MAT8 Q2W4D2: Discovering the Volume of a Sphere

🎯 Learning Goals

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

1. Define a sphere and identify its parts (center, radius, diameter) with at least 90% accuracy during guided class discussion.
2. Demonstrate, through video observation or hands-on activity, that the volume of a sphere is related to the volume of cones, and correctly state the derived formula.
3. Apply the volume formula of a sphere to solve at least 4 out of 5 given real-world and mathematical problems within 20 minutes.

🧩 Key Ideas & Terms

• Sphere - a set of all points in space equidistant from a fixed point called the center.
• Center - the fixed point from which all points on the sphere are equidistant.
• Radius (r) - the distance from the center to any point on the surface of the sphere.
• Diameter (d) - the longest distance across the sphere, equal to twice the radius $d=2r$.
• Volume of a Sphere - the space inside the sphere, calculated using the formula $V=\frac{4}{3}\pi r^3$.

🔄 Prior Knowledge

1. Recall the definition of a cone and its parts (base, radius, height, vertex).
2. State the formula for the volume of a cone:
   $$V=\frac{1}{3}\pi r^{2}h$$
3. Solve this quick item: A cone has a radius of 6 cm and a height of 9 cm. Find its volume.

Show Answer

$V=\frac{1}{3}\pi \left(6\right)2^{\mathrm{}}\left(9\right)$ = $\frac{324}{3}\pi$ = 108π ≈ 339.12 cm³

📖 Explore the Lesson

Part 1 - Revisiting Solids - From Cones to Spheres

In the previous lesson, we discovered the formula for the volume of a cone. We learned that it takes 3 cones to fill a cylinder of the same height and radius, which led us to the formula:

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

Now, we move on to a new solid - the sphere. Unlike the cone or cylinder, a sphere has no edges, no flat faces, and no vertices. It is perfectly round in three-dimensional space.

Guiding Question: How can we determine the amount of space inside a sphere if it has no base or height like a cone or cylinder?

Checkpoint: Recall the definition of a sphere.

Show Answer

A sphere is the set of all points in space equidistant from a fixed point called the center.

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Part 2 - Key Features of a Sphere

• Center - the fixed point at the middle.
• Radius (r) - the distance from the center to any point on the surface.
• Diameter (d) - the longest distance across the sphere, passing through the center: $d=2r$.

Guiding Question: Why do you think the radius is more useful than the diameter in formulas?

Show Answer

Because formulas usually involve squared or cubed measures, and using radius directly makes calculation simpler.

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Part 3 - Hands-On Exploration - Relating Cones and Spheres

Mathematicians discovered the volume of a sphere by comparing it with cones and cylinders.

1. Take a cone and a sphere with the same radius.
2. Ensure the height of the cone equals the diameter of the sphere.
3. Fill the cone with water or sand, then pour into the sphere.
4. Observe how many times it takes to fill the sphere.

Learners will notice: it takes 2 cones to fill one sphere when the cone height equals the sphere diameter.

$$V\left(\mathrm{sphere}\right)=2V\left(\mathrm{cone}\right)$$

Since $V\left(\mathrm{cone}\right)=\frac{1}{3}\pi r^{2}h$ and $h=2r$, substitute to get:

$$V\left(\mathrm{sphere}\right)=2\times \frac{1}{3}\pi r^2\left(2r\right)$$

Simplifying yields the sphere formula:

$$V=\frac{4}{3}\pi r^3$$

Checkpoint: Derive the formula step by step using substitution.

Show Answer

The cone volume gives $\frac{1}{3}\pi r^2\left(2r\right)$ which is $\frac{2}{3}\pi r^3$. Since the sphere equals 2 such cones, $V=2\times \frac{2}{3}\pi r^3=\frac{4}{3}\pi r^3$.

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Part 4 - Illustrative Example 1 - Basic Computation

Problem: A sphere has a radius of 5 cm. Find its volume.

$$V=\frac{4}{3}\pi 5^3$$ $$=\frac{4}{3}\pi \left(125\right)$$ $$=\frac{500}{3}\pi$$

Show Answer

The volume of the sphere is approximately 523.6 cm³.

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Part 5 - Illustrative Example 2 - Using Diameter

Problem: A spherical ball has a diameter of 14 cm. Find its volume.

Radius $r=7$.

$$V=\frac{4}{3}\pi 7^3$$ $$=\frac{4}{3}\pi \left(343\right)$$ $$=\frac{1372}{3}\pi$$

Show Answer

The sphere volume is approximately 1436.8 cm³.

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Part 6 - Real-World Applications of Sphere Volumes

1. Sports - calculating the volume of balls such as basketballs or tennis balls.
2. Astronomy - estimating the volume of planets or moons.
3. Food industry - measuring the amount of ice cream in a spherical scoop.
4. Engineering - determining fuel storage in spherical tanks.
5. Medicine - estimating dosage containers shaped like spheres.

Checkpoint: Why do engineers prefer spherical tanks for storing gases under pressure?

Show Answer

Spheres distribute pressure evenly, reducing stress points and maximizing volume for minimal surface area.

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Part 7 - Deep Dive - Comparison with Other Solids

• Cylinder: $V=\pi r^{2}h$
• Cone: $V=\frac{1}{3}\pi r^{2}h$
• Sphere: $V=\frac{4}{3}\pi r^3$

Checkpoint: Which solid’s volume depends only on the radius, not on height?

Show Answer

The sphere, because its volume formula uses only r cubed.

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Part 8 - Advanced Challenge - Planetary Example

The Earth is roughly spherical with a radius of about 6371 km. Estimate its volume.

$$V=\frac{4}{3}\pi {6371}^3$$

Computation idea: 6371³ ≈ 2.58 × 10¹¹ km³, then multiply by $\frac{4}{3}\pi$ to get approximately 1.083 × 10¹² km³.

Show Answer

The Earth volume is approximately 1.083 × 10¹² km³.

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Part 9 - Reflection Task (Integrated)

1. How is deriving the sphere formula similar to deriving the cone formula?
2. Which real-life application of spheres do you find most interesting and why?
3. Explain to a younger student why the formula is $\frac{4}{3}\pi r^3$.

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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). Visualizing the volume of a sphere formula [Video]. YouTube. https://www.youtube.com/watch?v=YNutS8eIhEs
• Serra, M. (2008). Discovering Geometry: An Investigative Approach. Key Curriculum Press.
• NASA Earth factsheet. Planetary science data on Earth radius and volume estimation.

💡 Example in Action

Now You Try - 5 items

1. A sphere has a radius of 3 cm. Find its volume.
2. A spherical balloon has a diameter of 10 cm. Find its volume.
3. A spherical scoop of ice cream has a radius of 4 cm. Find its volume.
4. A spherical ornament has a diameter of 18 cm. Find its volume.
5. A tennis ball has a radius of 3.5 cm. Find its volume.

Show Answer

1. 113.1 cm³
2. 523.6 cm³
3. 268.1 cm³
4. 3053.6 cm³
5. 179.6 cm³

📝 Try It Out

Practice - 5 items

1. A sphere has a radius of 6 cm. Find its volume.
2. A globe has a diameter of 20 cm. Find its volume.
3. A spherical orange has a radius of 4.5 cm. Find its volume.
4. A basketball has a diameter of 24 cm. Find its volume.
5. A pearl has a radius of 1.2 cm. Find its volume.

Show Answer

1. 904.8 cm³
2. 4188.8 cm³
3. 381.7 cm³
4. 7238.2 cm³
5. 7.2 cm³

✅ Check Yourself

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

1. MCQ: Which of the following best describes a sphere?
   a) A solid with two circular bases
   b) A solid with one circular base and a vertex
   c) A set of all points equidistant from a center in three-dimensional space
   d) A solid with six rectangular faces
2. T/F: The diameter of a sphere is equal to twice its radius.
3. Short Answer: Write the formula for the volume of a sphere.
4. MCQ: The radius of a sphere is tripled. Its volume will…
   a) Double
   b) Increase 9 times
   c) Increase 27 times
   d) Stay the same
5. Short Answer: A sphere has a radius of 8 cm. Find its volume.
6. T/F: The formula for the volume of a sphere is derived using cones and cylinders.
7. MCQ: A spherical ball has a diameter of 12 cm. Its radius is…
   a) 3 cm
   b) 6 cm
   c) 12 cm
   d) 24 cm
8. Short Answer: A spherical planet has a radius of 5000 km. Estimate its volume in km³.
9. MCQ: Which of the following is a real-life example of a sphere?
   a) Ice cream cone
   b) Dice
   c) Basketball
   d) Cylinder
10. Short Answer: Compare the formulas for the volume of a cone and a sphere. State one similarity and one difference.

Show Answer Key

1. c
2. True
3. V = (4/3)πr³
4. c (increase 27 times)
5. 2144.7 cm³
6. True
7. b (6 cm)
8. ≈ 5.24 × 10¹¹ km³
9. c (Basketball)
10. Both use π and radius; cone also uses height while sphere depends only on r cubed.

🚀 Go Further

Day 2 - 5 Activities

Activity 1 - Real-Life Connection
List five objects in your surroundings shaped like a sphere. Estimate their radii and compute their approximate volumes.

Show Example

Examples: Basketball (r = 12 cm → V ≈ 7238.2 cm³), Tennis ball (r = 3.5 cm → V ≈ 179.6 cm³).

Activity 2 - Compare Volumes
Compare the volume of a sphere with that of a cube that encloses it (cube side length equals the sphere diameter). Which is greater?

Show Answer

The cube volume is greater. Ratio: sphere to cube ≈ 0.52.

Activity 3 - Design Challenge
A company wants to design spherical perfume bottles of 8 cm diameter. Calculate the capacity in milliliters.

Show Answer

r = 4 cm → V ≈ 268.1 cm³ which is about 268.1 mL.

Activity 4 - Astronomy Application
The Moon has an average radius of 1737 km. Estimate its volume and compare with Earth volume of approximately 1.083 × 10¹² km³.

Show Answer

Moon volume ≈ 2.2 × 10¹⁰ km³. Earth is about 49 times larger in volume.

Activity 5 - Creative Project
Create a short illustrated guide showing how the formula for the volume of a sphere is derived using cones and cylinders. Present it as a poster or digital slideshow.

Show Example

Illustration shows: 2 cones (height equals diameter) fill one sphere which supports V = (4/3)πr³.

🔗 My Reflection

Option A - Write 3 to 5 sentences

In my notebook, I will write a short reflection about what I learned today. I will explain how the volume of a sphere is derived and why it is related to cones and cylinders. I will also reflect on how this knowledge applies to real-life situations.

Option B - 3-2-1 format

• 3 things I learned today: __________
• 2 interesting facts about spheres: __________
• 1 question I still have: __________

Instruction: Answer these in your notebook.