Volume - Square Pyramid And Triangular Pyramid

🔑 Key Concepts

• The volume of any pyramid can be found using \(\displaystyle V=\frac{1}{3}Bh\).
• The capital letter \(B\) represents the area of the pyramid’s base, not a single side length.
• For a triangular pyramid, first find the area of the triangular base using \(\displaystyle B=\frac{1}{2}ba\).
• For a square pyramid, find the base area using \(B=s^2\).
• The pyramid height must be perpendicular to the base.
• The factor \(\displaystyle \frac{1}{3}\) means a pyramid has one-third the volume of a prism with the same base area and height.

📝 Notation Used in This Video

In the triangular-pyramid example, the video uses capital \(H\) for the perpendicular height of the pyramid and lowercase \(h\) for the height of the triangular base. This notation is valid as long as each variable is clearly defined, but using \(H\) and \(h\) in the same problem can be visually confusing.

In other textbooks or lessons, the altitude of the triangular base may instead be labeled \(a\), while \(h\) is reserved for the vertical height of the pyramid. In the worked example below, \(a\) is used for the triangle’s altitude and \(H\) is used for the pyramid’s perpendicular height.

✏️ Worked Examples

Comparing Two Pyramid Bases

Both solids use \(\displaystyle V=\frac{1}{3}BH\), but the value of \(B\) is found differently because the bases have different shapes.

Example 1: Triangular Pyramid

A triangular pyramid has a triangular base with a base length of \(8\) feet and a triangle altitude of \(10\) feet. The perpendicular height of the pyramid is \(6\) feet. Find its volume.

Triangular base:
\(b=8\text{ ft}\)
\(a=10\text{ ft}\)

Pyramid height:
\(H=6\text{ ft}\)

First, find the area of the triangular base:

\(\displaystyle B=\frac{1}{2}ba\)

\(\displaystyle B=\frac{1}{2}(8)(10)\)

\(B=40\text{ ft}^2\)

Now substitute the base area and pyramid height into the pyramid volume formula:

\(\displaystyle V=\frac{1}{3}BH\)

\(\displaystyle V=\frac{1}{3}(40)(6)\)

\(\displaystyle V=\frac{240}{3}\)

\(V=80\)

Triangular pyramid volume:
\(\boxed{80\text{ ft}^3}\)

Example 2: Square Pyramid

A square pyramid has a base with side lengths of \(12\) feet and a perpendicular height of \(9\) feet. Find its volume.

Square base:
\(s=12\text{ ft}\)

Pyramid height:
\(H=9\text{ ft}\)

First, find the area of the square base:

\(B=s^2\)

\(B=(12)^2\)

\(B=144\text{ ft}^2\)

Now substitute the base area and pyramid height:

\(\displaystyle V=\frac{1}{3}BH\)

\(\displaystyle V=\frac{1}{3}(144)(9)\)

\(\displaystyle V=\frac{1296}{3}\)

\(V=432\)

Square pyramid volume:
\(\boxed{432\text{ ft}^3}\)

Same Volume Formula, Different Base-Area Formula

Both pyramids use the general formula:

\(\displaystyle V=\frac{1}{3}BH\)

Triangular pyramid

\(\displaystyle B=\frac{1}{2}ba\)

\(\displaystyle V=\frac{1}{3}\left(\frac{1}{2}ba\right)H\)

\(\displaystyle \boxed{V=\frac{1}{6}baH}\)

Square pyramid

\(B=s^2\)

\(\displaystyle V=\frac{1}{3}(s^2)H\)

\(\displaystyle \boxed{V=\frac{1}{3}s^2H}\)

The overall pyramid formula does not change. What changes is the method used to calculate \(B\), because \(B\) depends on the shape of the base.

The Triangular Pyramid Uses Two Different Heights

The triangular-pyramid example includes two measurements that may both be described as heights:

• The triangle altitude, labeled \(a=10\) feet here, is used to find the area of the triangular base.
• The pyramid height, labeled \(H=6\) feet, is the perpendicular distance from the pyramid’s vertex to the plane of the base.

\(\displaystyle V= \frac{1}{3} \left( \frac{1}{2} \cdot \text{triangle base} \cdot \text{triangle altitude} \right) \cdot \text{pyramid height} \)

These measurements perform different jobs in the calculation. The triangle altitude helps determine \(B\), while the pyramid height is multiplied by \(B\) to determine the three-dimensional volume.

🧠 Math Vocabulary

• Pyramid: A three-dimensional solid with one polygonal base and triangular faces that meet at a common vertex.
• Triangular pyramid: A pyramid with a triangular base.
• Square pyramid: A pyramid with a square base.
• Base area: The area of the polygon forming the base of the pyramid, represented by \(B\).
• Perpendicular height: The shortest distance from the pyramid’s vertex to the plane containing its base.
• Altitude of a triangle: A perpendicular segment from a vertex of a triangle to the opposite side or its extension.
• Vertex: The point where the triangular lateral faces of a pyramid meet.
• Slant height: The distance measured along a lateral face from the vertex toward the base; it is not used in the pyramid volume formula.
• Volume: The amount of three-dimensional space contained inside a solid.
• Cubic units: Units used to measure volume, such as cubic feet or \(\text{ft}^3\).

💡 Main Idea

The formula \(\displaystyle V=\frac{1}{3}Bh\) can be used for every pyramid. The value of \(B\), however, depends on the shape of the base. A triangular base requires the triangle-area formula, while a square base requires the square-area formula. After finding \(B\), multiply by the perpendicular height of the pyramid and then multiply by \(\displaystyle \frac{1}{3}\).

📚 What You Should Already Know

Students should know how to find the area of a triangle and square, distinguish perpendicular height from slant height, substitute measurements into formulas, multiply fractions and whole numbers, and label volume answers using cubic units.

🚀 What Comes Next

Students can extend this reasoning to pyramids with rectangular or other polygonal bases, solve for missing pyramid dimensions, compare pyramids and prisms with matching bases and heights, and apply the formula to composite three-dimensional figures.

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