Unlocking the Secrets of Spheres: Volume, Surface Area, and Cavalieri's Principle

Volume of a Cylinder	Volume of a Sphere
$V_{C} = π r^{2} h$	$V_{S} = \frac{4}{3} π r^{3}$

Volume of a Cylinder

Volume of a Sphere

V_{C} = π r^{2} h

V_{S} = \frac{4}{3} π r^{3}

Volume of a Cone	Volume of a Cylinder	Volume of a Hemisphere
$V_{1} = \frac{1}{3} π r^{2} h$	$V_{2} = π r^{2} h$	$V_{3} = \frac{2}{3} π r^{3}$

Volume of a Cone

Volume of a Cylinder

Volume of a Hemisphere

V_{1} = \frac{1}{3} π r^{2} h

V_{2} = π r^{2} h

V_{3} = \frac{2}{3} π r^{3}

Circle	Sphere
A circle is the set of all the points in a plane that are equidistant from a given point.	A sphere is the set of all points in the space that are equidistant from a given point called the center of the sphere.

Circle

Sphere

A circle is the set of all the points in a plane that are equidistant from a given point.

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

A spherical lune is the region we find between two great circles of a sphere.

a spherical lune with a general central angle and a radius of r

Find a general formula for the area of a lune when

θ

is measured in degrees.

The surface area of a sphere can be calculated by multiplying 4, π, and the square of the radius. S = 4π r^2 Let's investigate a few spherical lunes with different central angles.

From the diagram above, we can describe the portion of a sphere that the spherical lune represents by using the central angle. Half of a Sphere:& 1/2=180^(∘)/360^(∘) [1em] Third of a Sphere:& 1/3=120^(∘)/360^(∘) [1em] Quarter of a Sphere:& 1/4=90^(∘)/360^(∘) The general formula for a spherical lune with a central angle of θ then becomes as follows. A = θ/360^(∘) * 4π r^2

A Dyson sphere is a hypothetical spherical superstructure that completely encompasses a star for the purpose of capturing the star's total power output.

A scientist has calculated that, in order for a Dyson sphere built around our Sun to not burn up, it has to be at a distance of at least $20$ million kilometers from the surface of the Sun. Suppose we decide to harvest material from the Moon to build one (not a good idea, by the way). How many full size Moons would we need to make a Dyson sphere $1$ kilometer thick?

The radius of the Moon and of the Sun are approximately

1700

kilometers and

700000

kilometers, respectively. Round the answer to three significant digits.

There are a couple of things we must figure out:

The volume of the Dyson sphere.
The volume of the Moon.

Volume of the Dyson Sphere

To determine the volume of the Dyson sphere, we need to know its radius. The radius to the Dyson sphere's inner wall is the sum of the Sun's radius and the minimum distance from the Sun's surface. Let's illustrate this.

If we add these distances, we get the distance from the Dyson sphere's center to its inner wall. 20* 10^6+0.7* 10^6=20.7 * 10^6 km To calculate the volume of the Dyson sphere, we will subtract the volume of the sphere created by the inner wall that faces the Sun from the volume of the sphere created by the outer wall that faces the Earth.

Let's label the volumes of the outer and inner spheres as V_O and V_I. Consequently, we will label their corresponding radii as r_O and r_I. The difference between these volumes is the volume of the Dyson sphere V_(DS). We can write the following expression. V_(DS)= V_O- V_I ⇓ V_(DS)= 4/3π (r_O)^3- 4/3π (r_I)^3 ⇓ V_(DS)=4/3π ((r_O)^3-(r_I)^3) We know that the inner radius is r_I= 20.7* 10^6 kilometers. We also know that the outer radius is 1 kilometer greater than this. Therefore, the outer radius r_O= 20.7* 10^6+1. Now we can form an expression for the volume of the Dyson sphere.

Let's keep the volume in exact form for now.

Volume of the Moon

We know the radius of the Moon. With this information, we can figure out its volume.

We will keep the volume of the Moon in exact form as well.

Number of Moons Needed

Finally, we will divide the volume of the Dyson sphere by the volume of the Moon to calculate how many Moons we would need to make our Dyson sphere. Remember that we will be rounding our number to three significant digits.

We would need about 262 000 Moons to complete the Dyson sphere.

	14 Theory slides
	10 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions

Properties of Spheres

Catch-Up and Review

Proof

Finding the Hemisphere's Cross-Sectional Area

Finding the Cylinder's Cross-Sectional Area

Conclusion

Hint

Solution

Hint

Solution

Hint

Solution

Hint

Solution

Volume of the Pencil's Tip

Volume of the Pencil's Body

Volume of the Pencil's Eraser

Pencil's Volume

Proof

Hint

Solution

Covering of the Baseball

Why

Volume of the Dyson Sphere

Volume of the Moon

Number of Moons Needed

Properties of Spheres

Recommended exercises