MAT8 Q2W3D2: Volume of triangular pyramids

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

1. Derive the formula for finding the volume of triangular pyramids using the general pyramid volume formula.
2. Solve problems involving the volume of triangular pyramids with given dimensions accurately.
3. Apply the concept of triangular pyramid volume in real-life situations, such as modeling tents or roof structures.

• Triangular Pyramid (Tetrahedron) – a pyramid with a triangular base.
• Base Area (B) – the area of the triangular base.
• Height (h) – the perpendicular distance from the apex to the base.
• Volume of a Triangular Pyramid – $V=\frac{1}{3}Bh$

Activity: What Do You Still Remember?

1. Recall the formula for the area of a triangle.
2. Compute the area of the following triangles:
   • Base = 10 cm, Height = 6 cm
   • Base = 12 m, Height = 9 m
   • Base = 8 in, Height = 15 in

Show Answer

• Formula: $A=\frac{1}{2}bh$

• (10 × 6) ÷ 2 = $30{\mathrm{cm}}^2$
• (12 × 9) ÷ 2 = $54m^2$
• (8 × 15) ÷ 2 = $60{\mathrm{in}}^2$

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🔹 Part 1: From General Formula to Triangular Pyramids

In the previous lesson, we discovered that the volume of a pyramid is always one-third of the product of its base area and height:

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

This formula is true for all types of pyramids — whether the base is a square, rectangle, pentagon, or triangle.

👉 If the base is a triangle, then the base area $B$ is computed using the area of a triangle formula:

$$B=\frac{1}{2}bh$$

So the formula for a triangular pyramid’s volume becomes:

$$V=\frac{1}{3}\left(\frac{1}{2}bh\right)H$$

Where:

• $b$ = base of the triangle
• $h$ = height of the triangular base
• $H$ = height of the pyramid

👉 Checkpoint:
Why do you think the “double height” is needed (base height vs. pyramid height)?

Show Answer

Because the triangular base has its own height to calculate its area, while the pyramid has a different height that measures how tall the pyramid is. Both are needed.

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🔹 Part 2: Understanding with a Tetrahedron

A tetrahedron is the simplest triangular pyramid — it has 4 triangular faces, 6 edges, and 4 vertices.

Imagine a tetrahedron built with equilateral triangles. If each side measures 1 unit, its volume can be derived using the formula:

$$V=\frac{\sqrt{2}}{12}{a}^3$$

👉 Activity: If each side = 6 cm, find the tetrahedron’s volume.

Show Answer

$$V=\frac{\sqrt{2}}{12}{6}^3=\frac{216}{\times }\approx 25.46{\mathrm{cm}}^3$$

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🔹 Part 3: Worked Example with Triangular Base

Example 1:
A triangular pyramid has a base triangle with base = 12 m, height = 8 m, and pyramid height = 15 m. Find its volume.

Solution:

• Step 1: Base area = $\frac{1}{2}12\times 8=48m^2$
• Step 2: Multiply by pyramid height = $48\times 15=720$
• Step 3: Take one-third = $240m^3$

👉 Final Answer: $240m^3$

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🔹 Part 4: Guided Practice

Try the following:

1. Base triangle: b = 10 cm, h = 6 cm; Pyramid height = 12 cm
2. Base triangle: b = 15 m, h = 10 m; Pyramid height = 18 m
3. Base triangle: b = 9 in, h = 8 in; Pyramid height = 20 in

Show Answer

1. Base area = 30 cm² → V = (1/3)(30 × 12) = 120 cm³
2. Base area = 75 m² → V = (1/3)(75 × 18) = 450 m³
3. Base area = 36 in² → V = (1/3)(36 × 20) = 240 in³

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🔹 Part 5: Real-World Connections

• Tents – many camping tents are shaped like triangular pyramids.
• Roof trusses – architectural designs often use triangular pyramid shapes for support.
• Crystals – certain molecular structures (like methane, CH₄) form tetrahedral shapes.

👉 Activity: Research one real-life triangular pyramid (tent, roof, crystal, etc.). Describe how volume matters for its design.

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🔹 Part 6: Explore Through Discovery

Consider this puzzle:
If the base area doubles, but the pyramid’s height remains the same, what happens to the volume?

Show Answer

The volume also doubles, because volume is directly proportional to base area.

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🔹 Part 7: Critical Thinking Challenge

Suppose you have a triangular pyramid made of glass to be used as an aquarium.

• Base = triangle with b = 40 cm, h = 30 cm
• Pyramid height = 60 cm

1. Compute its volume in cm³.
2. Convert to liters.
3. Discuss: Would this design be practical for holding fish?

Show Answer

1. Base area = (1/2)(40 × 30) = 600 cm²
2. V = (1/3)(600 × 60) = 12,000 cm³ → since 1,000 cm³ = 1 L, that’s 12 L.
3. Yes, it could hold fish, but shape may be impractical (narrow apex reduces usable space).

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🔹 Part 8: Advanced Exploration

For equilateral triangles, the base area can also be expressed as:

$$A=\frac{\sqrt{3}}{4}{a}^2$$

This can simplify computations for tetrahedrons.

👉 Activity: If a tetrahedron has side length 10 cm, use this formula to find its volume.

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🔹 Part 9: Summary of Key Points

• All pyramids share the same general formula: $V=\frac{1}{3}Bh$
• For triangular pyramids, compute triangle area first.
• Real-world examples include tents, crystals, roofs, and aquariums.
• Volume grows proportionally with base area and height.

Illustration of a triangular pyramid with base 12 cm, base height 8 cm, and pyramid height 15 cm — showing step-by-step solution to find its volume (240 cm³).

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🔗 References

• Math is Fun (2024). “Volume of a Pyramid.” https://www.mathsisfun.com/geometry/pyramids.html
• Cuemath (2023). “Triangular Pyramid – Volume Formula, Examples.” https://www.cuemath.com/geometry/triangular-pyramid/
• Wikipedia (2024). “Tetrahedron.” https://en.wikipedia.org/wiki/Tetrahedron

🔹 Worked Example 1

A triangular pyramid has a base with b = 12 cm, h = 10 cm, and pyramid height H = 15 cm. Find its volume.

Solution:

• Base area = (1/2)(12 × 10) = 60 cm²
• Volume = (1/3)(60 × 15) = 300 cm³

👉 Final Answer: $300{\mathrm{cm}}^3$

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🔹 Worked Example 2

A triangular pyramid has a base area of 25 m² and height = 18 m. Find its volume.

Solution:
Volume = (1/3)(25 × 18) = 150 m³

👉 Final Answer: $150m^3$

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🔹 Worked Example 3

The base of a triangular pyramid is a triangle with b = 20 in, h = 12 in. The pyramid height is H = 24 in.

Solution:

• Base area = (1/2)(20 × 12) = 120 in²
• Volume = (1/3)(120 × 24) = 960 in³

👉 Final Answer: $960{\mathrm{in}}^3$

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🔹 Worked Example 4

A triangular pyramid has a base side = 8 m, base height = 7 m, and pyramid height = 10 m.

Solution:

• Base area = (1/2)(8 × 7) = 28 m²
• Volume = (1/3)(28 × 10) = 93.3 m³

👉 Final Answer: $93.3m^3$

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🔹 Worked Example 5

The base of a triangular pyramid is an equilateral triangle with side = 9 cm, height of triangle = 7.8 cm, and pyramid height = 20 cm.

Solution:

• Base area = (1/2)(9 × 7.8) = 35.1 cm²
• Volume = (1/3)(35.1 × 20) = 234 cm³

👉 Final Answer: $234{\mathrm{cm}}^3$

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👉 Now You Try (Mini-Tasks)

1. Base: b = 14 m, h = 10 m; Pyramid height = 21 m
2. Base: b = 16 cm, h = 12 cm; Pyramid height = 25 cm
3. Base area = 40 in²; Pyramid height = 18 in
4. Base: b = 30 dm, h = 24 dm; Pyramid height = 40 dm
5. Base: b = 50 ft, h = 35 ft; Pyramid height = 60 ft

Show Answers

1. Base area = 70 m² → V = (1/3)(70 × 21) = 490 m³
2. Base area = 96 cm² → V = (1/3)(96 × 25) = 800 cm³
3. V = (1/3)(40 × 18) = 240 in³
4. Base area = 360 dm² → V = (1/3)(360 × 40) = 4800 dm³
5. Base area = 875 ft² → V = (1/3)(875 × 60) = 17,500 ft³

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

1. A triangular pyramid has base b = 18 cm, base height = 12 cm, and pyramid height = 20 cm.
2. The base is a triangle with b = 15 m, h = 14 m; pyramid height = 30 m.
3. A triangular pyramid has base area = 48 in² and height = 27 in.
4. A triangular pyramid has base b = 22 ft, h = 15 ft; pyramid height = 40 ft.
5. A tetrahedron has equilateral sides of 10 cm. Use $V=\frac{\sqrt{2}}{12}{a}^3$.
6. The base of a triangular pyramid is b = 9 dm, h = 8 dm; pyramid height = 25 dm.
7. A triangular pyramid has base b = 35 m, h = 28 m; pyramid height = 50 m.

Show Answers

1. Base area = (1/2)(18 × 12) = 108 cm² → V = (1/3)(108 × 20) = 720 cm³
2. Base area = (1/2)(15 × 14) = 105 m² → V = (1/3)(105 × 30) = 1050 m³
3. V = (1/3)(48 × 27) = 432 in³
4. Base area = (1/2)(22 × 15) = 165 ft² → V = (1/3)(165 × 40) = 2200 ft³
5. V = (√2/12)(10³) ≈ 117.9 cm³ ≈ 118 cm³
6. Base area = (1/2)(9 × 8) = 36 dm² → V = (1/3)(36 × 25) = 300 dm³
7. Base area = (1/2)(35 × 28) = 490 m² → V = (1/3)(490 × 50) = 8167 m³

1. Multiple Choice
   The formula for the volume of a triangular pyramid is:
   A. $V=Bh$   B. $V=\frac{1}{2}Bh$   C. $V=\frac{1}{3}Bh$   D. $V=\frac{1}{4}Bh$
2. True or False
   All pyramids, regardless of base shape, use the same general volume formula.
3. Short Answer
   A triangular pyramid has base b = 10 cm, h = 8 cm, and pyramid height = 12 cm. Find its volume.
4. Multiple Choice
   The base of a triangular pyramid is b = 16 m, h = 12 m, and the pyramid height = 20 m. What is its volume?
   A. 640 m³   B. 1920 m³   C. 6400 m³   D. 19200 m³
5. True or False
   A tetrahedron is a triangular pyramid with four triangular faces.
6. Short Answer
   Write the formula for the area of a triangle.
7. Multiple Choice
   The base area of a triangular pyramid is 36 in², and the pyramid height is 15 in. What is its volume?
   A. 180 in³   B. 540 in³   C. 720 in³   D. 1800 in³
8. True or False
   If the pyramid height doubles, the volume also doubles (assuming base area remains the same).
9. Short Answer
   A triangular pyramid has base area = 50 dm² and height = 24 dm. Find its volume.
10. Multiple Choice
    Which real-world object best represents a triangular pyramid?
    A. A cube of ice   B. A camping tent   C. A cylinder   D. A cone

Show Answer Key

1. C
2. True
3. Base area = (1/2)(10 × 8) = 40 cm² → V = (1/3)(40 × 12) = 160 cm³
4. Base area = (1/2)(16 × 12) = 96 m² → V = (1/3)(96 × 20) = 640 m³ → Answer: A
5. True
6. $A=\frac{1}{2}bh$
7. (1/3)(36 × 15) = 180 in³ → Answer: A
8. True
9. V = (1/3)(50 × 24) = 400 dm³
10. B (A camping tent)

🔹 Activity 1: Scale Tent Model

An engineer designs a camping tent shaped like a triangular pyramid.

• Actual dimensions: base = b = 3 m, h = 2.5 m, pyramid height = 4 m.
• A model is built at 1:20 scale.

👉 Find the volume of the model tent.

Show Answer

Base area = (1/2)(3 × 2.5) = 3.75 m²
Actual volume = (1/3)(3.75 × 4) = 5 m³
Scale factor = (1/20)³ = 1/8000
Model volume = 5 ÷ 8000 = 0.000625 m³ = 625 cm³

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🔹 Activity 2: Volume Comparison

Complete the table below.

Base (b × h)	Pyramid Height	Base Area	Volume
12 cm × 10 cm	15 cm	?	?
20 m × 14 m	18 m	?	?
30 in × 24 in	40 in	?	?

Show Answer

• Base area 60 cm² → V = (1/3)(60 × 15) = 300 cm³
• Base area 140 m² → V = (1/3)(140 × 18) = 840 m³
• Base area 360 in² → V = (1/3)(360 × 40) = 4800 in³

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🔹 Activity 3: Aquarium Design

A glass aquarium is shaped like a triangular pyramid.

• Base b = 50 cm, h = 40 cm, pyramid height = 60 cm.

👉 Find how many liters of water it can hold.

Show Answer

Base area = (1/2)(50 × 40) = 1000 cm²
Volume = (1/3)(1000 × 60) = 20,000 cm³
1 L = 1000 cm³ → 20 L capacity

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🔹 Activity 4: Critical Thinking

A triangular pyramid has base b = 20 m, h = 12 m, and height = 25 m.

• If the base length doubles but the base height stays the same, what happens to the volume?

Show Answer

Original base area = 120 m² → V = 1000 m³
New base area = (1/2)(40 × 12) = 240 m² → V = 2000 m³
The volume doubles.

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🔹 Activity 5: Real-Life Research Task

Search for a modern building or structure shaped like a triangular pyramid (e.g., Louvre Pyramid in Paris).

• Find its dimensions (approximate).
• Estimate its volume using the formula.
• Reflect on how volume might affect its construction or purpose.

Show Example

Louvre Pyramid: base length = 35.4 m, height = 21.6 m
Base area (treating as right isosceles estimate) = (1/2)(35.4 × 35.4) ≈ 626 m²
Volume = (1/3)(626 × 21.6) ≈ 4498 m³

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

Today, I learned how to compute the volume of triangular pyramids using the general pyramid formula. I understood the difference between the base height of the triangle and the overall pyramid height. I practiced solving problems and realized how this applies in real-world structures like tents, aquariums, and buildings.

✅ Checklist for Self-Reflection

• I can explain the formula for the volume of a triangular pyramid.
• I can compute the base area of a triangle correctly.
• I can distinguish between base height and pyramid height.
• I can solve real-life problems involving triangular pyramids.
• I can give examples of real-world objects shaped like triangular pyramids.