MAT Q2W4D1: Discovering the volume of a square

🎯 Learning Goals

1. By the end of the lesson, students will be able to define a cone and identify its parts (base, radius, height) with 100% accuracy during class discussion.
2. Within the lesson period, students will demonstrate through a hands-on activity that the volume of a cone is one-third the volume of a cylinder with the same height and radius.
3. After guided practice, students will be able to apply the derived formula for the volume of a cone to solve at least 3 out of 4 problems correctly within 15 minutes.

🧩 Key Ideas & Terms

• Cone – a three-dimensional solid with a circular base and one vertex not in the same plane as the base.
• Base – the circular face of a cone.
• Radius (r) – the distance from the center of the base to any point on the circle.
• Height (h) – the perpendicular distance from the base to the vertex.
• Volume of a cone – the space inside the cone, calculated using the formula $V=\frac{1}{3}\pi r^{2}h$.

🔄 Prior Knowledge

1. Recall the definition of a cylinder and its parts (base, radius, height).
2. State the formula for the volume of a cylinder:
   $$V=\pi r^{2}h$$
3. Solve this quick item: A cylindrical drum has a radius of 7 cm and a height of 10 cm. Find its volume.

Show Answer

$V=\pi \left(7\right)2^{\mathrm{}}\left(10\right)$ = $1540\pi$ ≈ 4830.67 cm³

📖 Explore the Lesson

Part 1: Warming Up – From Cylinders to Cones

Imagine holding a cylindrical water tumbler and an ice cream cone. Both are solids, both have circular bases, yet their structures are different. The tumbler has two parallel circular bases; the ice cream cone has just one circular base, with its sides slanting towards a single point called the vertex.

Guiding Question: How do we measure how much ice cream a cone can hold compared to how much water a tumbler can hold?

Checkpoint: Recall the formula for the volume of a cylinder.

$$V=\pi r^{2}h$$

Show Answer

The volume of a cylinder depends on the radius ($r$) and the height ($h$). It is calculated as $\pi r^{2}h$.

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Part 2: Defining a Cone

A cone is a solid figure with a circular base and a vertex that is not in the same plane as the base.

• Base (circle): The flat circular region.
• Radius (r): Distance from the center of the base to the circle.
• Height (h): Perpendicular distance from the base to the vertex.
• Vertex: The point not in the same plane as the base.

Real-life examples: Ice cream cone, birthday party hat, traffic cone.

Checkpoint: Which part of a cone determines its “thickness” at the base?

Show Answer

The radius determines how wide or thick the base of the cone is.

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Part 3: Discovery Activity – Filling a Cylinder with Cones

Now, let’s explore with a thought experiment (or hands-on activity if you have materials).

Materials: A cone and a cylinder with the same radius and height; sand, rice, or water.

1. Fill the cone completely with sand.
2. Pour it into the cylinder.
3. Repeat until the cylinder is full.

Observation: It takes 3 cones to fill 1 cylinder of the same radius and height.

This means: $$V\left(\mathrm{cone}\right)=\frac{1}{3}V\left(\mathrm{cylinder}\right)$$

Substituting the cylinder’s formula:

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

Checkpoint: If the cylinder has a volume of 300 cm³, what is the volume of the cone with the same radius and height?

Show Answer

$\frac{1}{3}\left(300\right)$ = 100 cm³

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Part 4: Illustrative Example

Problem: A cone has a radius of 3 cm and a height of 4 cm. Find its volume.

Solution:

$$V=\frac{1}{3}\pi \left(3\right)2^{\mathrm{}}\left(4\right)$$ $$=\frac{1}{3}\pi \left(9\right)\left(4\right)$$ $$=\frac{36}{3}\pi$$ $$=12\pi$$

Show Answer

The volume of the cone is $12\pi$ ≈ 37.70 cm³.

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Part 5: Real-Life Applications

1. Construction: Calculating the volume of concrete for a conical roof.
2. Food: Measuring the capacity of an ice cream cone.
3. Storage: Determining space inside conical containers.

Checkpoint: Why is it important to know how to compute the volume of cones in real life?

Show Answer

It helps in estimating materials (like food, cement, or liquids) and making efficient use of space in daily life.

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Part 6: Guided Practice – Solve Together

Task: Find the volume of a cone with radius = 5 cm and height = 10 cm.

Show Answer

$V=\frac{1}{3}\pi r^{2}h$ = $\frac{1}{3}\pi \left(5\right)2^{\mathrm{}}\left(10\right)$ = $\frac{250}{3}\pi$ ≈ 261.80 cm³

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Part 7: Reflection Activity

Write a short summary in your notebook:

• How does the cone’s volume compare with the cylinder’s volume?
• Which step in the discovery activity helped you understand the formula the most?

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References

• Burger, E., Chard, D. J., Hall, E. J., Kennedy, P. A., & Leinwand, S. J. (2008). Holt California Geometry. Holt, Rinehart and Winston.
• Kyle Pearce. (2014, September 28). Cones and Spheres [Act 3]: How many cones does it take to fill a sphere? [Video]. YouTube. https://www.youtube.com/watch?v=PaA-g_z_E2E
• Serra, M. (2008). Discovering Geometry: An Investigative Approach. Key Curriculum Press.

💡 Example in Action

Now You Try – 5 items

1. A cone has a radius of 2 cm and a height of 6 cm. Find its volume.
2. A cone has a diameter of 8 cm and a height of 9 cm. Find its volume.
3. An ice cream cone has a radius of 3.5 cm and a height of 12 cm. How much ice cream can it hold?
4. A conical flask has a height of 15 cm and a radius of 4 cm. Find its volume.
5. A cone-shaped cap has a base radius of 7 cm and a height of 10 cm. Find its volume.

Show Answer

1. 25.13 cm³
2. 150.80 cm³
3. 153.94 cm³
4. 251.33 cm³
5. 513.13 cm³

📝 Try It Out

Practice – 5 items

1. A cone has a radius of 5 cm and a height of 8 cm. Find its volume.
2. A cone has a diameter of 10 cm and a height of 12 cm. Find its volume.
3. A cone has a radius of 6 cm and a height of 15 cm. Find its volume.
4. A cone has a base radius of 4.5 cm and a height of 10 cm. Find its volume.
5. A cone has a radius of 2 cm and a height of 7 cm. Find its volume.

Show Answer

1. 209.44 cm³
2. 314.16 cm³
3. 565.49 cm³
4. 212.06 cm³
5. 29.32 cm³

✅ Check Yourself

10 Mixed Items (MCQ, T/F, Short Answer)

1. MCQ: Which of the following best describes a cone?
   a) A solid with two parallel circular bases
   b) A solid with a circular base and one vertex
   c) A solid with all points equidistant from a center
   d) A solid with a rectangular base
2. T/F: The height of a cone is measured along its slant side.
3. Short Answer: Write the formula for the volume of a cone.
4. MCQ: A cone and a cylinder have the same radius and height. The cone’s volume is…
   a) Equal to the cylinder’s volume
   b) Twice the cylinder’s volume
   c) One-half the cylinder’s volume
   d) One-third the cylinder’s volume
5. Short Answer: A cone has a radius of 3 cm and a height of 9 cm. Find its volume.
6. T/F: The base of a cone is always a circle.
7. MCQ: The radius of a cone is doubled, and the height stays the same. Its volume will…
   a) Double
   b) Triple
   c) Increase four times
   d) Remain the same
8. Short Answer: Give one real-life example of a cone-shaped object.
9. MCQ: Which formula correctly represents the volume of a cone?
   a) $\pi r^{2}h$
   b) $\frac{1}{3}\pi r^{2}h$
   c) $\frac{4}{3}\pi r^3$
   d) $2\pi rh$
10. Short Answer: If a cone has a diameter of 10 cm and a height of 15 cm, find its volume.

Show Answer Key

1. b
2. False (height is perpendicular, not slanted)
3. V = (1/3)πr²h
4. d (one-third)
5. 84.82 cm³
6. True
7. c (volume increases four times since radius is squared)
8. Ice cream cone, party hat, traffic cone, etc.
9. b
10. 392.70 cm³

🚀 Go Further

Day 1 – 5 Activities

Activity 1 – Real-Life Connection
Find three real-life objects shaped like cones (home, school, or community). Estimate their dimensions (radius and height) and compute their approximate volumes.

Show Example

Examples: Ice cream cone, traffic cone, funnel. An ice cream cone with r = 3 cm, h = 12 cm → V ≈ 113 cm³.

Activity 2 – Compare with Cylinders
Choose a cone and a cylinder that have the same radius and height. Compute both volumes and compare.

Show Example

If r = 5 cm and h = 10 cm → V(cylinder) = 785.4 cm³, V(cone) = 261.8 cm³. Cone is 1/3 of cylinder.

Activity 3 – Design Challenge
Draw a cone-shaped container that could hold 500 cm³ of liquid. Determine at least one possible set of dimensions (radius and height).

Show Example

Using V = (1/3)πr²h. If r = 5 cm, solving gives h ≈ 19.1 cm.

Activity 4 – Puzzle Problem
A cone and a hemisphere (half-sphere) have the same radius and height. Which one has the greater volume?

Show Answer

The hemisphere has greater volume because V(hemisphere) = (2/3)πr³ while V(cone) = (1/3)πr²h = (1/3)πr³. So, hemisphere volume is twice that of the cone.

Activity 5 – Group Task
Work in pairs or groups to create a short video or presentation showing how to derive the formula for the volume of a cone using real objects or digital animation.

Show Example

Students may demonstrate filling a cylinder with 3 cones of sand to illustrate V(cone) = (1/3)πr²h.

🔗 My Reflection

Option A – Guiding Questions

1. What did I learn about the relationship between the volume of a cone and a cylinder?
2. How did the discovery activity help me understand the formula for the volume of a cone?
3. In what real-life situations might I need to calculate the volume of a cone?
4. Which part of today’s lesson was the most challenging, and how did I overcome it?
5. How confident am I now in solving cone volume problems?

Option B – Checklist

• ☐ I can define a cone and its parts.
• ☐ I can explain why the volume of a cone is one-third that of a cylinder with the same radius and height.
• ☐ I can write and use the formula for the volume of a cone.
• ☐ I can solve word problems involving cones.
• ☐ I can explain real-life applications of cone volumes.

Instruction: Answer these in your notebook.