The Yurt Mathematics of a Modern structure
A yurt begins not with wood or fabric, but with a circle drawn on the earth. Somewhere between wind, sunlight, rope, and gravity, a quiet language reveals itself — mathematics. Not the kind we memorised in classrooms, but the kind that exists in shadows, distances, angles, and balance. In this essay, I share the applied geometry of a modern yurt as I have learned it through hands, soil, and structure. With just one diameter, two gentle angles, and a few simple formulas, you can design a real, standing home using bamboo, eucalyptus poles, rope, and whatever nature offers freely. This is not engineering from books. This is mathematics as a living craft.
Seen through the eyes of a desert architect
I was never “good at maths” in school.
In class, numbers lived on a blackboard.
In real life, numbers were water levels in a well, the shade of a tree at noon, the distance to the next village, the angle of the sun on a hot roof.
Years later, while working with circles, domes and tents in the desert, I realised something important:
Maths is not a subject. Maths is a language that structures use to talk to us.
A yurt is a beautiful example of this.
If you look carefully at a yurt, you can see the maths hiding inside it:
- in the circle of the floor
- in the rhythm of the lattice wall
- in the angle of the roof
- in the opening of the crown ring
You don’t have to be an engineer.
You don’t have to be “good at maths”.
You just need:
- one circle
- two angles
- a few simple formulas
And you can design your own yurt with bamboo, dowels, eucalyptus poles, rope, or steel cables – whatever mother nature gives you for free.
This is the maths of a modern yurt, the way I use it in my work.
1. Everything starts with one number: the diameter
Every yurt begins as a circle scratched on the ground.
You choose how wide you want your round home to be. That gives you the diameter.
Let:
\(D = \text{diameter of the yurt (in feet)}\)From this we get the radius (half the width of your home):
\(r = \frac{D}{2}\)If you like working in millimetres in the workshop, use:
\(D_\text{mm} = D \times 304.8\) \(r_\text{mm} = \frac{D_\text{mm}}{2}\)Emotionally: you decide how generous you want the circle to be, and the radius is just the halfway mark from center to wall.
Floor area of the yurt
The floor is a simple circle. Its area is:
\(A_\text{floor} = \pi r^2\)If you keep D in feet:
\(A_\text{floor (ft}^2\text{)} = \pi \left(\frac{D}{2}\right)^2\)This tells you how much floor space, carpet, or mud plaster you’ll need.
2. The hidden length of the circle: your wall
If you cut the circle and unroll it into a straight line, you get the circumference. That is the total wall length around the yurt.
The universal formula:
\(C = 2\pi r\)In millimetres, using the radius in mm:
\(C_\text{mm} = 2\pi r_\text{mm}\)In feet:
\(C_\text{ft} = \frac{C_\text{mm}}{304.8}\)In words: if you straighten your round yurt like a fence, this is how long that fence will be.
This wall length will carry:
- the lattice framework
- the fabric or cladding
- the tension cable
3. The 10-inch rhythm: how many lattice joints?
Now we give the wall a rhythm.
In our system, we place a lattice joint every 10 inches along the wall.
10 inches ≈ 254 mm. Let:
\(s = 254 \text{ mm (spacing between lattice joints)}\)Number of lattice joints around the circle:
\(N_\text{joints} = \frac{C_\text{mm}}{s}\)This is what I call the “lattice count”.
You normally round this to the nearest whole number:
\(N_\text{joints (rounded)} \approx \text{round}\left(\frac{C_\text{mm}}{s}\right)\)Rafters from lattice joints
Each joint at the top of the lattice is a place where a roof rafter can sit.
- If you place one rafter on every joint:
- If you place one rafter on every second joint:
Again, round to a friendly whole number.
One simple division — wall length divided by 10 inches — quietly decides how many sticks, ties, and rafters your yurt will need.
Adjusting for the door opening (optional but useful)
If your door is, for example, 3 feet wide, it removes some length from the lattice.
Door width in mm:
\(W_\text{door mm} = 3 \times 304.8\)Effective wall length for lattice:
C’ mm = C mm − W_door mm
New lattice joint count:
N’ joints = C’ mm / s
This is a nice way to show that the door is part of the mathematics of the circle, not an afterthought.
4. The flowering wall: 60° lattice geometry
A yurt wall is not a vertical fence.
It leans out slightly and forms diamonds when the sticks cross.
We use a powerful and simple angle:
\(\alpha = 60^\circ\)Now pick your wall height:
Let:
\(h = \text{wall height in feet}\) \(h_\text{mm} = h \times 304.8\)Each lattice stick runs diagonally, and its vertical component is the wall height.
The relationship is:
\(h_\text{mm} = L \cdot \sin(\alpha)\)Where LLL is the lattice stick length. So:
\(L = \frac{h_\text{mm}}{\sin(60^\circ)}\)Since sin(60°) ≈ 0.866
\(L \approx h_\text{mm} \times 1.1547\)Convert back to feet if you like:
\(L_\text{ft} = \frac{L}{304.8}\)This is the cut length of every wall stick: bamboo, dowel, eucalyptus, anything.
Approximate diamond size
If you’re curious about the size of each diamond in the lattice:
Horizontal distance between joints (projection):
Diamond width ≈ 2 × horizontal distance
Vertical distance between crossing points:
Diamond height ≈ 2 × vertical distance
Each lattice diamond is roughly:
Twice the horizontal spacing in width
Twice the vertical spacing in height
With one angle (60°) and one stick length, the wall automatically arranges itself into a repeating pattern of triangles and diamonds. That’s the yurt’s secret strength.
5. The calm roof: 28° rafter geometry
Now we look up, to the roof.
We choose a calm, low, wind-friendly angle:
\(\beta = 28^\circ\)This angle allows:
- rain to slide
- wind to glide over
- the yurt to stay visually grounded
We start with the radius at the top of the wall:
\(r_\text{outer} = r_\text{mm}\)In the simplest version, ignoring crown ring size, the horizontal run of the rafter is the radius.
Ideal rafter length:
\(L_\text{rafter mm} = \frac{r_\text{outer}}{\cos(\beta)}\)Convert to feet:
\(L_\text{rafter ft} = \frac{L_\text{rafter mm}}{304.8}\)In practice, we subtract a little for how the rafter sits into the ring and on the wall. In my working system I often use:
\(L_\text{rafter cut ft} = L_\text{rafter ft} – 1.5\)That “– 1.5 ft” is not a law of nature – it’s craft knowledge.
More accurate version with crown ring
If you want to include the crown ring radius:
Let:
\(r_\text{inner} = \text{crown ring radius (in mm)}\)Then the horizontal run of the rafter is:
\(\text{run} = r_\text{outer} – r_\text{inner}\)And:
\(L_\text{rafter mm} = \frac{r_\text{outer} – r_\text{inner}}{\cos(\beta)}\) \(L_\text{rafter ft} = \frac{L_\text{rafter mm}}{304.8}\)This lets you play with different crown ring sizes while keeping the roof angle the same.
6. Centre height: how high the sky is inside
The rise of the roof, from the top of the wall to the centre, comes from simple trigonometry again.
With a crown ring:
\(\Delta h_\text{roof mm} = (r_\text{outer} – r_\text{inner}) \cdot \tan(\beta)\)If you ignore the ring (or assume it is small):
\(\Delta h_\text{roof mm} = r_\text{mm} \cdot \tan(28^\circ)\)Convert to feet:
\(\Delta h_\text{roof ft} = \frac{\Delta h_\text{roof mm}}{304.8}\)Total height at the centre, measured from the ground, is then:
\(H_\text{total ft} = h + \Delta h_\text{roof ft}\)This number controls how the space feels: low and intimate, or tall and temple-like.
7. Wall area: fabric, insulation, and cladding
Once you know the wall height and circumference, wall surface area is easy.
If you use feet for both:
\(A_\text{wall (ft}^2\text{)} = C_\text{ft} \times h\)This tells you how much you need of:
- outer fabric or cladding
- insulation layer (wool, felt, recycled fibre)
- inner lining (if you use one)
8. Roof area: covering the cone
The roof of a yurt is roughly a cone. If you have a crown ring, it becomes a truncated cone (a cone with the top cut off).
Let:
- \(R = r_\text{outer}\) (outer radius at wall, in meters or feet – just be consistent)
- \(r = r_\text{inner}\) (crown ring radius)
- \(L_\text{roof}\) = slant length of the roof (same as rafter length from wall to ring)
Approximate roof area with ring:
\(A_\text{roof} \approx \pi (R + r),L_\text{roof}\)If the ring is small, a simpler version:
\(A_\text{roof} \approx \pi R,L_\text{roof}\)This is useful for:
- fabric ordering
- waterproof outer skin
- insulation over the roof
9. Volume: the amount of air inside
For a quick sense of air volume and heating needs, treat the yurt as:
- a cylinder (walls)
- plus a cone (roof)
Using feet:
- \(r = \frac{D}{2}\)
- \(h\) = wall height (ft)
- \(\Delta h_\text{roof}\) = roof rise (ft)
Volume of walls (cylinder):
\(V_\text{cyl} = \pi r^2 h\)Volume of roof (cone):
\(V_\text{cone} = \frac{1}{3}\pi r^2 \Delta h_\text{roof}\)Total:
\(V_\text{total} = V_\text{cyl} + V_\text{cone}\)This is enough to understand:
- how much air you are heating
- how wood-efficient your stove must be
- how ventilated the space will feel
10. Crown ring spacing and rafter count (a little extra)
Once you know how many rafters you want, you can check if your crown ring size makes sense.
Ring circumference:
\(C_\text{ring} = 2\pi r_\text{inner}\)Spacing between rafter ends at the ring:
If you already know how many rafters you want, and you prefer the rafter ends to be about 80–120 mm apart, you can reverse the calculation to find the crown ring size.\(p_\text{ring} = \frac{C_\text{ring}}{N_\text{rafters}}\)
\(r_\text{inner} = \frac{N_\text{rafters} \cdot p_\text{desired}}{2\pi}\)This is how craft and maths meet:
“I want around 60 rafters and about 10 cm between them” becomes a clear ring size.
Closing: math as a shared language
All the formulas we used are very simple:
- \(C = 2\pi r\)
- \(A = \pi r^2\)
- \(L = \frac{h}{\sin(60^\circ)}\)
- \(R = \frac{r}{\cos(28^\circ)}\)
- \(\Delta h = r\tan(28^\circ)\)
But they are not here to impress anyone.
They are here so that:
- a child with a compass and a stick
- a farmer with bamboo and rope
- a maker with dowels and jute
- a desert architect with eucalyptus and steel cable
…can all build something round, balanced and alive.
Maths is the quiet language that lets people, materials and gravity agree on one thing:
“This will stand.”
Someday, we might turn all of this into a simple tool where you type:
“20 ft yurt, 6 ft wall”
…and it gives you every length, every angle, every quantity.
But before the tool, I wanted to show you the way of seeing:
A yurt is not magic.
It is a circle, two angles, and a handful of kind formulas.
How to use this blog
- Treat the formulas as a cheat sheet.
- Treat the sketches as your visual teacher.
- And treat the desert as the classroom.
If you sit with a pencil, a rope, a tape measure, and these equations, you can start designing your own round shelter — anywhere on Earth.
The full calculation of a 20 feet yurt
A yurt is not held together by wood, rope, or fabric alone. It is held together by quiet agreements between angles, lengths, and gravity. When you draw a circle on the earth and follow its mathematics, you are not just building a shelter — you are joining an ancient conversation between human hands and natural law. At Houbara Outdoors, this is the same language we use to shape our domes and spaces: refined by experience, simplified by craft, and guided by nature. Whether you build with bamboo in a forest, with eucalyptus in a desert, or someday with us, may these simple formulas help you create spaces that feel grounded, alive, and deeply human.
Best regards
Mahendra
Founder
Houbara Outdoors
+919079656429
This work is entirely based on my own observations, calculations, and hands-on experience in the field. It has not been copied from any book, research paper, or external source. The mathematics shared here is my personal way of understanding and explaining the yurt through practice. You are free to use, share, and apply this knowledge for learning, building, and exploration — no permission needed, no credit required. May it travel freely and help many hands build meaningful spaces.