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 cap is a portion of a sphere cut off by a plane.

To calculate the volume of the spherical cap, we can use the following formula.

V = \frac{π h}{6} (3 a^{2} + h^{2})

Find the volume of a spherical cap given the following information. Answer in exact form.

r = 5 ft and a = 4 ft

r = 17 ft and h = 9 ft

To calculate the volume of the spherical cap, we need to know the height h and the distance a. V=π h/6(3 a^2+ h^2) We already know the value of a, so we need to determine the height h.

Finding the Height of the Spherical Cap

From the exercise, we have only been given the value of a and the radius of the sphere. Let's add this information to the diagram. We will also label the distance between the center of the sphere and the center of the spherical cap as x.

Notice the right triangle with a hypotenuse of 5 and legs of 4 and x. This is a Pythagorean Triple. In this particular triangle, the shorter leg is 3 feet long. Let's add this information to the diagram.

Now we can determine the value of h as the difference between the sphere's radius and 3. h=5-3= 2

Finding the Volume of the Spherical Cap

Now we have everything we need to calculate the volume of the spherical cap.

The volume of the spherical cap is 52π3 cubic feet.

This time we know the height of the spherical cap. Since we also know the sphere's radius, we can find the distance between the center of the sphere and the center of the spherical cap. 17-9 = 8 ft Let's add this information to the diagram.

Again, we see a right triangle. This time, it has a hypotenuse of 17 and legs of 8 and a. This is also a Pythagorean Triple with legs of 8 and 15 feet. Let's add this information to the diagram.

Now we have everything we need to calculate the volume of the spherical cap.

The volume of the spherical cap is 1134π cubic feet.

The following sphere has a diameter of $4 (x + 3)$ centimeters and a surface area of $1024 π$ square centimeters.

Determine

x .

From the exercise, we know that the sphere has a surface area of 1024 π square centimeters. Let's recall how we calculate the surface area of a sphere. S=4π r^2 We also know from the exercise that the sphere has a diameter of 4(x+3) centimeters. However, we need the radius which is half the diameter. We can find it by dividing the given diameter by 2. r=4(x+3)/2=2(x+3) To find x, we will substitute the radius and surface area into the formula for the surface area of a sphere and then solve for x.

Since 4(x+3) is the diameter of a sphere, it must be nonnegative. Therefore, the value of x must be 5.

A sphere is inscribed in a cube with a volume of $216$ cubic inches.

What is the sphere's volume? Leave your answer in terms of

π .

To determine the volume of a sphere, we use the following formula. V=4/3π r^3 As we can see, we need to know the sphere's radius. We know that the sphere is inscribed in the cube. This means that the side of the cube equals the diameter of the sphere. If we do a cross-section across the midpoint of opposite sides of one of the cube's surfaces, it would look something like below.

Therefore, to find the radius of the sphere, we must first find the side of the cube.

Finding the Side of the Cube

Remember that we have been given the cube's volume. To calculate the volume of a cube, we cube its side length. V=s^3 By substituting 216 for V and solving for s, we can determine the cube's side.

The side of the cube is 6 inches.

Finding the Volume of the Sphere

Since we know the side of the cube, we also know the diameter of the sphere. With this information, we can calculate the radius of the sphere. r=6/2=3 in Now we have everything we need to find the volume of the sphere.

The volume is 36π cubic inches.

Find the radius of a sphere with the given properties.

The surface area in square meters equals the volume in cubic meters.

The ratio of surface area in square meters to volume in cubic meters is

1 : 6 .

Let r be the radius of a sphere where the surface area of the sphere equals its volume in cubic meters.

Let's substitute the formulas for finding the surface area and volume of a sphere in the equation S=V and solve for r.

The radius of a sphere with equal surface area in square meters and volume in cubic meters is 3 meters.

This time, we want to determine the radius of a sphere where the ratio of the surface area to the volume is 1:6. This means that if we divide the surface area by the volume, we should get a ratio of 16.

As in the previous section, we will substitute the formulas used to find the surface area and the volume into this proportionality and solve for r.

Therefore, the radius of the sphere is 18 meters.

	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

Finding the Height of the Spherical Cap

Finding the Volume of the Spherical Cap

Finding the Side of the Cube

Finding the Volume of the Sphere

Properties of Spheres

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