“Mathematical Analysis of Disphenoid (Isosceles Tetrahedron)”
(Derivation of volume, surface area, radii of inscribed & circumscribed spheres, coordinates of vertices, incentre, circumcentre & centroid of a disphenoid)
Mr Harish Chandra Rajpoot June, 2016
M.M.M. University of Technology, Gorakhpur273010 (UP), India
1. Introduction: We very well know that disphenoid is a tetrahedron having four congruent faces each as an acuteangled triangle. Each pair of opposite edges of a disphenoid is equal in length hence it is also called isosceles tetrahedron (as shown in the figure1). Solid angle subtended by a disphenoid at each of its vertices is equal i.e. solid angle subtended by each face at its opposite vertex is equal because each face has the same area & is at an equal normal distance from its opposite vertex. Here we are interested to derive the general formula to compute volume, surface area, radii of inscribed & circumscribed spheres, coordinates of four vertices & the coordinates of incentre, circumcentre & centroid of a disphenoid using 3D coordinate geometry.
Figure 1: A disphenoid ABCD has each pair of opposite edges equal in length (,
2. Analysis of disphenoid (isosceles tetrahedron): Consider a disphenoid ABCD having four congruent triangular faces each of sides in 3D space optimally such that its triangular face ABC lies on XYplane so that vertex A is at origin & vertex B is on the xaxis. (As shown in the figure 2)
Applying cosine formula in as follows
Now, drop a perpendicular CN from vertex C to the side AB, the area of is as
Figure 2: A disphenoid ABCD optimally has its vertex A at origin, vertex B on the xaxis, vertex C on the XYplane & vertex D at in the first octant in 3D space
Now, coordinates of vertex C are . Let the coordinates of vertex D be
Now, using distance formula, the distance between the vertices is given as
Similarly, the distance between the vertices is given as
Setting the value of from (1),
Similarly, the distance between the vertices is given as
Setting the value of from (1) & the value of from (2),
But from Heron’s formula, Setting the value of in above equation, we should get
Now, setting the values of from (2) & (3) respectively into (1), we should get
But from Heron’s formula, Setting the value of in above equation, we should get
Thus, setting the values of from (2), (3) & (4), the coordinates of vertex D are given as
The coordinates of four vertices of disphenoid (isosceles tetrahedron): The coordinates of all four vertices A, B, C & D of disphenoid ABCD with three unequal edges for the optimal configuration in 3D space are given as &
Where, is the area of each acute triangular face with sides of the disphenoid.
The vertical height of disphenoid (isosceles tetrahedron): The disphenoid has four congruent acutetriangular faces thus by symmetry, the vertical height of each vertex from its opposite face (base) is equal & it is equal to the vertical height DE of vertex D from triangular face ABC lying on the XYplane
The volume of disphenoid (isosceles tetrahedron): The volume () of disphenoid ABCD (see figure 2 above) is given as
Above is the general formula to compute the volume of a disphenoid having four congruent faces each as an acuteangled triangle with sides
Now, the vertical height () of disphenoid in terms of volume & area of face is given as
The surface area of disphenoid (isosceles tetrahedron): The disphenoid consists of four congruent triangular faces hence the surface area of disphenoid
Where, is the semiperimeter of any of four congruent acutetriangular faces of a disphenoid
The radius of sphere inscribed by the disphenoid (isosceles tetrahedron): Let be the radius of sphere inscribed by the disphenoid ABCD, the centre O of inscribed sphere is at an equal normal distance from all four congruent triangular faces (As shown in the figure 3) Drop the perpendiculars from incentre O to four faces to get four congruent elementary tetrahedrons each of base area & normal height . If we add the volumes of these four congruent elementary tetrahedrons then we get the volume of original disphenoid thus the volume of disphenoid
Figure 3: The centre O of inscribed sphere is at an equal normal distance from all four congruent acutetriangular faces of disphenoid ABCD.
The radius of sphere circumscribing the disphenoid (isosceles tetrahedron): Let be the radius of sphere circumscribing the disphenoid ABCD, the centre P of the circumscribed sphere is at an equal distance from all four vertices A, B, C & D (As shown in the figure 4). Let the coordinates of circumcentre P be in 3D space.
Now, using distance formula, the distance between the circumcentre of disphenoid ABCD is given as
Figure 4: The centre P of circumscribed sphere is at an equal distance from all four vertices A, B, C & D of disphenoid ABCD.
Similarly, the distance between is given as
p<>{color:#000;}.
Setting the value of from (5),
Similarly, the distance between is given as
Setting the value of from (5) & the value of from (6),
But from Heron’s formula, Setting the value of in above equation, we should get
Similarly, the distance between & D is given as
Setting the value of from (5), the value of from (6) & the value of from (7),
The above result shows that
#
p<>{color:#000;}. Distance of circumcentre P from the acutetriangular face ABC (lying on the XYplane) is equal to the radius of inscribed sphere of disphenoid ABCD
#
p<>{color:#000;}. The perpendicular drawn from the circumcentre P of disphenoid ABCD falls at the circumcentre of acutetriangular face ABC
#
p<>{color:#000;}. By symmetry of disphenoid that four faces are acute angled triangles, the circumcentre of disphenoid is at an equal normal distance from each acutetriangular face.
#
p<>{color:#000;}. The normal distance of circumcentre from each face is equal to the inradius of disphenoid, hence by symmetry the circumcentre must be coincident with the incentre of a disphenoid.
#
p<>{color:#000;}. Inscribed sphere touches each of four congruent acutetriangular faces at circumcentre of that face
Now, substituting the values of from (6), (7) & (8) respectively into (5), the circumradius is given as
For a disphenoid (isosceles tetrahedron) having three unequal edges , radius of circumscribed sphere
The coordinates of circumcentre or incentre of disphenoid (isosceles tetrahedron): The coordinates of incentre or circumcentre of a disphenoid ABCD having vertices (for the optimal configuration in 3D space) & is given by substituting the values of from (6), (7) & (8) respectively as follows
Where, is the volume of disphenoid & is the area of each acuteangled triangular face with sides
The indentre & the circumcentre of a disphenoid (isosceles tetrahedron) are coincident
The coordinates of centroid of disphenoid (isosceles tetrahedron): The coordinates of centroid of a disphenoid ABCD having vertices (for the optimal configuration in 3D space) & is given as follows
But from Heron’s formula, Setting the value of in above equation, we should get
The above result shows that the coordinates of centroid are same as that of incentre or circumcentre of a disphenoid hence we can conclude that incentre, circumcentre & centroid of a disphenoid (isosceles tetrahedron with four congruent faces each as an acuteangled triangle) are always coincident.
The solid angle subtended by disphenoid (isosceles tetrahedron) at its vertex: All four acuteangled triangular faces of disphenoid are congruent hence the solid angle subtended by each face at its opposite vertex will be equal to the solid angle subtended by disphenoid at any of its four vertices. The solid angle subtended by any tetrahedron (or disphenoid) at its vertex is given by the general formula
Where, are the angles between consecutive lateral edges meeting at the concerned vertex of disphenoid. The angles meeting at any vertex of a disphenoid can be easily computed by using cosine formula as follows
Conclusion: Let there be a disphenoid (isosceles tetrahedron) having four congruent faces each as an acuteangled triangle with sides then all the important parameters of disphenoid can be computed as tabulated below
table<>.
<>. <>.
p<>{color:#000;}.
Volume
<>. p<>{color:#000;}.
 <>. <>. p<>{color:#000;}.
Surface area
<>. p<>{color:#000;}.  <>. <>. p<>{color:#000;}.
Vertical height
<>. p<>{color:#000;}.  <>. <>. p<>{color:#000;}.
Inradius (radius of inscribed sphere)
<>. p<>{color:#000;}.
 <>. <>. p<>{color:#000;}.
circumradius (radius of circumscribed sphere)
<>. p<>{color:#000;}.
 <>. <>. p<>{color:#000;}.
Coordinates of four vertices
(Optimal configuration of a disphenoid in 3D space)
<>.
p<>{color:#000;}.
 <>. <>. p<>{color:#000;}.
Coordinates of coincident incentre, circumcentre & centroid
<>. p<>{color:#000;}.
Where, is the volume of disphenoid &
is the area of each acuteangled triangular face with sides

Note: Above articles had been derived & illustrated by Mr H.C. Rajpoot (B Tech, Mechanical Engineering)
M.M.M. University of Technology, Gorakhpur273010 (UP) India June, 2016
Email:[email protected]
Author’s Home Page: https://notionpress.com/author/HarishChandraRajpoot
©3D Geometry by H. C. Rajpoot