Triangle Area Calculator
Calculate triangle area, perimeter, angles, altitudes, circumradius, inradius, and classification. Supports base-height, Heron's formula, and SAS methods with unit selection and a complete 15-prop..
Triangular Pyramid Volume Calculator
Calculate the volume and surface area of a triangular pyramid (tetrahedron). Supports regular and irregular modes with base triangle dimensions or full vertex coordinates.
Volume⧉
25.455844
V = ⅓ × base area × height
Total Surface Area⧉
62.353829
Sum of all four triangular face areas
Base Area⧉
15.588457
Area of the base triangle
Lateral Surface Area⧉
46.765372
Sum of the three lateral face areas
Height⧉
4.898979
Perpendicular distance from apex to base
Base Perimeter⧉
18.000000
Sum of the base triangle's sides
Face Areas
Base: 15.5885
Lateral 1: 15.5885
Lateral 2: 15.5885
Lateral 3: 15.5885
Volume / Surface Area Ratio
V/SA = 0.408248
ratio
Pyramid Type Reference
| Type | Base | Volume | Faces |
|---|---|---|---|
| Regular Tetrahedron | Equilateral triangle | a³√2 / 12 | 4 congruent equilateral |
| Right Triangular Pyramid | Right triangle | ⅓ × ½ab × h | 1 right + 3 lateral |
| Isosceles Triangular Pyramid | Isosceles triangle | ⅓Bh | 1 iso + 3 lateral |
| General Tetrahedron | Any triangle | |det| / 6 | 4 different triangles |
Key Formulas
| Formula | Expression |
|---|---|
| General Volume | V = ⅓ × Base Area × Height |
| Regular Tetrahedron Volume | V = a³√2 / 12 |
| Regular Tetrahedron SA | S = a²√3 |
| Regular Tetrahedron Height | h = a√(2/3) |
| Heron's Base Area | B = √(s(s−a)(s−b)(s−c)) |
| Vertex Volume | V = |det[AB, AC, AD]| / 6 |
| Euler's Formula | V − E + F = 2 → 4 − 6 + 4 = 2 |
Planning notes, formulas, and examples
About the Triangular Pyramid Volume Calculator
A triangular pyramid — also known as a tetrahedron — is a three-dimensional solid with four triangular faces, six edges, and four vertices. It is the simplest polyhedron and appears throughout mathematics, chemistry (molecular geometry), architecture, and engineering.
The volume of any triangular pyramid is one-third the base area times the perpendicular height: V = ⅓ × B × h. For a regular tetrahedron where all four faces are equilateral triangles with edge length a, the volume simplifies to V = a³√2 / 12 and the surface area to S = a²√3.
This calculator operates in three modes. **Regular mode** computes all properties from a single edge length. **Base + height mode** lets you specify the three sides of the triangular base plus the pyramid's perpendicular height. **Vertex mode** accepts the full 3D coordinates of all four vertices and computes volume using the scalar triple product, along with all face areas and the total surface area.
Eight presets provide instant examples including regular tetrahedra of various sizes, classic right-angled pyramids, and elongated forms. The results include volume, lateral and total surface area, base area, base perimeter, and a reference table comparing pyramid types. Visual bars show proportional face areas. Whether you're solving a geometry homework problem, designing a structural element, or exploring solid geometry, this calculator delivers detailed, verified results.
When This Page Helps
A triangular pyramid can be described in several fundamentally different ways: by one edge for a regular tetrahedron, by a base triangle plus perpendicular height, or by four vertices in 3D space. Switching among those models by hand is slow and easy to get wrong. This calculator keeps the formulas aligned with the chosen mode and reports consistent geometric outputs such as face areas, surface area, and height, making it useful for solid-geometry problems, modeling, and coordinate-based verification.
How to Use the Inputs
- Select a mode: Regular Tetrahedron, Base + Height, or Vertex Coordinates.
- For regular mode, enter the edge length.
- For base + height mode, enter the three sides of the base triangle and the perpendicular height.
- For vertex mode, enter the (x, y, z) coordinates of all four vertices.
- Click a preset to quickly load a common configuration.
- Review volume, surface area, and other properties in the output cards.
Formula used
General: V = ⅓ × Base Area × Height.
Regular tetrahedron (edge a): V = a³√2 / 12, Surface Area = a²√3.
Base area (Heron): s = (a+b+c)/2, B = √(s(s−a)(s−b)(s−c)).
Vertex mode: V = |det[AB, AC, AD]| / 6.Example Calculation
Result: For edge length 6, the regular tetrahedron volume is about 25.46 and total surface area is about 62.35.
Regular tetrahedron with edge a = 6: V = 6³×√2/12 = 216×1.4142/12 ≈ 25.46. Surface area = 6²×√3 = 36×1.7321 ≈ 62.35. Each face area = 9√3 ≈ 15.59.
Tips & Best Practices
- The regular tetrahedron has the maximum volume-to-surface-area ratio among all tetrahedra with a given edge length.
- For the base + height mode, height must be perpendicular to the base plane.
- In vertex mode, the volume is always positive (absolute value of the determinant).
- A triangular pyramid has exactly 4 faces, 6 edges, and 4 vertices.
- Check that the three base sides satisfy the triangle inequality before computing.
One Solid, Multiple Input Models
A triangular pyramid is simple in face count but flexible in how it is specified. In some problems you get a regular tetrahedron with all edges equal, in others you get a base triangle and a perpendicular height, and in coordinate geometry you may get four full 3D points. These descriptions look different, but they should agree on core outputs such as volume and total surface area when they describe the same solid.
Why the Base Area Matters
For any pyramid, the volume is controlled by the base area and the perpendicular height through the formula $V = frac13Bh$. That means two pyramids with very different slanted faces can share the same volume if those two quantities match. The calculator makes that relationship explicit by reporting the base area separately from the lateral area, which is useful when comparing shape efficiency or checking homework solutions.
Regular vs Irregular Tetrahedra
The regular tetrahedron is highly symmetric, with four congruent equilateral faces and clean closed-form formulas. General tetrahedra are less tidy, so coordinate methods and determinant-based volume formulas become more important. Having both options in one tool helps students see how special symmetry simplifies geometry while also giving them a way to handle the broader class of irregular triangular pyramids.
Sources & Methodology
Last updated:
Frequently Asked Questions
They are the same shape. "Triangular pyramid" emphasizes it has a triangular base, while "tetrahedron" (Greek for "four faces") describes it by its four triangular faces.
A regular tetrahedron has all four faces as congruent equilateral triangles, all six edges of equal length, and all four solid angles equal. It is one of the five Platonic solids.
If you know the volume V and base area B, then h = 3V / B. For a regular tetrahedron with edge a, h = a√(2/3).
Yes. Form three edge vectors from one vertex to the other three, then V = |det[v₁, v₂, v₃]| / 6. This calculator's vertex mode does exactly this.
The calculator is unit-agnostic. If you enter side lengths in centimeters, the volume will be in cubic centimeters and area in square centimeters.
Four. Every face is a triangle. The base is one triangular face, and the three lateral faces connect the base edges to the apex.
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