To be a theoretical structural archaeologist, you are going to have to understand (or at least believe in) Pythagoras, the sine rule, and pi; but there is no need to panic -- I think they still teach this to children, so it might be useful to have one to hand.

Digging holes for a living is an ideal place to hide from maths, not to mention reading and writing, but, unfortunately, Iron Age roundhouses only really make sense if you understand their geometry. Maths is important to help us model the nearly 2-dimensional evidence of prehistoric postholes, and encourages us to think 3-dimensionally about structures.

Digging holes for a living is an ideal place to hide from maths, not to mention reading and writing, but, unfortunately, Iron Age roundhouses only really make sense if you understand their geometry. Maths is important to help us model the nearly 2-dimensional evidence of prehistoric postholes, and encourages us to think 3-dimensionally about structures.

**The geometry of a cone and frustum of a cone**

Roofs are made using tree trunks, which, theoretically at least, start off as cones, but when the thin pointy end is removed, the trunk becomes a ‘frustum’ of a cone (plural: frusta or frustums) -- which is one for the pub quiz.

I want to introduce you to one of the most important graphs in the study of the prehistoric built environment. I first published it, but it’s not mine; it is just an equation. [1]

**The relationship between the diameter of a circular building and its roof area**

The roofs of prehistoric roundhouses in Britain are assumed to have been conical; as a result, their surface area, and therefore their weight, increases disproportionately with roof diameter. To put this graph in perspective, roundhouses are said to average about 10m in diameter, and 17m would be considered about the limit. [2]

So when we compare an average roundhouse with a large one, although only 70% larger in diameter, the latter’s roof is 3 times the size, with 3 times the weight and thrust of the former. So large roundhouses are much larger buildings than the average circular building, and more structurally challenging to construct.

So when we compare an average roundhouse with a large one, although only 70% larger in diameter, the latter’s roof is 3 times the size, with 3 times the weight and thrust of the former. So large roundhouses are much larger buildings than the average circular building, and more structurally challenging to construct.

**Simple roundhouse geometry**

Two different roof pitches are shown, since this changes the surface area of the roof. Reconstructions have followed Peter Reynolds’ assumption that a 45 ° thatched roof was probably used. I have discussed this previously,[3] but Reynolds' reasons serve to illustrate the mindset underlying his approach:

- Simple angle to calculate
- Simple to construct
- Produces a balanced roof
- Minimum angle for thatch
- Uses the minimum amount of materials
- Used in Africa

Since, I reject minimalism and primitivism, and do not regard roof pitch as some sort of genetic memory brought out of Africa by our ancestors, I note with interest that traditional English building and thatching tends to use a steeper angle, such as 53°, which derives from a 3:4:5 triangle.

A steeper angle of the thatch is better at shedding water, and this increases the life of the roof surface. An additional complication is junctions between roofs, such as where a porch joins the main roof. These ‘valleys’, which channel water, are slacker than the main pitch. A steeper main roof pitch helps ensure that valleys are at the minimum pitch of 45 degrees.

If you imagine the object of prehistoric building is to create the simplest, most basic structure, using the minimum of materials, by all means use 45°, but I prefer 53° for the same reasons builders (in England) have traditionally used it: because it produces an effective and long lasting roof surface.

The next problem is to try and appreciate what this surface graph may represent in terms of roof weight. This can be calculated theoretically, but luckily experimental archaeology can provide some ‘real’ figures.

The next problem is to try and appreciate what this surface graph may represent in terms of roof weight. This can be calculated theoretically, but luckily experimental archaeology can provide some ‘real’ figures.

**Theoretical roof weight for different diameters of roundhouses**

The Great House reconstruction at Butser, based on a building from Cow Down (Longbridge Deverill house 3), has a 25 tonne roof, from which we can calculate a figure of 88 Kg m² of roof surface, which saves an awful lot of mucking about with maths. [4]

Ignoring the porch in the calculations, the roof of our average 10m roundhouse comes in about 10 tonnes, compared with 25 tonnes for the large roundhouse. This is the ‘static’ roof load, but we can also calculate the effect of snow and ice.[5]

Ignoring the porch in the calculations, the roof of our average 10m roundhouse comes in about 10 tonnes, compared with 25 tonnes for the large roundhouse. This is the ‘static’ roof load, but we can also calculate the effect of snow and ice.[5]

**The effect of snow and ice on the weight of a theoretical 53° roundhouse roof**

**Rafter length for roundhouse roofs of different diameters**

The pitch of a circular roof is a simple calculation, and unlike roof area, rafter length is proportional to the roundhouse width. Our average roundhouse entails 7--8m rafters, about 24’ long and quite large by modern domestic standards, but a very large example would require rafters in the 14m region. That’s 46’ in old money, a very long piece of timber.

As we have discussed in a previous article, [7] trees were grown to order to fit a particular need, and woods would be managed to produce what builders required. It’s clear that roundhouses need long straight timber from several hundred trees of specific sizes, so someone better have planned ahead – at least 60 years ahead!

As we have discussed in a previous article, [7] trees were grown to order to fit a particular need, and woods would be managed to produce what builders required. It’s clear that roundhouses need long straight timber from several hundred trees of specific sizes, so someone better have planned ahead – at least 60 years ahead!

It is worth a brief venture into the largely unfamiliar world of the traditional woodsman. The growth of each tree will be unique, depending on its local environment and situation. Some of these factors can be controlled by woodsmen, who can also intervene in the process to maximise the potential of the tree to produce the desired timbers, be they withies for baskets, or rafters for roundhouses.

The usable trunk of a tree is known as the ‘butt'. Its diameter is measured 1.2m from the ground, where it is more typical than at the base of the trunk. This is known as the ‘DBH’ or ‘diameter at breast height’. We can think of a tree trunk as a cone. In reality the growth pattern of individual species is considerably more complex, but at some point in a tree like oak the trunk will start to branch in its search for light, forming the ‘crown’ of the tree. Above this point the trunk is too irregular to be of practical value.

**The growth pattern of a theoretical oak tree**

In most systems of forestry, trees are periodically thinned, the weaker specimens being weeded out to allow light and space for the ‘main crop’ to grow. Foresters use tables of figures, gathered through observation, to predict the growth of their wood, and its potential yield in terms of volumes of usable timber. These figures can be adapted to model the growth of a ‘theoretical oak tree’. Using figures for a good 'yield class’ of oak tree, [7] we can plot the height of the tree, its ‘top height’, against its DBH, to help visualise how a tree grows. Lower ‘yield classes’ may produce similar shaped trees; they just take longer for them to grow.

To be useful for prehistoric rafters, the butt must be thick enough at the ‘thin’ end to be rigid, but not too massive and heavy at the ‘thick’ end. If we decide that the trees we need have a butt thinner than 0.3m, but at least 0.1m thick, then our theoretical tree must be harvested between 35 and 60 years old.

To be useful for prehistoric rafters, the butt must be thick enough at the ‘thin’ end to be rigid, but not too massive and heavy at the ‘thick’ end. If we decide that the trees we need have a butt thinner than 0.3m, but at least 0.1m thick, then our theoretical tree must be harvested between 35 and 60 years old.

*(Figures are for illustrative purposes only; actual growth may vary -- this is an unregulated market!)***Trunk dimensions for a theoretical oak trees**

Forestry yield tables can also be used to model the length of the butt by plotting the changing trunk diameter against height. This tells us that at 60 years old, our theoretical trunk has reached our maximum thickness, and has a top height of 23m, and that the trunk reaches our minimum diameter of 0.1m about 15m from the ground.

The graph demonstrates that oak trees can only produce rafters within certain limits, and while it is clearly difficult to be specific, the maximum usable rafter is probably around 14 – 15 m in length, which corresponds to those required for the largest roundhouses found.

This suggests that the largest roundhouses required the longest timbers available, and given the exponential increase in weight for conical roofs, were probably at the limits of what it was considered safe to build.

This is what our graphs have told us: Not all roundhouses are equal -- some are considerably more equal than others, and if you live in one that is the biggest available and three times as large as the average, then you are probably not an average person.

However, in the next article we will look at some real examples, and I will show something truly remarkable about prehistoric buildings and the people who designed and built them.

The graph demonstrates that oak trees can only produce rafters within certain limits, and while it is clearly difficult to be specific, the maximum usable rafter is probably around 14 – 15 m in length, which corresponds to those required for the largest roundhouses found.

This suggests that the largest roundhouses required the longest timbers available, and given the exponential increase in weight for conical roofs, were probably at the limits of what it was considered safe to build.

This is what our graphs have told us: Not all roundhouses are equal -- some are considerably more equal than others, and if you live in one that is the biggest available and three times as large as the average, then you are probably not an average person.

However, in the next article we will look at some real examples, and I will show something truly remarkable about prehistoric buildings and the people who designed and built them.

Sources & Further Reading:

Sources & Further Reading:

[1] G. A. Carter 1998: Excavations at the Orsett ‘Cock’ enclosure, Essex, 1976. East Anglian Archaeology Report No 86.

[2] see: F Pryor 2004: Britain BC: Life in Britain and Ireland Before the Romans Harper Perennial [pp 235 – 238]

[3 ] D W Harding, I M Blake, and P J Reynolds 1993 An Iron Age settlement in Dorsett: Excavation and reconstruction. University of Edinburgh. Department of Archaeology Monograph series No. 1.

[4] http://structuralarchaeology.blogspot.com/2008/10/8-who-would-live-in-house-like-this.html

[5] http://www.butser.org.uk/iaflphd_hcc.html http://www.butser.org.uk/index_sub.html

S. C. Hawkes 1994. Longbridge Deverill Cow Down, Wiltshire, House 3: A Major Round House of the Early Iron Age. Oxford Journ. Archaeol. 13(1), 49-69.

[6] Karl VanDevender Ice and Snow Accumulations on Roofs, in Disaster Response Handbook

University of Arkansas, United States Department of Agriculture, and County Governments Cooperating http://www.aragriculture.org/disaster/ice_snow/ice_snow_accumulation.pdf

[7] http://structuralarchaeology.blogspot.com/2009/01/17-not-seeing-wood-or-trees.html

[8] James. 1989, Forester's Companion Cambridge University Press, ISBN 0631108114

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