MAT8 Q2W3D1: Volume of square and rectangular pyramids

At the end of the lesson, the learners will be able to:

1. Derive the formula for finding the volume of square and rectangular pyramids using the cube as a reference.
2. Accurately calculate the volume of pyramids with square and rectangular bases given specific measurements.
3. Compare and explain the relationship between the volume of a cube and a pyramid to justify why the volume of a pyramid is one-third that of a cube.

• Volume – the amount of space occupied by a three-dimensional object.
• Pyramid – a solid figure with a polygonal base and triangular lateral faces that meet at an apex.
• Square Pyramid – a pyramid with a square base.
• Rectangular Pyramid – a pyramid with a rectangular base.
• Base Area (B) – the area of the base polygon of a pyramid.
• Height (h) – the perpendicular distance from the apex to the base.

Activity: How Much Do You Remember?

1. Recall the formula for the volume of a cube.
2. Solve for the volume of a cube with the following side lengths:
   • 4 cm
   • 5 in
   • 6 dm
   • 7 cm
   • 8 in

Show Answer

• Formula: $s^3$, where $s$ is the side of the cube.

• 4 cm → $64{\mathrm{cm}}^3$
• 5 in → $125{\mathrm{in}}^3$
• 6 dm → $216{\mathrm{dm}}^3$
• 7 cm → $343{\mathrm{cm}}^3$
• 8 in → $512{\mathrm{in}}^3$

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🔹 Part 1: Starting with the Cube

Think back to the cube you worked with in Grade 6. A cube has all sides equal. If each edge of the cube measures $s$, then its volume is:

$$V=s^3$$

👉 Question: Why do you think multiplying the side three times gives the volume of a cube?

Show Answer

Because volume measures space in three dimensions (length × width × height), and all three are equal to $s$, so $s^3$.

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🔹 Part 2: From Cube to Pyramid

Now imagine cutting a cube into three identical pyramids. When joined, the three pyramids perfectly fill the cube. This means the volume of one pyramid is one-third of the cube.

So, the general formula for the volume of a pyramid becomes:

$$V=\frac{1}{3}Bh$$

Where $B$ is the area of the base and $h$ is the perpendicular height of the pyramid.

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🔹 Part 3: Real-World Connection – The Great Pyramid of Giza

The Great Pyramid of Giza is one of the most famous examples of a square pyramid. Built around 2600 BC, it originally stood at 146.6 m with a base length of 230.3 m. Using the formula $\frac{1}{3}Bh$, its volume is approximately 2.6 million m³.

👉 Activity:

1. Use the given base and height to compute its volume yourself.
2. Compare your answer to the historical estimate of 2.6 million cubic meters.

Show Answer

$$B={230.3}^2=53038.09m^2$$

$$V=\frac{1}{3}(53038.09\times 146.6)=2592672m^3$$

Yes ✅, it is close to the known volume of about 2.6 million m³.

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🔹 Part 4: Square Pyramid Formula

If the pyramid’s base is a square with side length $s$, then:

$$B=s^2$$

So the volume is:

$$V=\frac{1}{3}{s}^{2}h$$

Guided Check: Find the volume of a square pyramid with side $8m$ and height $15m$.

Show Answer

$$B=8\times 8=64m^2$$

$$V=\frac{1}{3}(64\times 15)=320m^3$$

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🔹 Part 5: Rectangular Pyramid Formula

If the pyramid’s base is a rectangle with length $l$ and width $w$, then:

$$B=lw$$

So the volume is:

$$V=\frac{1}{3}lwh$$

Example Problem: A rectangular pyramid has base length = 6 in, width = 8 in, height = 16 in. Find its volume.

Show Answer

$$B=6\times 8=48{\mathrm{in}}^2$$

$$V=\frac{1}{3}(48\times 16)=256{\mathrm{in}}^3$$

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🔹 Part 6: Discovery Questions

1. Why is the volume of a pyramid always one-third of the corresponding prism or cube?
2. If the height is doubled, what happens to the volume of the pyramid?
3. How do changes in base area affect the volume of the pyramid?

Show Answers

1. Because three identical pyramids fit exactly into the prism/cube.
2. Volume doubles, since height is directly proportional when base area is fixed.
3. Increasing the base area increases the volume proportionally (with fixed height).

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🔹 Part 7: Real-Life Applications

• Engineers use pyramid formulas when designing roof structures and monuments.
• Architects calculate volumes to estimate materials and costs.
• Historians and archaeologists calculate pyramid volumes to understand ancient engineering.

Think: If you were tasked to build a miniature model of the Great Pyramid at 1:100 scale, what would be its volume?

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🔗 References

• Cuemath (2013). Area of Equilateral Triangle – Formula, Derivation, Examples. https://www.cuemath.com/measurement/area-of-equilateral-triangle/
• Pierce, R. (2024). Math is Fun: Pyramid vs. Cube. https://www.mathsisfun.com/geometry/pyramid-vs-cube.html
• Wikipedia Contributors (2024). Great Pyramid of Giza. https://en.wikipedia.org/wiki/Great_Pyramid_of_Giza
• Math Monks (2023). Trapezoidal Pyramid – Formulas, Examples & Diagrams. https://mathmonks.com/pyramid/trapezoidal-pyramid

Worked Example 1

A square pyramid has a base side of 6 cm and a height of 9 cm. What is its volume?

Show Answer

$$B=6^2=36{\mathrm{cm}}^2$$ $$V=\frac{1}{3}(36\times 9)=108{\mathrm{cm}}^3$$

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Worked Example 2

A rectangular pyramid has a base 12 m by 10 m and a height of 15 m. Find its volume.

Show Answer

$$B=12\times 10=120m^2$$ $$V=\frac{1}{3}(120\times 15)=600m^3$$

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Worked Example 3

A pyramid has a square base with side 20 in and height 27 in. What is its volume?

Show Answer

$$B={20}^2=400{\mathrm{in}}^2$$ $$V=\frac{1}{3}(400\times 27)=3600{\mathrm{in}}^3$$

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👉 Now You Try (Mini-Tasks)

1. A square pyramid has a base side of 9 m and a height of 14 m. Find its volume.
2. A rectangular pyramid has dimensions 7 cm by 11 cm at the base and a height of 18 cm. Find its volume.
3. The base of a pyramid is a square with side 25 ft and height 30 ft. Find its volume.

Show Answer

1. $378m^3$
2. $462{\mathrm{cm}}^3$
3. $6250{\mathrm{ft}}^3$

Solve for the volume of the following pyramids. Express answers in the nearest whole number.

1. A square pyramid with a base side of 10 cm and height of 12 cm.
2. A rectangular pyramid with base dimensions 5 m by 7 m and a height of 9 m.
3. A square pyramid with side 15 in and height 20 in.
4. A rectangular pyramid with base 14 cm by 8 cm and height 11 cm.
5. A square pyramid with side 9 dm and height 16 dm.
6. A rectangular pyramid with base 18 ft by 12 ft and height 25 ft.
7. A square pyramid with side 6 m and height 9 m.
8. A rectangular pyramid with base 13 cm by 9 cm and height 15 cm.

Show Answer

1. $400{\mathrm{cm}}^3$
2. $105m^3$
3. $1500{\mathrm{in}}^3$
4. $410.67\approx 411{\mathrm{cm}}^3$
5. $432{\mathrm{dm}}^3$
6. $1800{\mathrm{ft}}^3$
7. $108m^3$
8. $585{\mathrm{cm}}^3$

1. Multiple Choice
   The base of a square pyramid has a side length of 12 m, and the height is 9 m. What is its volume?
   A. 432 m³   B. 518 m³   C. 4320 m³   D. 5184 m³
2. True or False
   The volume of a pyramid is always one-half the volume of the prism with the same base and height.
3. Short Answer
   Write the general formula for the volume of any pyramid.
4. Multiple Choice
   A rectangular pyramid has a base of 10 cm × 8 cm and a height of 15 cm. What is its volume?
   A. 400 cm³   B. 1200 cm³   C. 3600 cm³   D. 4000 cm³
5. True or False
   If the height of a square pyramid doubles, then its volume also doubles.
6. Short Answer
   A square pyramid has a base side of 18 in and height of 27 in. Find its volume.
7. Multiple Choice
   Which of the following real-world objects best models a square pyramid?
   A. A soda can   B. The Great Pyramid of Giza   C. A rectangular box   D. A cone

Show Answer Key

1. A ($432m^3$)
2. False — it is one-third, not one-half.
3. $V=\frac{1}{3}Bh$
4. C ($3600{\mathrm{cm}}^3$)
5. True — with base area fixed, volume is directly proportional to height.
6. $2916{\mathrm{in}}^3$
7. B (The Great Pyramid of Giza)

Activity 1: Scale Models

An architect creates a 1:50 scale model of a square pyramid whose actual base side is 30 m and height is 45 m.

• Find the volume of the scale model.
• Compare it with the actual volume.

Show Answer

Actual volume = $\frac{1}{3}{30}^2\times 45=13500m^3$

Scale side = $\frac{30}{50}=0.6$ m, height = $\frac{45}{50}=0.9$ m

Scale model volume = $\frac{1}{3}{0.6}^2\times 0.9=0.108m^3$

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Activity 2: Compare Solids

Complete the table by comparing the volumes of a cube and a square pyramid with the same base side and height.

Shape	Base side (cm)	Height (cm)	Volume Formula	Volume (cm³)
Cube	10	10	${10}^3$	1000
Square Pyramid	10	10	$\frac{1}{3}{10}^2\times 10$	333.33

👉 Question: What fraction of the cube’s volume is the pyramid’s volume?

Show Answer

The pyramid’s volume is one-third of the cube’s volume.

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Activity 3: Real-Life Problem

A glass company wants to design a rectangular pyramid-shaped container with dimensions: length = 24 cm, width = 18 cm, height = 30 cm.

• Find the maximum volume the container can hold.
• If the company produces 200 containers, what is the total volume?

Show Answer

Volume of one container = $\frac{1}{3}(24\times 18\times 30)=4320{\mathrm{cm}}^3$

Total for 200 containers = $200\times 4320=864000{\mathrm{cm}}^3$

✍️ In your notebook, write a short reflection (3–5 sentences):

Today, I learned about the volume of square and rectangular pyramids. I discovered how the formula is derived from a cube and why a pyramid’s volume is only one-third of a prism’s with the same base and height. I also practiced solving real-life problems, which helped me see the importance of this concept in engineering and architecture.

✅ Checklist for Self-Reflection

• I can explain how the volume of a pyramid is related to the volume of a cube.
• I can state and apply the formula $\frac{1}{3}Bh$.
• I can solve problems involving square pyramids.
• I can solve problems involving rectangular pyramids.
• I can connect the concept of volume to real-life structures like the Great Pyramid of Giza.