Three dimensions describe the plan, and people quote whichever flatters them
A regular polygon has a side length, an apothem — the distance from the centre square out to the middle of a side — and a circumradius, the distance from the centre out to a corner. Across the flats is twice the apothem and across the corners is twice the circumradius, and for a hexagon those two differ by 15.5 percent. A twelve foot gazebo measured across the corners fits in a smaller space than a twelve foot gazebo measured across the flats, which is why the dimension mode on this page comes before anything else.
For a hexagon of 12 ft across the flats the apothem is 6 ft, each side is 6 ft 11-1/8 in, the circumradius is the same 6.9282 ft as the side, and the floor area inside the plate is 124.71 sq ft. That equality of side and radius is a hexagon party trick and it is the reason the shape is so easy to set out: swing the radius round the circle six times and the sixth mark lands on the first.
| Sides | Side, for 12 ft across the flats | Across the corners | Floor area |
|---|---|---|---|
| 4 | 12 ft | 16 ft 11-5/8 in | 144.00 sq ft |
| 6 | 6 ft 11-1/8 in | 13 ft 10-1/4 in | 124.71 sq ft |
| 8 | 4 ft 11-5/8 in | 12 ft 11-15/16 in | 119.29 sq ft |
| 12 | 3 ft 2-9/16 in | 12 ft 5-1/16 in | 115.75 sq ft |
More sides at the same across-the-flats dimension means less floor, because the shape is converging on a circle inscribed in the same square. It also means more corners to cut, more posts to set and more hips to fit. Six is the common gazebo for a reason, and eight is the point at which most people building one on site stop enjoying it.
Two pitches in one roof
Every face of a polygon roof is a triangle with the plate as its base and the peak as its apex. A common rafter runs square off the middle of a side, so its run is the apothem. A hip runs to a corner, so its run is the circumradius, which is longer. Both reach the same peak height. That forces the hip to a shallower pitch, in exactly the ratio of the apothem to the circumradius — which is the cosine of half the interior angle at the centre.
On a hexagon that ratio is cos 30, or 0.8660. A common at 6 in 12 gives hips at 5.196 in 12. On an octagon the ratio is cos 22.5, or 0.9239, so the same 6 in 12 common gives hips at 5.543. The plumb cuts follow: on the hexagon the common is 26.57 degrees off square and the hip is 23.41, a difference of a little over three degrees. Three degrees over a 7 ft rafter is more than four inches of error at the peak, and it does not go away by pushing harder.
Working the default hexagon at 6 in 12 all the way through: the rise is 6 ft times 0.5, or 3 ft. The common rafter to the plate is 6 times 1.11803, or 6.708 ft, and to the end of a 12 inch level overhang it is 7 times 1.11803, or 7.826 ft. The hip to the corner is the square root of 6.9282 squared plus 3 squared, which is 7.5498 ft, and out to the overhang it is 8.808 ft. The plan area out to the eaves is 169.74 sq ft, and multiplying by the 1.11803 slope factor gives a roof surface of 189.78 sq ft in six triangular faces.
Setting it out on the ground
The stake-out figure that saves the job is the diagonal from one post to the post after next, because it catches a polygon that has drifted out of regular in a way that measuring sides never does. On a hexagon that diagonal is the side times the square root of three; on the default 12 ft gazebo it is 12 ft exactly, which happens to equal the across-the-flats dimension, and that coincidence is another reason hexagons get built. Set the six corners, measure all six of those diagonals, and if they are not equal the shape is out however good the sides look.
For the miter itself, a flat frame around an N-sided figure takes 180 divided by N at each end — 30 degrees for a hexagon, 22.5 for an octagon. Where the piece is standing up rather than lying flat, as fascia around a sloping roof is, the cut becomes a compound one and the miter angle calculator handles the pair of settings. The straight rafter geometry, if you want to check a single member, is on the rafter length calculator, and roof pitch converts between pitch, degrees and slope multiplier.
Once the shape is settled, the covering area feeds the roofing squares calculator or, for panel, the corrugated panel calculator — with the warning that triangular faces waste panel badly, roughly half of every full width once the rake cuts are made. Posts and footings are their own job: concrete footings for the pads and post concrete for buried posts.
Questions people ask
What is the side length of a hexagonal gazebo?
For a regular hexagon the side equals the circumradius, so it is exactly half the across-the-corners dimension. Across the flats it is different: the side is the across-the-flats dimension times the tangent of 30 degrees, or 0.5774 of it. A 12 ft hexagon measured across the flats has sides of 6 ft 11-1/8 in and measures 13 ft 10-1/4 in across the corners. Confirm which dimension a kit or a drawing is quoting before ordering anything, because the two differ by more than 15 percent.
Why is the hip rafter a different pitch from the common?
Because it has further to travel across the plan to reach the same peak. The common runs square off a side, a distance equal to the apothem. The hip runs diagonally to a corner, a distance equal to the circumradius, which on a hexagon is 15.5 percent longer. Same rise, longer run, shallower pitch. The hip pitch is the common pitch times the apothem divided by the circumradius, so a 6 in 12 common gives 5.196 in 12 hips on a hexagon and 5.543 on an octagon. Cut the hips on the common angle and the peak will not close.
How many rafters does a gazebo roof need?
The hips are fixed by the shape: one per corner, so six on a hexagon and eight on an octagon. The commons are one per side. Everything else is jacks, and how many of those there are is a spacing decision that comes from what the roof deck needs, which comes from the deck material and the span table, not from geometry. This page counts them at whatever spacing you enter and reports each pair getting shorter as they walk out towards the corner. Small gazebos with structural sheet decking are often built with hips and commons only.
Do I add the overhang before or after working out the side length?
After, and the outer shape is a scaled copy of the inner one rather than the same shape with a constant added to each side. A level overhang measured square off each side pushes the apothem out by that amount, and the outer side length grows in the same ratio as the apothem, not by twice the overhang. On the default hexagon a 12 inch overhang takes the apothem from 6 to 7 ft and the side from 6.928 to 8.083 ft. The calculator does this for you; doing it by hand is where the fascia ends up short.
Are these rafter lengths what I cut to?
They are lengths to the theoretical peak, where the rafter centrelines meet. Whatever actually sits in the middle — a king post, a fabricated hub, a peak block, or the hips notched into each other — has to be deducted along the slope of each rafter, and because the hips and the commons run at different pitches the deduction is a different number for each. The reliable method is to cut one hip and one common long, offer them to whatever the peak really is, mark the deduction on the timber, and only then cut the rest.