MAT8 Q2W3D4: Application of pyramid volume

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

1. Apply the volume formula of pyramids in solving real-world word problems involving space, capacity, and construction.
2. Analyze and interpret problem situations to determine the correct base area and height to use in the formula $V=\frac{1}{3}Bh$.
3. Evaluate solutions to ensure they are reasonable and connected to practical situations such as tents, monuments, aquariums, and architectural designs.

• Word Problem - a practical or real-life situation expressed in mathematical form.
• Capacity - the amount of space inside a 3D object, measured in cubic units.
• Reasonableness of Answer - checking if the computed volume makes sense in the real-world context.
• Pyramid Volume Formula - $V=\frac{1}{3}Bh$

Activity: Connecting Previous Knowledge to Real-Life Problems

1. Recall the formula for the volume of a pyramid:

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

2. Think back:
   • What is the base area in the formula, and how do we compute it for different polygons?
   • What is the height of a pyramid, and how is it different from the base’s height?
3. Warm-up Problem:
   A square pyramid has base side = 12 m and height = 15 m. Find its volume.

Show Answer

Base area = 12 × 12 = 144 m²
Volume = (1/3)(144 × 15) = 720 m³

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🔹 Part 1: Why Do We Apply Volume in Real-Life?

The volume of pyramids helps us reason about everyday contexts, such as:

• The amount of air in a tent ⛺
• The storage capacity of a pyramid-shaped container 📦
• The volume of water an aquarium shaped like a pyramid can hold 🐟
• The interior space of a pyramid-like monument or roof 🏛️

Question: Why is it important to check if the computed answer is reasonable in real life?

Show Answer

Because math solutions must reflect reality. If a tent volume is calculated as 10,000 m³ but the tent is small, the answer is unrealistic. Always check context.

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🔹 Part 2: Review of the Formula

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

Where $B$ is the base area and $h$ is the pyramid height (perpendicular from apex to base).

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🔹 Part 3: Steps in Solving Word Problems

1. Understand the Problem - Identify what is given and what is asked; sketch if necessary.
2. Plan a Solution - Compute the base area; use $V=\frac{1}{3}Bh$.
3. Solve - Perform calculations carefully with correct units.
4. Check and Interpret - Does the answer make sense? Are the units correct?

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🔹 Part 4: Worked Example 1 – Tent Design

A camping tent is shaped like a triangular pyramid. Its triangular base has base = 4 m and height = 3 m. The pyramid’s vertical height is 2.5 m. Find the air space inside the tent.

Solution:

• Base area = (1/2)(4 × 3) = 6 m²
• Volume = (1/3)(6 × 2.5) = 5 m³

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🔹 Part 5: Worked Example 2 – Storage Box

A souvenir box is shaped like a square pyramid. Its base has side = 10 cm, and the height = 12 cm. Find its capacity.

Solution:

• Base area = 10 × 10 = 100 cm²
• Volume = (1/3)(100 × 12) = 400 cm³

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🔹 Part 6: Guided Activity – Solve These

1. A pyramid-shaped candle has a square base with side = 6 cm and height = 9 cm. Find its volume.
2. A triangular pyramid has base = 5 m, height of base = 4 m, and pyramid height = 10 m.
3. An Egyptian pyramid has a square base with side = 200 m and height = 150 m. Find its volume.

Show Answers

1. Base = 36 cm² → V = (1/3)(36 × 9) = 108 cm³
2. Base area = (1/2)(5 × 4) = 10 m² → V = (1/3)(10 × 10) = 33.3 m³
3. Base = 40,000 m² → V = (1/3)(40,000 × 150) = 2,000,000 m³

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🔹 Part 7: Real-World Application – Architecture

The Louvre Pyramid in Paris has a square base of side = 35.4 m and height = 21.6 m. Estimate its volume.

• Base area = 35.4 × 35.4 = 1253.16 m²
• Volume = (1/3)(1253.16 × 21.6) ≈ 9017 m³

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🔹 Part 8: Critical Thinking

If the height of a pyramid doubles while the base remains the same, how does the volume change?

Show Answer

Volume is directly proportional to height. Doubling the height doubles the volume.

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🔹 Part 9: Problem-Solving Challenge

A pyramid-shaped water tank has a hexagonal base with side = 3 m, apothem = 2.6 m, and height = 8 m. How many liters of water can it hold?

• Perimeter = 18 m
• Base area = (1/2)(2.6 × 18) = 23.4 m²
• Volume = (1/3)(23.4 × 8) = 62.4 m³ = 62,400 L

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🔹 Part 10: Reflection Prompt

What kinds of objects in your daily life resemble pyramids, and why is it important to know their volume?

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🔗 References

• National Council of Teachers of Mathematics (2023). Geometry in the Real World.
• Cuemath (2024). Volume of Pyramids.
• Math is Fun (2024). Solid Geometry – Pyramids.

🔹 Worked Example 1 – Monument Volume

A monument is shaped like a square pyramid with side length = 18 m and height = 25 m. Find its volume.

• Base area = 18 × 18 = 324 m²
• Volume = (1/3)(324 × 25) = 2700 m³

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🔹 Worked Example 2 – Storage Silo

A grain silo is shaped like a hexagonal pyramid with side = 5 m, apothem = 4.3 m, and height = 12 m.

• Perimeter = 6 × 5 = 30 m
• Base area = (1/2)(4.3 × 30) = 64.5 m²
• Volume = (1/3)(64.5 × 12) = 258 m³

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🔹 Worked Example 3 – Aquarium

An aquarium is shaped like a triangular pyramid with base = 1 m, height of base = 0.8 m, and pyramid height = 1.2 m. Find its water capacity in liters.

• Base area = (1/2)(1 × 0.8) = 0.4 m²
• Volume = (1/3)(0.4 × 1.2) = 0.16 m³ = 160 L

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🔹 Worked Example 4 – Tent Air Space

A pyramid-shaped tent has a square base of side = 4 m and height = 3 m.

• Base area = 16 m²
• Volume = (1/3)(16 × 3) = 16 m³

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🔹 Worked Example 5 – Jewelry Box

A jewelry box is in the shape of a pentagonal pyramid with side = 6 cm, apothem = 4.1 cm, and height = 10 cm.

• Perimeter = 30 cm
• Base area = (1/2)(4.1 × 30) = 61.5 cm²
• Volume = (1/3)(61.5 × 10) = 205 cm³

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👉 Now You Try (Mini-Tasks)

1. A small monument is a square pyramid with side = 10 m and height = 18 m. Find its volume.
2. A sugar storage bin is a hexagonal pyramid with side = 6 m, apothem = 5.2 m, and height = 15 m. Find its volume.
3. A paperweight is a triangular pyramid with base = 8 cm, base height = 7 cm, and pyramid height = 12 cm. Find its volume.
4. A pyramid-shaped candle has a square base of side = 5 cm and height = 9 cm. Find its volume.
5. An ancient pyramid has a square base of side = 200 m and height = 150 m. Find its volume.

Show Answers

1. Base area = 100 m² → V = (1/3)(100 × 18) = 600 m³
2. P = 36 m → A = (1/2)(5.2 × 36) = 93.6 m² → V = (1/3)(93.6 × 15) = 468 m³
3. Base area = (1/2)(8 × 7) = 28 cm² → V = (1/3)(28 × 12) = 112 cm³
4. Base area = 25 cm² → V = (1/3)(25 × 9) = 75 cm³
5. Base area = 40,000 m² → V = (1/3)(40,000 × 150) = 2,000,000 m³

Solve the following. Round answers to the nearest whole number when needed.

1. A display case in the shape of a square pyramid has side = 15 cm and height = 20 cm. Find its volume.
2. A fruit basket is designed like a pentagonal pyramid with side = 8 cm, apothem = 5.5 cm, and height = 12 cm. Find its capacity.
3. A giant tent is shaped like a triangular pyramid with base = 10 m, base height = 8 m, and pyramid height = 15 m. Find the air space inside the tent.
4. A pyramid-shaped water tank has a hexagonal base with side = 7 m, apothem = 6.1 m, and height = 10 m. How many cubic meters of water can it hold?
5. A small chocolate container is shaped like a square pyramid with base side = 5 cm and height = 7 cm. Find its volume.
6. An octagonal pyramid-shaped lantern has side = 12 in, apothem = 14.5 in, and height = 20 in. Find its volume.
7. A museum model of the Great Pyramid has a square base of side = 30 m and height = 25 m. Find its volume.

Show Answers

1. Base = 225 cm² → V = (1/3)(225 × 20) = 1500 cm³
2. P = 40 cm → A = (1/2)(5.5 × 40) = 110 cm² → V = (1/3)(110 × 12) = 440 cm³
3. Base area = (1/2)(10 × 8) = 40 m² → V = (1/3)(40 × 15) = 200 m³
4. P = 42 m → A = (1/2)(6.1 × 42) = 128.1 m² → V = (1/3)(128.1 × 10) = 427 m³
5. Base area = 25 cm² → V = (1/3)(25 × 7) = 58 cm³
6. P = 96 in → A = (1/2)(14.5 × 96) = 696 in² → V = (1/3)(696 × 20) = 4640 in³
7. Base = 900 m² → V = (1/3)(900 × 25) = 7500 m³

1. Multiple Choice
   The volume of a square pyramid with side = 12 m and height = 9 m is:
   A. 324 m³   B. 432 m³   C. 1296 m³   D. 648 m³
2. True or False
   The formula for the volume of a pyramid can be used for all pyramids, no matter the base shape.
3. Short Answer
   Write the steps in solving a word problem involving the volume of pyramids.
4. Multiple Choice
   A hexagonal pyramid has side = 10 m, apothem = 8.7 m, and height = 15 m. Its volume is:
   A. 261 m³   B. 1305 m³   C. 130.5 m³   D. 391.5 m³
5. True or False
   If the height of a pyramid doubles while the base area stays the same, the volume also doubles.
6. Short Answer
   An Egyptian-style pyramid has base side = 120 m and height = 90 m. Find its volume.
7. Multiple Choice
   Which of the following is the best interpretation of “reasonableness of answer” in solving pyramid volume problems?
   A. Checking if the answer matches the units only.   B. Checking if the answer makes sense in the real-world context.   C. Checking if the answer is exact and without decimals.   D. Checking if the answer uses the correct formula.
8. True or False
   A pyramid-shaped aquarium with a volume of 1.5 m³ can hold 1500 liters of water.
9. Short Answer
   A triangular pyramid has base = 5 m, base height = 4 m, and pyramid height = 12 m. Find its volume.
10. Multiple Choice
    Which real-life structure is shaped like a pyramid?
    A. A basketball   B. The Louvre in Paris   C. A cylinder of water   D. A cone

Show Answer Key

1. B — Base = 144 m² → V = (1/3)(144 × 9) = 432 m³
2. True
3. (a) Understand the problem; (b) Plan (find base area, use formula); (c) Solve; (d) Check and interpret answer.
4. B — P = 60 m → A = (1/2)(8.7 × 60) = 261 m² → V = (1/3)(261 × 15) = 1305 m³
5. True
6. Base = 14,400 m² → V = (1/3)(14,400 × 90) = 432,000 m³
7. B — Checking if the answer makes sense in the real-world context.
8. True — 1 m³ = 1000 L → 1.5 m³ = 1500 L
9. Base area = (1/2)(5 × 4) = 10 m² → V = (1/3)(10 × 12) = 40 m³
10. B — The Louvre Pyramid in Paris

🔹 Activity 1: Comparing Storage Capacities

A company wants to build two pyramid-shaped containers:

• Container A: square base, side = 8 m, height = 12 m
• Container B: hexagonal base, side = 6 m, apothem = 5.2 m, height = 10 m

👉 Which container has the larger volume?

Show Answer

Container A: Base = 64 m² → V = (1/3)(64 × 12) = 256 m³
Container B: P = 36 m → A = (1/2)(5.2 × 36) = 93.6 m² → V = (1/3)(93.6 × 10) = 312 m³
✅ Container B (hexagonal base) holds more.

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🔹 Activity 2: Tent Capacity

A large tent is shaped like a triangular pyramid. Its base is 12 m wide with height = 10 m, and the tent’s vertical height = 8 m.

👉 How much air space does the tent provide?

Show Answer

Base area = (1/2)(12 × 10) = 60 m²
Volume = (1/3)(60 × 8) = 160 m³

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🔹 Activity 3: Scale Model Problem

A model of a pyramid has a base side of 30 cm and height = 24 cm. The actual pyramid has a base side of 60 m and height = 48 m.

👉 By what scale factor is the model made, and what is the ratio of their volumes?

Show Answer

Scale factor (linear) = 30 cm : 6000 cm = 1:200
Volume ratio = (1/200)³ = 1:8,000,000

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🔹 Activity 4: Doubling Dimensions

An octagonal pyramid has a volume of 2400 m³. If all its dimensions are doubled, what will be the new volume?

Show Answer

Doubling dimensions multiplies volume by 2³ = 8.
New volume = 2400 × 8 = 19,200 m³.

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🔹 Activity 5: Research & Report

Research an ancient or modern pyramid-shaped structure (e.g., Egyptian pyramids, Mayan temples, Louvre Pyramid).

1. Approximate its base and height dimensions.
2. Estimate its volume using the pyramid formula.
3. Write 3–4 sentences explaining how its volume relates to its purpose (e.g., religious, storage, tourist attraction).

Show Example

The Great Pyramid of Giza: Base ≈ 230 m, Height ≈ 146 m.
Volume = (1/3)(230² × 146) ≈ 2.6 million m³.
✅ Its massive volume shows its function as both a tomb and a symbol of power.

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

Today, I learned how to apply the formula for the volume of pyramids in solving practical word problems. I understood the importance of carefully identifying the base area and height before applying the formula. Solving real-world problems like tents, monuments, and storage containers made me realize how mathematics connects directly to daily life.

✅ Checklist for Self-Reflection

• I can set up and solve word problems involving pyramid volumes.
• I can correctly compute base areas of different polygons.
• I can apply the formula $V=\frac{1}{3}Bh$ in real-life situations.
• I can check the reasonableness of my answers.
• I can explain why solving volume problems is useful in real-world contexts.