Internal surface area = area of circle = π r², where r = radius
Surface area of roof = area of curved surface of a cone
= ½(circumference of base) X (slant height)
= ½(2πr X l)
= πr √ (r²+h²) [l = √ (r²+h²), for 45°pitch roof, h = r]
= πr √(2r²)
= π r √2 r
≈ Approx. 4.44r²
Maximum rafter length
= √(2²), where r is radius or half the breadth
To compare rectilinear and circular roofs over a cellular building
Say 2 by 6 rectangular building = 12 square metres
For same circular area, pi r squared so r = 1.95
Roof of circular with 1.95 radius ≈ Approx. 4.44r² = 16.9
Roof rectangular building 2 by 6 = 5.66 + 2 ((4) root 2 = 16.97
Therefore a circular and a rectangular building of same area use almost exactly the same roof materials, but not timber length.
Check this for other sizes.
10 by 2 = 20
Roof area = 5.66 + 23.04 = 28.7
For a circular building of area 20, r = 2.5
Roof area = 27.75
Dun Cuier, Barra (Branigan and Foster 2002, 98)
Approximately 7.1m diameter
Internal surface area = area of circle = πr² = 44.2 square metres
Surface area of roof = 4.44r² = 62.4 square metres
Max rafter length = 5.3m
Bostadh, Great Bernera, Lewis (Neighbour and Crawford 2001)
House 1 with two cells, one of 6m diameter, one approx 3 by 1.5m
Internal surface area = 28.3 + 4.5 = 32.8 square metres.
Surface area of two separate roofs (both feasibly conical) = 40 + 22.5 = approximately 62.5 square metres
One rectilinear roof for 9m length by average 5.5m width = 5.66 r² + 2 ((L-B) X √2 r)
= 42.8 + 27.2 = 70 square metres
Max rafter length = 4.2
House 3 with two cells, one of 6m diameter, one of 4.5m diameter
Internal surface area = 28.3 + 15.9 = 44.2 square metres
Roof areas both identical to House 1
Max rafter length identical
Cnip, Lewis (Armit 1996)
Rectilinear building 7 by 2.2m
Internal area = 14.4
Roof surface area = ≈ 5.66 r² + 2 ((L-B) X √2 r)
= 6.9 + 15.4 = 22.3
Rafter = 1.6
Beirgh, Lewis (Harding and Gilmour 2000)
Central cell of 6.5m diameter. Additional cell approx. 10 by 1.5m
Internal area = 33.2 + approx. 15 = 48.2
Roof surface area = 46.9
Rafter = 4.6
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