Triangular pyramid volume calculator

What is a triangular pyramid?

A triangular pyramid, also known as a tetrahedron, is a three-dimensional geometric figure with a triangular base and three triangular faces converging at a singular vertex point, not on the plane of the base. The triangular pyramid is a type of polyhedron, specifically comprising four triangular faces, six edges, and four vertices.

Formula for triangular pyramid volume

The volume $V$ of a triangular pyramid can be found using various methods depending on the known parameters of the pyramid:

1. Volume based on base area and height

$V = \frac{1}{3} \times A_{\text{base}} \times H$ Where:

• $A_{\text{base}}$ is the area of the triangular base
• $H$ is the height of the pyramid from the base to the apex

2. Volume with known three sides of the base

When the three sides $a$, $b$, and $c$ of the triangular base are known, and $H$, the height of the pyramid, is provided, we compute the base area using Heron’s Formula:

1. Compute the semi-perimeter $s$: $s = \frac{a + b + c}{2}$
2. Use Heron’s Formula for base area $A_{\text{base}}$: $A_{\text{base}} = \sqrt{s(s-a)(s-b)(s-c)}$
3. Substitute $A_{\text{base}}$ into the volume formula: $V = \frac{1}{3} \times \sqrt{s(s-a)(s-b)(s-c)} \times H$

3. Volume with two sides and the included angle

When two sides $a$ and $b$ of the base and the included angle $\alpha$ are known: $A_{\text{base}} = \frac{1}{2} \times a \times b \times \sin(\alpha)$ Then use the area in the volume formula.

4. Volume with one side and two adjacent angles

When the side $b$ of the base and its two adjacent angles, $\alpha$ and $\beta$, are known, you can use the Sine Rule to find the area of the base: $A_{\text{base}} = \frac{b^2 \times \sin(\alpha) \times \sin(\beta)}{2 \times \sin(\alpha + \beta)}$ Utilize this $A_{\text{base}}$ in the volume formula.

5. Volume with known base height and side

If the base height $h_{\text{base}}$ and the side $b$ of the triangular base are given: $A_{\text{base}} = \frac{1}{2} \times b \times h_{\text{base}}$ Incorporate into the same volume equation.

Understanding correct and incorrect triangular pyramid

Regular triangular pyramid (tetrahedron)

A regular tetrahedron is a triangular pyramid where all edges are equal, and all faces are regular triangles. If the edge length is $a$, the volume is calculated using the formula: $V = \frac{\sqrt{2}}{12} \times a^3$

Note: In some sources, the term “regular triangular pyramid” refers to a pyramid with a regular triangle in the base and equal side edges, but not necessarily with equal base edges and side edges. In this case, the volume formula will depend on the height of the pyramid and the area of the base.

Irregular (or incorrect) triangular pyramid

An irregular triangular pyramid has sides of varying lengths and does not exhibit uniformity in angles or edge measurements. The volume calculation relies on known measurements such as different side lengths and corresponding heights.

If the coordinates of the vertices of a triangular pyramid are known

If the coordinates of the vertices of a triangular pyramid are known, you can use an alternative method by using the tetrahedron volume calculator. By determining the coordinates of the vertices in three-dimensional space, it becomes possible to calculate using vector mathematics. This tool is useful when the pyramid does not match the clear measurements of height and base area.

Examples of volume calculation

Example 1: Known base area and height

Let’s calculate the volume for a triangular base area of $6 \, \text{cm}^2$ and pyramid height $9 \, \text{cm}$. $V = \frac{1}{3} \times 6 \times 9 = 18 \, \text{cm}^3$

Example 2: Volume with three known sides

Given side lengths $a = 3 \, \text{cm}$, $b = 4 \, \text{cm}$, $c = 5 \, \text{cm}$, and pyramid height $10 \, \text{cm}$:

1. Calculate semi-perimeter $s = \frac{3 + 4 + 5}{2} = 6$
2. Base area $A_{\text{base}} = \sqrt{6(6-3)(6-4)(6-5)} = \sqrt{6 \times 3 \times 2 \times 1} = \sqrt{36} = 6 \, \text{cm}^2$
3. Volume $V = \frac{1}{3} \times 6 \times 10 = 20 \, \text{cm}^3$

Example 3: Known two sides and included angle

For a triangular base with $a = 5 \, \text{cm}$, $b = 6 \, \text{cm}$, angle $\theta = 60^\circ$, and pyramid height $8 \, \text{cm}$:

1. Base area $A_{\text{base}} = \frac{1}{2} \times 5 \times 6 \times \sin(60^\circ) = \frac{15\sqrt{3}}{2} \, \text{cm}^2$
2. Volume $V = \frac{1}{3} \times \frac{15\sqrt{3}}{2} \times 8 = 20\sqrt{3} \, \text{cm}^3$

Frequently Asked Questions

What is the volume of a triangular pyramid if the base area and height are known?

The volume of a triangular pyramid is one-third of the product of the base area and height.

How many triangular faces does a pyramid have?

A triangular pyramid consists of four triangular faces: the base and three side faces.

Can a triangular pyramid have a horizontal base?

Yes, the base of a triangular pyramid is often horizontal in conventional illustrations, although in reality it can be oriented in any position relative to another reference plane.

What is the difference between a triangular pyramid and a tetrahedron?

A tetrahedron is a polyhedron with four triangular faces, which can be regular (all edges and angles are equal) or irregular. A triangular pyramid is a special case of a tetrahedron, where one face is the base, and the other three are side faces. Therefore, all triangular pyramids are tetrahedrons, but not all tetrahedrons necessarily have a designated base.

What is the volume of a regular triangular pyramid if the base edge length is 3?

For a regular tetrahedron or a regular triangular pyramid (where all edges are equal), the volume is calculated using the formula: $V = \frac{\sqrt{2}}{12} \times a^3$ Substituting $a = 3$: $V = \frac{\sqrt{2}}{12} \times 3^3 = \frac{\sqrt{2}}{12} \times 27 = \frac{27\sqrt{2}}{12} = \frac{9\sqrt{2}}{4}$

The volume of a regular triangular pyramid is 3.182 cm³.

Note: If the term “regular triangular pyramid” refers to a pyramid with a regular triangle in the base and equal side edges, but not necessarily with equal base edges and side edges, then the volume formula will depend on the height of the pyramid and the area of the base. In this case, the volume formula will depend on the height of the pyramid and the area of the base.