117.8511
units³
173.2051
units²
43.3013
units²
8.165
units
2.0412
units
6.1237
units
117.8511
units³
173.2051
units²
43.3013
units²
8.165
units
2.0412
units
6.1237
units
A regular tetrahedron is the simplest of all Platonic solids — a polyhedron composed of four congruent equilateral triangular faces, six equal edges, and four vertices. It is the three-dimensional analog of the equilateral triangle, possessing the highest degree of symmetry among all triangular pyramids. The Tetrahedron Volume Calculator computes all essential geometric properties from a single measurement: the edge length $$a$$.
The volume of a regular tetrahedron is given by:
$$V = \frac{a^3}{6\sqrt{2}} = \frac{a^3 \sqrt{2}}{12}$$
This can be derived by treating the tetrahedron as a pyramid with an equilateral triangular base (area $$\frac{\sqrt{3}}{4}a^2$$) and height $$h = a\sqrt{\frac{2}{3}}$$, then applying $$V = \frac{1}{3}Ah$$.
The total surface area is simply four times the area of one equilateral triangular face:
$$A_{\text{total}} = 4 \times \frac{\sqrt{3}}{4}a^2 = \sqrt{3}\,a^2$$
The height of a regular tetrahedron (the perpendicular distance from one vertex to the opposite face) is:
$$h = a\sqrt{\frac{2}{3}}$$
Two important spheres are associated with a regular tetrahedron. The insphere (the largest sphere that fits inside, tangent to all four faces) has radius:
$$r_{\text{in}} = \frac{a}{2\sqrt{6}} = \frac{a\sqrt{6}}{12}$$
The circumsphere (the smallest sphere that contains all four vertices) has radius:
$$R_{\text{circ}} = \frac{a\sqrt{6}}{4}$$
A remarkable property is that the circumsphere radius is exactly three times the insphere radius: $$R = 3r$$.
Regular tetrahedra appear throughout chemistry (methane CH₄ has tetrahedral molecular geometry), crystallography (carbon atoms in diamond form tetrahedra), structural engineering (tetrahedral trusses are used in space frames), and game design (four-sided dice are tetrahedra). In computational geometry, tetrahedral meshes are fundamental for finite element analysis. In architecture, tetrahedral structures provide exceptional strength-to-weight ratios. Alexander Graham Bell experimented extensively with tetrahedral kite frames, demonstrating their structural efficiency. Understanding the precise measurements of a regular tetrahedron is essential across all these disciplines, and this calculator makes those computations effortless.
Enter the edge length $$a$$. Every measurement of a regular tetrahedron is determined by this single value. The calculator computes the volume, total surface area, individual face area, height, insphere radius, and circumsphere radius using the closed-form formulas for regular tetrahedra.
The volume is the enclosed space. The surface area is the total area of all four equilateral triangular faces. The height is the perpendicular distance from any vertex to the opposite face. The insphere radius is the radius of the largest sphere fitting inside the tetrahedron. The circumsphere radius is the radius of the sphere passing through all four vertices. Note that $$R_{\text{circ}} = 3 r_{\text{in}}$$ for every regular tetrahedron.
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A tetrahedron with edge 10 has volume ~117.85 cubic units. The circumsphere radius (6.12) is exactly 3× the insphere radius (2.04).
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At molecular scale (C-C bond length in diamond ≈ 1.54 Å), the tetrahedral unit cell volume is ~0.43 ų.
A Platonic solid is a convex polyhedron where every face is an identical regular polygon and the same number of faces meet at every vertex. There are exactly five Platonic solids: tetrahedron (4 triangles), cube (6 squares), octahedron (8 triangles), dodecahedron (12 pentagons), and icosahedron (20 triangles). The tetrahedron is the simplest.
This is a unique property of the regular tetrahedron. The centroid (center of mass) divides the line from a vertex to the center of the opposite face in the ratio 3:1. Since the insphere touches faces and the circumsphere passes through vertices, this 3:1 ratio directly gives $$R = 3r$$.
No. Regular tetrahedra cannot tessellate three-dimensional space by themselves. However, a combination of regular tetrahedra and regular octahedra can fill space perfectly — this is called the octet truss or tetrahedral-octahedral honeycomb, widely used in structural engineering.
In methane (CH₄), the carbon atom sits at the center and the four hydrogen atoms occupy the four vertices of a regular tetrahedron. The H-C-H bond angle is approximately 109.47° — this is the tetrahedral angle, equal to $$\arccos(-1/3)$$. This geometry minimizes electron pair repulsion (VSEPR theory).
The dihedral angle (the angle between two adjacent faces measured along their shared edge) is $$\arccos(1/3) \approx 70.53°$$. This is a fundamental constant in solid geometry and explains why regular tetrahedra cannot tile space — the angle does not divide 360° evenly.
For an irregular tetrahedron with vertices at known coordinates $$(x_i, y_i, z_i)$$, use the formula: $$V = \frac{1}{6}|\det[\vec{AB}, \vec{AC}, \vec{AD}]|$$, where $$\vec{AB}$$, $$\vec{AC}$$, $$\vec{AD}$$ are edge vectors from one vertex to the other three. This calculator is designed for regular tetrahedra only.
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