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Subsection 2.8.4 Volume of Pyramids

Learner Experience 2.8.25.

"Pyramid City"
  • Materials
    🔹 Pictures of famous pyramids (e.g., Egyptian Pyramids, Mayan Pyramids)
    🔹 Measuring tape or rulers (for estimating dimensions)
    🔹 A small model or LEGO pyramid

Key Takeaway 2.8.26.

What is a Pyramid? A pyramid is a three-dimensional solid with a polygonal base and triangular faces that meet at a single point called the apex.
Formula:
Volume of a Pyramid \(= \frac{1}{3} \times\) Base Area \(\times h\)
where:
  1. Base Area is the area of the polygonal base
  2. \(h\) is the height of the pyramid (perpendicular distance from base to apex)
Why \(\frac{1}{3}\) ?
Three pyramids with the same base and height fit exactly inside a prism with the same base and height. Therefore, the volume of a pyramid is one-third the volume of a prism.
Types of Pyramids Based on Base Shape:
  1. Square pyramid: Base is a square.
  2. Rectangular pyramid: Base is a rectangle.
  3. Triangular pyramid: Base is a triangle.
  4. Pentagonal pyramid: Base is a pentagon.
Scaffolding Strategies to Address Misconceptions:
Misconception Clarification
A pyramid is the same as a prism No, a pyramid has triangular faces that meet at an apex, while a prism has two parallel bases
The volume formula is Base Area \(\times\) height No, you must multiply by \(\frac{1}{3}\) The formula is \(\frac{1}{3} \times\) Base Area \(\times\) height.
The height is the slant height No, the height is the perpendicular distance from the base to the apex, not the slant height along the face
All pyramids have square bases No, pyramids can have any polygonal base (triangle, rectangle, pentagon, etc.)
A pyramid has a polygonal base and triangular faces that meet at the apex.

Example 2.8.27.

Find the volume of a square pyramid with a height of 6 cm and a side length of 10cm.
Solution.
Step 1: Select the correct formula and substitute the given values.
We are given b = 10 and H = 6, therefore
\begin{align*} V = \amp \frac{1}{3} \times \text{base Area} \\ \text{Base Area} =\amp (10 \text{cm} \times 10 \text{cm}) \\ = \amp \frac {1}{3} \times (10 \times 10) \text{cm}^2 \times 6\text{cm} \\ = \amp 100 \text{cm}^2 \times 2 \text{cm} \\ = \amp 200 \text{cm}^3 \end{align*}
The volume of the square pyramid is \(200 \text{cm}^3\text{.}\)

Example 2.8.28.

A square pyramid has a base of 6 cm × 6 cm and a height of 9 cm. Find it’s volume.
Solution.
\begin{align*} V = \amp \frac{1}{3} \times \text{base Area} \times h\\ \text{Base Area} =\amp (6 \text{cm} \times 6 \, \text{cm}) \\ = \amp \frac {1}{3} \times (6 \times 6) \, \text{cm} \times 9 \, \text{cm}\\ = \amp \frac{1}{3} \times 36\, \text{cm}^2 \times 9 \, \text{cm} \\ = \amp 108 \, \text{cm}^3 \end{align*}

Example 2.8.29.

A triangular pyramid has a base of 5 cm × 8 cm and a height of 10 cm.
Solution.
\begin{align*} V = \amp \frac{1}{3} \times \text{base Area} \times h\\ \text{Base Area} = \amp(\frac {1}{2} \times 5\, \text{cm}\times 8 \,\text{cm}) \\ = \amp \frac {1}{3} \times ( \frac {1}{2} \times 5 \,\text{cm} \times 8 \, \text{cm}) \times 10\, \text{cm} \\ = \amp \frac{1}{3} \times 20 \,\text{cm}^2 \times 10 \,\text{cm} \\ = \amp 66.67\, \text{cm}^3 \end{align*}

Example 2.8.30.

A pyramid has a rectangular base of 4 m by 6 m and a height of 12 m.
Solution.
\begin{align*} V = \amp \frac{1}{3} \times \text{base Area} \times h\\ \text{Base Area} = \amp( \times 4\, \text{m}\times 6\,\text{m}) \\ = \amp \frac {1}{3} \times\times( 4 \,\text{m} \times 6 \,\text{m}) \times 12 \\ = \amp \frac{1}{3} \times 24\, \text{m}^2 \times 12 \,\text{m} \\ = \amp 96 \,\text{m}^3 \end{align*}

Checkpoint 2.8.31.

Checkpoint 2.8.32.

Exercises Exercises

1.

A pyramid has a square base with a side length of \(6\) cm. The height of the pyramid is \(9\) cm.
Answer.
\(108 \;\text{cm}^3\)

2.

A pyramid-shaped tent has a rectangular base of 8 m by 6 m and a height of 5 m. Find the volume of air inside the tent.
Answer.
\(80 \;\text{m}^3\)

3.

A pyramid has a square base with each side measuring 10 cm. The height of the pyramid is 15 cm.
Answer.
\(500 \;\text{cm}^3\)

4.

A pyramid has a triangular base where the base of the triangle is 8 cm and the height of the triangle is 6 cm. The height of the pyramid is 10 cm.
Answer.
\(80 \;\text{cm}^3\)

5.

A decorative garden pyramid has a square base with each side measuring 4 m. The height of the pyramid is 3 m.
\(16 \;\text{m}^3\)
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