Problem 52 - Calculus: Early Transcendentals

Question

Find the volume of the described solid \( S \). A pyramid with height \( h \) and base an equilateral triangle with side \( a \) (a tetrahedron).

Step-by-Step Solution

Answer

The volume of the tetrahedron is \( \frac{\sqrt{3}}{12} a^2 h \).

Step 1: Understand the Problem

We need to find the volume of a pyramid with an equilateral triangular base with side length \( a \) and height \( h \). This shape is also known as a tetrahedron.

Step 2: Identify the Formula for Volume of a Pyramid

The volume \( V \) of any pyramid is calculated by the formula: \[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]

Step 3: Calculate the Area of the Equilateral Triangle

The formula for the area of an equilateral triangle with side length \( a \) is: \[ A = \frac{\sqrt{3}}{4} a^2 \]. Substitute this formula into the pyramid's volume formula as the base area.

Step 4: Substitute the Base Area and Height into the Volume Formula

Using the formula for the pyramid's volume and the area of the equilateral triangle, we substitute to get: \[ V = \frac{1}{3} \times \frac{\sqrt{3}}{4} a^2 \times h \]

Step 5: Simplify the Expression for the Volume

Simplify the expression obtained in Step 4: \[ V = \frac{\sqrt{3}}{12} a^2 h \]. This is the volume of the tetrahedron.

Key Concepts

Volume Formula

To find the volume of a pyramid, you can use a simple yet powerful formula. This formula helps us understand how much space is inside a pyramid. The volume of any pyramid is calculated using the expression: \[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]

• Base Area: This is the area of the shape that forms the pyramid's base. For different types of bases, you'll have different formulas to calculate this area.
• Height: This is the perpendicular distance from the base to the topmost point of the pyramid, known as the apex.

This formula effectively means that the volume of a pyramid is a third of the volume of a prism with the same base and height.
Understanding this concept helps when working with shapes like tetrahedrons, which are specific types of pyramids with triangular bases. Once we've figured out the base area and know the height, finding the volume becomes straightforward by plugging values into the formula.

Pyramid with Triangular Base

A pyramid with a triangular base is a fascinating geometric shape. Specifically, when talking about a tetrahedron, we deal with a pyramid whose base is an equilateral triangle. Such a pyramid is very symmetrical and can be seen as a polyhedron with four triangular faces.
Here's what makes these pyramids interesting:

• They have 4 faces in total, with 3 triangular faces converging to a single point called the vertex or apex above the base.
• The base itself is formed by an equilateral triangle, which means all sides of this triangle are of equal length.

For any pyramid's volume, knowing the base's area is crucial. In our case with a triangular base, the calculations involve determining the area of this triangle before applying the formula. This seamless integration of the base into the pyramid's architecture is what makes geometrical calculations both ingenious and satisfying.

Equilateral Triangle

Equilateral triangles are unique and harmoniously balanced, possessing some easy-to-use geometric properties. Every side of an equilateral triangle is equal in length, making all three sides harmonious and congruent.
To find the area of such a triangle with side length \( a \), you can use this formula:\[ A = \frac{\sqrt{3}}{4} a^2 \]

• This formula uses the square of the side length, ensuring all properties of the triangle are captured geometrically.
• The \( \sqrt{3} \) factor comes from using height derived from dividing the triangle into two right triangles.

This formula is fundamental for proceeding to calculate volumes when the equilateral triangle serves as a base in a tetrahedron. The regularity of the equilateral triangle makes calculations predictable and straightforward, allowing for seamless substitution into the pyramid's volume formula.