Tetrahedron Volume Calculator
Solve tetrahedron volume problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
Formula
V = a^3 / (6 * sqrt(2))
For a regular tetrahedron with edge length a, the volume equals a cubed divided by six times the square root of two. For a general tetrahedron, use V = (1/3) * Base Area * Height. The surface area of a regular tetrahedron is sqrt(3) * a squared.
Worked Examples
Example 1: Regular Tetrahedron with Edge Length 10
Problem: Find the volume, surface area, height, inradius, and circumradius of a regular tetrahedron with edge length 10 units.
Solution: Volume = 10^3 / (6 * sqrt(2)) = 1000 / 8.485 = 117.8511\nSurface Area = sqrt(3) * 10^2 = 173.2051\nHeight = 10 * sqrt(2/3) = 8.1650\nInradius = 10 / (2 * sqrt(6)) = 2.0412\nCircumradius = 10 * sqrt(6) / 4 = 6.1237
Result: Volume: 117.8511 cubic units | Surface Area: 173.2051 sq units | Height: 8.1650 units
Example 2: General Tetrahedron from Base and Height
Problem: A tetrahedron has a triangular base with area 50 square cm and a height of 12 cm. Find its volume.
Solution: Using the general pyramid formula:\nVolume = (1/3) * Base Area * Height\nVolume = (1/3) * 50 * 12\nVolume = (1/3) * 600\nVolume = 200 cubic cm
Result: Volume: 200.0000 cubic cm
Frequently Asked Questions
What is a tetrahedron and how is its volume calculated?
A tetrahedron is a three-dimensional solid with four triangular faces, six edges, and four vertices. It is the simplest of all the Platonic solids and serves as a fundamental shape in geometry, chemistry, and structural engineering. For a regular tetrahedron where all edges have equal length a, the volume formula is V = a cubed divided by six times the square root of two. This formula is derived from the general pyramid volume formula V = (1/3) times base area times height, combined with the specific geometric properties of equilateral triangular faces.
What is the difference between a regular and irregular tetrahedron?
A regular tetrahedron has all four faces as congruent equilateral triangles, meaning all six edges are the same length and all four vertices are equidistant from each other. An irregular tetrahedron has faces that are not all identical, so edge lengths can vary and the faces can be different types of triangles including scalene or isosceles. The volume calculation for a regular tetrahedron only requires one measurement (edge length), while an irregular tetrahedron typically requires knowing the base area and height, or the coordinates of all four vertices to compute the volume using the scalar triple product.
How do you find the height of a regular tetrahedron?
The height of a regular tetrahedron is the perpendicular distance from one vertex to the opposite face. For an edge length of a, the height equals a times the square root of two-thirds, which simplifies to a times the square root of six divided by three. This can be derived by placing the tetrahedron with one face on a flat surface and computing the vertical distance to the apex. For a tetrahedron with edge length 10, the height is approximately 8.165 units. Understanding the height is crucial because it connects the regular tetrahedron formula to the general pyramid formula V = (1/3) times base times height.
What is the inradius and circumradius of a regular tetrahedron?
The inradius is the radius of the largest sphere that fits inside the tetrahedron, tangent to all four faces. For a regular tetrahedron with edge length a, the inradius equals a divided by two times the square root of six, or equivalently a times the square root of six divided by twelve. The circumradius is the radius of the smallest sphere that passes through all four vertices, and equals a times the square root of six divided by four. The ratio of circumradius to inradius for a regular tetrahedron is always exactly 3:1, which is a distinctive geometric property of this Platonic solid.
How is the surface area of a regular tetrahedron calculated?
The total surface area of a regular tetrahedron equals the square root of three times the edge length squared, because it has four equilateral triangular faces each with area equal to the square root of three divided by four times a squared. For an edge length of 10, each face has an area of approximately 43.30 square units, giving a total surface area of about 173.21 square units. The surface-area-to-volume ratio decreases as the tetrahedron gets larger, which has practical implications in fields like chemistry where molecular surface interactions depend on this ratio.
How does tetrahedron volume scale with edge length?
Volume scales with the cube of the edge length, meaning that doubling the edge length increases the volume by a factor of eight (two cubed). For example, a regular tetrahedron with edge length 5 has a volume of approximately 14.73 cubic units, while one with edge length 10 has a volume of about 117.85 cubic units, exactly eight times larger. This cubic scaling relationship is fundamental to all three-dimensional geometry and explains why small changes in linear dimensions produce dramatic changes in volume. The surface area, by contrast, scales with the square of the edge length.