# Calculate pyramid - volume of a pyramid, formulas, 3D interactive

## Calculate pyramid - The designations of a pyramid

### The pyramid

**Calculate pyramid **

Before we start with calculations on a pyramid, let us first define what a pyramid is.

In very general terms, a pyramid is a geometric figure consisting of a base and triangular sides that meet at a point, the top of the pyramid. A typical pyramid that we know from Egypt, for example, has a square base, called the base, and triangular sides that run from the vertices of the base to the top.

Later on, we will also consider pyramids with a triangular base and with regular polygons such as a pentagon or an octagon.

You can find more information about the pyramid calculation at Wikipedia!

## Pyramid 3D model - interactive

**Pyramid 3D Model **

Explore the pyramid in our 3D model. Use the mouse or your fingers to view the pyramid from all perspectives.

The 3D model of the pyramid was created with Plotly.com

Also visit the Plotly page of Mathefritz.

## Calculate the volume of a pyramid - The derivation

### Volume of a pyramid - The approximation with stacked cuboids.

We decompose the pyramid into a number of stacked cuboids as shown in the 3D animation next to it. The smaller the height of the cuboids, the closer we get to the actual volume of the pyramid.

Try it out by choosing the number of levels yourself!

**Derivation Calculate volume of a pyramid **

We start with our example and 5 planes. Then we want to derive the volume for any number of n planes.

First, the key figures of the pyramid with a square base:

- We call the edge length:
**a** - We call the height:
**h**

With 5 planes a cuboid has the height \(h_q=\frac{h}{5}\)

The half side length a is divided into 5 lengths of equal width.

The first (lowest) cuboid has the volume:

\(V_1 = a^2 \cdot \frac{h}{5} \)

The second cuboid has the volume:

\(V_2 = (\frac{4}{5} a)^2 \cdot \frac{h}{5} \)

The third cuboid has the volume:

\(V_3 = (\frac{3}{5} a)^2 \cdot \frac{h}{5} \)

The fourth cuboid has the volume:

\(V_4 = (\frac{2}{5} a)^2 \cdot \frac{h}{5} \)

The fifth cuboid has the volume:

\(V_5 = (\frac{1}{5} a)^2 \cdot \frac{h}{5} \)

If we summarize the elements as a sum and factor them out, we obtain as a first approximation for the volume:

\( V = a^2 \cdot \frac{1}{5} h \cdot (\frac{1}{25} + \frac{4}{25} + \frac{9}{25} + \frac{16}{25} + \frac{25}{25})=a^2 \cdot h \cdot \frac{11}{25} =0,44\cdot h\cdot a^2\)

The volume for 5 levels is larger than the actual volume, because all cuboids protrude slightly above the real pyramid. See picture.

Let's summarize the calculation with a summation formula:

We set n=5.

\( V = a^2 \cdot h \cdot \frac{1}{n^3}\cdot \sum_{i=1}^{n} i^2 \)

Let's increase the number of layers as in our 3D animation for the pyramid:

n = 50 : \(V=\frac {1717}{5000}\cdot a^2 \cdot h = 0,34 \cdot a^2 \cdot h \)

n = 1000 : \(V= 0.3338\cdot a^2 \cdot h \)

Wir sehen, dass sich das Volumen immer mehr dem Wert:

\(V=\frac {1}{3}\cdot a^2 \cdot h \) annähert.

At 1000 levels, we have more or less already reached the correct value.

Calculating the volume of a pyramid with the formula for 1000 levels with a calculator: Here a CASIO fx-991DE CW.

## Pyramid in 3D - Where to find a pyramid in buildings?

Calculate pyramid in practice:

If you look for pyramids in nature, you will find many pyramids in structures.

Pyramid-shaped structures can be found in different cultures and at different times around the world. Here are some well-known examples:

The Egyptian pyramids: The pyramids of Giza in Egypt are probably the most famous examples of pyramidal structures. The largest and most famous pyramid is the Pyramid of Khufu, a burial place of the pharaoh Khufu.

The Central American pyramids: Numerous pyramid-shaped structures are found in the Mayan, Aztec regions. Examples are the pyramids of Chichén Itzá in Mexico or the pyramids of Tikal in Guatemala.

The Pyramids of Teotihuacán: In Mexico there is the archaeological site of Teotihuacán, which contains the remains of an ancient city. There are the Pyramid of the Sun and the Pyramid of the Moon, which are among the most impressive pyramid-shaped structures of the pre-Columbian era.

These are just a few examples of pyramid-shaped structures. However, there are also modern buildings inspired by the traditional pyramid shape, such as the glass pyramid in the courtyard of the Louvre in Paris or the Luxor Hotel in Las Vegas.