with angle.
Is the speed figure the speed a child actually reaches?
No, it is a lower bound. It is the ideal pendulum result for a mass released from rest at that angle with no friction and no pumping. A rider pumping puts energy in on every cycle and starts from something other than rest, so the real speed is higher. It is included to show how sharply energy climbs with angle, not as a figure to quote about anything.
Can I use the sweep figure to decide how close a fence can be?
No. The sweep is where the seat goes. What has to be clear around a swing accounts for a rider leaving it, and that distance comes from the equipment manufacturer instructions and the current playground safety guidance rather than from geometry. Use the sweep as a floor — if something is inside the swept path it is definitely too close — and take the real clearance from those sources.