Copyright © Peter Wakefield Sault 1973-2005
All rights reserved worldwide
Volume of The Trapeza
The trapeza can be decomposed into nine separate modules, comprising four types identified as V_{1} through V_{4} in Figure F-2 below.
The heights, h (normal to the view in Figure F-2), of the modules are all equal. The lengths of the edges are:
a | Short side of the rectangle |
b | Long side of the rectangle |
c | Side of the square of equal area |
Where, by definition:
a/c | = | c/b | |
\ | c^{2} | = | ab |
The volumes of the module types are:
Cuboid | V_{1} | = | ach | |
Triangular prism | V_{2} | = | ch(c-a)/4 | |
= | ah(b-c)/4 | |||
Triangular prism | V_{3} | = | ah(b-c)/4 | |
Tetrahedron | V_{4} | = | h(b-c)(c-a)/12 |
from which it can be seen that V_{2} = V_{3}.
Therefore the total volume of the trapeza:
V_{T} | = | V_{1} + 2V_{2} + 2V_{3} + 4V_{4} |
= | h[ac + a(b-c) + (b-c)(c-a)/3] | |
= | h(ab + ac + bc)/3 |
Volume of The Rectangle-Based Axehead
The rectangle-based axehead can be decomposed into three separate modules, comprising two types identified as V_{5} and V_{6} in Figure F-3 below.
The heights, 2h (normal to the view shown in Figure F-3), of the modules are all equal, where h is the height of the trapeza. The lengths of the edges are:
a | Short side of the rectangle |
b | Long side of the rectangle |
d | Edge of the axehead |
Where d = b + c
The volumes of the module types are:
Triangular prism | V_{5} | = | abh | |
Tetrahedron | V_{6} | = | ah(d-b)/6 | |
= | ach/6 |
Therefore the total volume of the axehead:
V_{RA} | = | V_{5} + 2V_{6} |
= | abh + ach/3 | |
= | ah(b + ^{c}/_{3}) |
Volume of The Square-Based Axehead
The square-based axehead can be decomposed into three separate modules, comprising two types identified as V_{7} and V_{8} in Figure F-4 below.
The heights, 2h (normal to the view shown in Figure F-4), of the modules are all equal, where h is the height of the trapeza. The lengths of the edges are:
c | Side of the square |
e | Edge of the axehead |
Where e = a + c
The volumes of the module types are:
Triangular prism | V_{7} | = | c^{2}h | |
= | abh | |||
Tetrahedron | V_{8} | = | ch(e-c)/6 | |
= | ach/6 |
Therefore the total volume of the axehead:
V_{SA} | = | V_{7} + 2V_{8} |
= | abh + ach/3 | |
= | ah(b + ^{c}/_{3}) |
from which it becomes clear that the volumes of the two axeheads are equal.
Volume of The Composite Tetrahedron
Copyright © Peter Wakefield Sault 1973-2005
All rights reserved worldwide