Polyhedron volume calculator

What is a polyhedron volume calculator?

The polyhedron volume calculator allows you to calculate the volume of a figure based on two different criteria:

1. The volume of a polyhedron whose vertices are points of a rectangular parallelepiped;
2. A composite figure made of two connected rectangular parallelepipeds; it calculates the total volume of the 3D shape formed by two rectangular prisms.

Formulas

Formula for a polyhedron inscribed in a parallelepiped

First, determine the type of polyhedron inscribed in the parallelepiped:

1. If the polyhedron is a pyramid (e.g., with a base on one face of the parallelepiped and a vertex at the opposite corner), the volume is calculated as:

where $A$ is the base area, and $h$ is the height (distance from the vertex to the base).

2. If the polyhedron is a prism (e.g., between two parallel faces), the volume is:

where $A$ is the base area, and $h$ is the prism height.

Formula for a composite polyhedron

The total volume $V$ of a composite polyhedron is calculated as:

Where:

• $L_1$ and $L_2$: lengths (long sides) of the first and second parallelepipeds.
• $W_1$ and $W_2$: widths (short sides) of the two parallelepipeds.
• $H$: common height.

Step-by-step examples

Example 1: Volume of a Polyhedron Based on the Vertices of a Parallelepiped

Find the volume of a polyhedron whose vertices are the points $A, D, A_1, B, C, B_1$ of a rectangular parallelepiped $ABCDA_1B_1C_1D_1$, where $AB = 3$, $AD = 4$, $AA_1 = 5$, where $ABCD$ is the base of the parallelepiped and $A_1B_1C_1D_1$ is the top base of the parallelepiped above the corresponding points of the base.

1. Determine that the figure inscribed in the parallelepiped is a triangular prism.

2. Calculate the area of the base of the prism:

$A = \frac{1}{2} \times AA_1 \times AD = \frac{1}{2} \times 4 \times 5 = 10$

3. Find the volume of the prism:

$V = A \times h = 10 \times 3 = 30$ In this example, the height of the prism equals the length of the side $AB$.

Note: In the examined example, the prism occupies exactly 1/2 of the volume of the parallelepiped, and the calculated result can be verified by finding the volume of the parallelepiped: $V = 3 \times 4 \times 5 = 60$, half of which is 30.

Example 2: Volume of an L-shaped table

A table has the parameters:

• Main part: $L_1 = 1.8\ \text{m}$, $W_1 = 0.7\ \text{m}$
• Extension: $L_2 = 1.2\ \text{m}$, $W_2 = 0.6\ \text{m}$
• Height $H = 0.75\ \text{m}$

Calculation:

Historical background

The study of polyhedrons began in Ancient Greece, where Euclid and Archimedes explored their properties. The term “polyhedron” derives from the Greek words poly (many) and hedra (face). Composite polyhedrons, such as connected prisms, gained importance during the Renaissance for analyzing complex architectural elements like arched vaults and buttresses.

Applications

1. Architecture: Calculating materials for multi-level structures.
2. Logistics: Designing containers with multiple compartments.
3. Manufacturing: Estimating space for equipment with complex shapes.

Notes

• All measurements must be in the same unit system (meters, feet, etc.).
• The formula for composite figures assumes a common height. If heights differ, calculate volumes separately and sum them:

• This calculator works only for rectangular parallelepipeds. For complex shapes, use our Volume Calculator.
• For polyhedrons inscribed in parallelepipeds, the calculator supports figures with 4–6 specific vertices if the parallelepiped’s dimensions are known.

FAQs

How to calculate volume if the prism heights differ?

For different heights $H_1$ and $H_2$, calculate volumes separately and add them:

Example: $L_1 = 4\ \text{m}$, $W_1 = 2\ \text{m}$, $H_1 = 3\ \text{m}$; $L_2 = 3\ \text{m}$, $W_2 = 1\ \text{m}$, $H_2 = 2\ \text{m}$:

Find the volume of the polyhedron whose vertices are the points $A, B, C, B_1$ of the rectangular parallelepiped $ABCDA_1B_1C_1D_1$, with $AB = 3$, $AD = 3$, $AA_1 = 4$.

In this case, we assume that $ABCD$ is the base of the parallelepiped, and $A_1B_1C_1D_1$ is the top base of the parallelepiped over corresponding points of the base.

Solution Steps:

1. Determine that the figure inscribed in the parallelepiped is a triangular pyramid with the following known values: AB = 3, BC = 3 (as a side parallel to AD), and the height BB1 = 4 (as a side parallel to AA1).

2. Calculate the area of the base of the pyramid:

$A = \frac{1}{2} \times AB \times BC = \frac{1}{2} \times 3 \times 3 = 4.5$

3. Find the volume of the pyramid:

$V = \frac{1}{3} \times A \times h = \frac{1}{3} \times 4.5 \times 4 = 6$

The volume of the polyhedron with vertices $A, B, C, B_1$ is 6.

How to use the calculator?

1. Select the polyhedron type: “Polyhedron inscribed in a parallelepiped” or “Composite polyhedron”.
2. Choose the number of vertices.
3. Enter the parallelepiped’s length, width, and height.
4. The calculator will automatically compute the volume.

Were composite polyhedrons used in ancient architecture?

Yes. For example, the Colosseum’s foundation in Rome combined trapezoidal and rectangular blocks to distribute load on uneven terrain.