# Mathematical analysis of disphenoid (isosceles tetrahedron)

Mathematical Analysis of Disphenoid (Isosceles Tetrahedron)”

(Derivation of volume, surface area, radii of inscribed & circumscribed spheres, coordinates of vertices, in-centre, circum-centre & centroid of a disphenoid)

Mr Harish Chandra Rajpoot June, 2016

M.M.M. University of Technology, Gorakhpur-273010 (UP), India

1. Introduction: We very well know that disphenoid is a tetrahedron having four congruent faces each as an acute-angled 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 in-centre, circum-centre & 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 XY-plane so that vertex A is at origin & vertex B is on the x-axis. (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 x-axis, vertex C on the XY-plane & 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 acute-triangular 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 XY-plane

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 acute-angled 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 semi-perimeter of any of four congruent acute-triangular 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 in-centre 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 acute-triangular 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 circum-centre P be in 3D space.

Now, using distance formula, the distance between the circum-centre 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 circum-centre P from the acute-triangular face ABC (lying on the XY-plane) is equal to the radius of inscribed sphere of disphenoid ABCD

#
p<>{color:#000;}. The perpendicular drawn from the circum-centre P of disphenoid ABCD falls at the circum-centre of acute-triangular face ABC

#
p<>{color:#000;}. By symmetry of disphenoid that four faces are acute angled triangles, the circum-centre of disphenoid is at an equal normal distance from each acute-triangular face.

#
p<>{color:#000;}. The normal distance of circum-centre from each face is equal to the in-radius of disphenoid, hence by symmetry the circum-centre must be coincident with the in-centre of a disphenoid.

#
p<>{color:#000;}. Inscribed sphere touches each of four congruent acute-triangular faces at circum-centre of that face

Now, substituting the values of from (6), (7) & (8) respectively into (5), the circum-radius is given as

For a disphenoid (isosceles tetrahedron) having three unequal edges , radius of circumscribed sphere

The coordinates of circum-centre or in-centre of disphenoid (isosceles tetrahedron): The coordinates of in-centre or circum-centre 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 acute-angled triangular face with sides

The in-dentre & the circum-centre 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 in-centre or circum-centre of a disphenoid hence we can conclude that in-centre, circum-centre & centroid of a disphenoid (isosceles tetrahedron with four congruent faces each as an acute-angled triangle) are always coincident.

The solid angle subtended by disphenoid (isosceles tetrahedron) at its vertex: All four acute-angled 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 acute-angled triangle with sides then all the important parameters of disphenoid can be computed as tabulated below

table<>. <>. |<>.
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Volume

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Surface area

|<>. p<>{color:#000;}.   | <>. |<>. p<>{color:#000;}.

Vertical height

|<>. p<>{color:#000;}.   | <>. |<>. p<>{color:#000;}.

|<>. p<>{color:#000;}.

| <>. |<>. p<>{color:#000;}.

|<>. 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 in-centre, circum-centre & centroid

|<>. p<>{color:#000;}.

Where, is the volume of disphenoid &

is the area of each acute-angled 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, Gorakhpur-273010 (UP) India June, 2016

Email:[email protected]