A regular tetrahedron is the simplest three-dimensional solid there is: four equilateral triangular faces, six equal edges, four vertices, and nothing else. This tetrahedron volume calculator takes any one of its measurements, an edge, a volume, a total surface area or a vertical height, and returns the complete set, along with the radii of the inscribed and circumscribed spheres and the capacity in litres.
Arb Digital built the reverse-solving modes in because tetrahedron questions rarely start from an edge. They start from a volume you need to hit, or a height you have to fit under, and the edge is the answer rather than the input. The substitution line shows the arithmetic either way, so a 6 cm edge reads V = a³ ÷ (6√2) = 216 ÷ 8.485 = 25.46 cm³.
What This Tetrahedron Calculator Does
The calculator handles the regular tetrahedron, the case where all six edges are the same length. That single edge determines every other property of the solid, which is why any one measurement is enough to recover all the others.
Whichever mode you pick, the tool solves back to the edge first. From a volume it takes a cube root. From a surface area it takes a square root. From a height it divides by the appropriate constant. Then it rebuilds volume, total surface area, the area of a single face, the vertical height, and the two sphere radii from that one edge, so the results are always internally consistent rather than independently rounded.
Capacity is reported in litres and US gallons alongside the cubic units. Tetrahedral shapes turn up in packaging, drinks cartons, structural nodes, molecular models and stability calculations, and in most of those contexts the useful number is a capacity or a material quantity rather than an abstract volume.
How to Use It
- Pick the measurement you already have from the selector. The field label changes to match.
- Type the value. When you enter a volume the unit is treated as cubed, and when you enter a surface area it is treated as squared.
- Choose your unit. Every result comes back in that unit with the correct exponent.
- Read the volume in the hero panel, then the edge, surface area, height and face area in the tiles below it.
- Check the working line to confirm the substitution matches what you meant to enter.
The Formula: How a Tetrahedron Is Calculated
For a regular tetrahedron with edge length a:
- Volume: V = a³ ÷ (6√2), equivalently a³√2 ÷ 12. With a = 6 cm: 216 ÷ 8.4853 = 25.456 cm³.
- Total surface area: A = √3 × a². Four equilateral faces, each of area (√3/4)a². For a = 6 cm: 1.7321 × 36 = 62.354 cm².
- Area of one face: (√3/4)a² = 15.588 cm².
- Vertical height: h = a × √(2/3) = 0.8165a. For a = 6 cm that is 4.899 cm.
- Insphere radius: a ÷ (2√6) = 1.2247 cm. Circumsphere radius: a√6 ÷ 4 = 3.674 cm.
Reversing each is straightforward. Edge from volume is the cube root of 6√2 times V. Edge from surface area is the square root of A divided by √3. Edge from height is h divided by 0.8165. The complete property list is set out at Wolfram MathWorld's regular tetrahedron entry.
Why the Height Is Not What People Expect
A regular tetrahedron with a 6 cm edge is only 4.899 cm tall. That is noticeably less than its edge length, and it catches people out constantly when they are checking clearance or designing packaging. The reason is that the apex does not sit above a corner or above the midpoint of an edge; it sits above the centroid of the base triangle, which is closer to the base than any vertex is.
There is a second sloping measurement that is easy to confuse with the height. The distance from the apex down the middle of a face to the midpoint of a base edge, the face median, is a√3/2, which for a 6 cm edge is 5.196 cm. That is the height of each triangular face, and it is what you use for surface area, not for volume or clearance.
The relationship between the two is a right triangle: the vertical height, the distance from the base centroid to the midpoint of a base edge, and the face median. Squaring and subtracting recovers whichever one you are missing, using the same logic as our Pythagorean theorem calculator. For the equilateral faces on their own, the triangle area calculator handles the two-dimensional part.
How Little Volume a Tetrahedron Holds
A tetrahedron is remarkably empty for its size. Compare it with a cube of the same edge: a 6 cm cube holds 216 cm³, while a 6 cm tetrahedron holds 25.456 cm³, which is under 12% of it. Even against the smallest box that fully encloses it, the tetrahedron uses only a small fraction of the space.
That ratio is the reason tetrahedral packaging is rare despite being cheap to fabricate from a folded tube: it wastes shipping volume and does not stack. The same property is an advantage structurally. A tetrahedral frame is the smallest rigid three-dimensional truss, using six members to brace four nodes, and rigidity per unit of enclosed volume is exactly what a space frame wants.
It is also worth knowing that regular tetrahedra do not tile space on their own. Unlike cubes, they cannot fill a volume without gaps, which matters in packing problems and in any attempt to build a solid from identical tetrahedral units. To convert a computed volume into other capacity units, use the volume converter.
Irregular Tetrahedra and When This Tool Does Not Apply
Every result on this page assumes a regular tetrahedron with six equal edges. Plenty of real tetrahedra are not regular. A triangular pyramid with an arbitrary base and an off-centre apex is still a tetrahedron, but none of the constants above hold for it.
For an irregular tetrahedron the universal formula still works: V = ⅓ × base area × perpendicular height. Measure the area of any one face, measure the perpendicular distance from the opposite vertex to the plane of that face, multiply, and divide by three. Any face can serve as the base, and all four choices give the same volume, which is a useful cross-check when the measurements are awkward.
If you only have the six edge lengths of an irregular tetrahedron and no heights, the volume can still be determined, but it requires the Cayley-Menger determinant rather than a simple formula, and it is beyond what a single-input calculator can do cleanly. Unit definitions used throughout this page follow NIST's guide to the SI.
Inspheres, Circumspheres and Practical Clearance
Two spheres describe the extremes of a tetrahedron. The insphere touches all four faces from the inside and has radius a ÷ (2√6). The circumsphere passes through all four vertices and has radius a√6 ÷ 4, exactly three times the insphere radius.
The circumsphere radius is the practical one for clearance questions: it tells you the smallest spherical space that a tetrahedron of a given size will pass through or occupy while rotating freely. For a 6 cm edge that is 3.674 cm, so the solid sweeps a sphere of 7.35 cm diameter when spun about its centre, well beyond its 4.899 cm standing height. The insphere matters when something has to fit inside the solid, or when you are estimating how far a coating can penetrate before the interior is filled. For the sphere arithmetic itself, the sphere volume calculator takes either radius and returns the volume and surface area.
Arb Digital designs, writes and ships free calculator pages built to rank, complete with schema markup and the technical SEO underneath them.
Browse All Free Tools Talk To Arb DigitalCommon Mistakes to Avoid
- Using the face median as the height. The face median a√3/2 is the height of a triangular face. The vertical height is a√(2/3), which is shorter.
- Applying regular-tetrahedron constants to an irregular one. If the six edges are not all equal, use one third of base area times perpendicular height instead.
- Assuming a tetrahedron holds roughly what a cube holds. At the same edge length it holds under 12% as much.
- Forgetting there are four faces, not three. The base is a face too, so total surface area is √3a², not three quarters of that.
- Taking a cube root of a capacity figure. Convert litres or gallons to a cubic unit before reverse-solving for the edge.
Related Free Tools From Arb Digital
The triangle area calculator covers the equilateral faces on their own, and the sphere volume calculator handles the inscribed and circumscribed spheres. For other solids, the cylinder volume calculator and circle calculator cover the round cases, and the volume converter moves any result between capacity units. Browse the full free online tools hub for the rest.
Frequently Asked Questions
Volume is the edge length cubed divided by six times the square root of two. A 6 cm edge gives 216 divided by 8.485, which is 25.46 cubic centimetres.
Total surface area is the square root of three times the edge squared, because there are four identical equilateral faces. A 6 cm edge gives 62.35 square centimetres, or 15.59 per face.
Its vertical height is the edge length times the square root of two thirds, about 0.8165 times the edge. A 6 cm edge stands 4.899 cm tall, noticeably shorter than the edge itself.
Because the apex sits above the centroid of the base triangle rather than above a vertex. The centroid is closer to the base plane than any corner, which pulls the overall height down.
Multiply the volume by six times the square root of two, then take the cube root. Convert any capacity figure such as litres into a cubic unit before doing this.
No. These constants assume six equal edges. For an irregular tetrahedron use one third of the base area times the perpendicular height from the opposite vertex, which works for any of the four faces.
At the same edge length, a regular tetrahedron holds under 12% of what a cube holds. A 6 cm cube contains 216 cubic centimetres while the tetrahedron contains 25.46.
Results are mathematical values rounded for display. Physical models carry material thickness and joint tolerance, so allow a margin before cutting or fabricating to these figures.
