Question

Compute the volume of a regular triangular pyramid whose lateral edge is \(l\) and the side of the base is \(a\).

Short answer

The volume is \( V = \frac{\sqrt{3}}{12} a^2 \sqrt{l^2 - \frac{a^2}{4}} \).

Step-by-step solution

Understand the Shape

A regular triangular pyramid (or tetrahedron) is a pyramid with a triangular base where all edges are equal. It has four triangular faces including the base, and six edges of the same length.

Formula for the Volume of a Pyramid

The volume of a pyramid is generally given by the formula: \[ V = \frac{1}{3} imes \text{Base Area} imes \text{Height} \] To compute the volume, we need both the area of the base and the height of the pyramid.

Calculate Base Area

The base of the pyramid is an equilateral triangle with side length \(a\). The area of an equilateral triangle is:\[ \text{Area} = \frac{\sqrt{3}}{4}a^2 \] This will give us the area of the base of the pyramid.

Find the Height of the Tetrahedron

Using the Pythagorean theorem in the face of the tetrahedron, the height \(h\) of the pyramid can be calculated. For a regular tetrahedron, \[ h = \sqrt{l^2 - \left(\frac{a}{2}\right)^2} \] where \(l\) is the lateral edge and \(\frac{a}{2}\) is half of the side of the equilateral base triangle.

Substitute in the Volume Formula

Substitute the area and height into the volume formula: \[ V = \frac{1}{3} \times \frac{\sqrt{3}}{4} a^2 \times \sqrt{l^2 - \left(\frac{a}{2}\right)^2} \] This expression will give us the volume of the pyramid in terms of \(a\) and \(l\).

Key concepts

Regular triangular pyramid

A regular triangular pyramid, also known as a regular tetrahedron, is a fascinating geometric shape. This pyramid has a triangular base and is characterized by having all edges of equal length. In simple terms, this means that not only are its three sides on the base equal, but the sides connecting the base to the apex are also of the same length.

The regular tetrahedron features:

• Four surfaces, with each surface being an equilateral triangle.
• Six equal-length edges.
• Four vertices, where the sides and base meet.

Understanding this symmetry and uniformity is crucial for calculating its properties, like volume, because these equilateral characteristics simplify computations and lead to unique simplifications not possible with other types of pyramids.

Equilateral triangle area

The base of our regular triangular pyramid is an equilateral triangle. This means each side of the base has the same length, which is denoted by the variable \(a\).

To find the area of an equilateral triangle, we use a specific formula: \[ \text{Area} = \frac{\sqrt{3}}{4} a^2 \]

This formula arises from the properties of equilateral triangles, where all angles are 60 degrees. Due to these constant angles, using basic trigonometric relationships allows us to express the area using only the side length \(a\). Knowing how to calculate this area is essential for applications beyond just pyramids, as equilateral triangles often appear in a variety of geometric contexts.

Tetrahedron height calculation

To calculate the volume of our regular triangular pyramid, we need the height. The height in this context is not the distance from the base to the apex along the pyramid's side, but rather perpendicular to the base. Thus, the height \(h\) can be found using the formula: \[ h = \sqrt{l^2 - \left(\frac{a}{2}\right)^2} \]

This formula utilizes the Pythagorean theorem. Here's how: imagine a slice through the pyramid that includes one vertex of the base, the apex, and the midpoint of the base, forming a right triangle. The lateral edge \(l\) and half the base \(\frac{a}{2}\) are the triangle's legs, and \(h\) is the altitude from the apex perpendicular to the base, allowing us to solve for \(h\).

By accurately determining this height, we effectively unlock the means to compute the pyramid's volume, since it's a pivotal part of the pyramid's volume formula.