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Extending to Three Dimensions

Properties of Spheres

Using the formulas for the volume of a cylinder or volume of a cone, the formula to find the volume of a sphere can be obtained. Even cooler, in this lesson, the formula for finding the surface area of a sphere will be developed.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Explore

Cone and Hemisphere Volumes

Consider a hemisphere and a cone that have the same radius and the height of the cone is also Use the cone to fill up the hemisphere with water.
Animation to fill a hemisphere using a cone
What is the relationship between the volume of the full sphere and the volume of the given cone?

Discussion

Spheres

A sphere is the set of all points in the space that are equidistant from a given point called the center of the sphere.
Sphere
The distance from the center to any point on the sphere is called the radius of the sphere.

Discussion

Volume of a Sphere: Developing its Formula

In the exploration, it was seen that it took two cones with radius and height to fill up a hemisphere of radius Therefore, the volume of the hemisphere is twice the volume of the cone. Since the hemisphere is half of a sphere, the volume of a sphere is twice the volume of the hemisphere. Therefore, the volume of a sphere with radius is four times the volume of a cone with radius and height
1. Volume of a hemisphere twice the volume of a cone. 2. Volume of a sphere is four times the volume of a cone
Additionally, the volume of the sphere can be connected to the volume of a cylinder. To do so, remember that the volume of a cone is one-third the volume of a cylinder with same radius and height. Consequently, the volume of a sphere is four-thirds the volume of a cylinder with radius and height
1. Volume of a cone is one-third the volume of a cylinder. 2. volume of a sphere is four-thirds the volume of a cylinder
Finally, the volume of a cylinder with height is given by the formula Substituting this expression into the last equation, an explicit formula to calculate the volume of a sphere is obtained.

Now, the formula for the volume of a sphere will be rigorously proven.

Discussion

Formula for the Volume of a Sphere

The volume of a sphere with radius is four-thirds the product of pi and the radius cubed.

Volume of a Sphere

Proof

Cavalieri's Principle will be used to show that the formula for the volume of a sphere holds. For this purpose, consider a hemisphere and a right cylinder with a cone removed from its interior, each with the same radius and height.

Cylinder, cone and hemisphere
Now, consider a plane that cuts the solids at a height and is parallel to the bases of the solids.
solids cut by a parallel plane
The area of each cross-section will be calculated one at a time.

Finding the Hemisphere's Cross-Sectional Area

Draw a right triangle with height base and hypotenuse Here, is the distance between the center of the base of the hemisphere and the center of the cross sectional circle, is the radius of the cross sectional circle, and is the radius of the hemisphere.

hemisphere and its horizontal cross-section
Using the Pythagorean Theorem, an expression for can be found.
Solve for
Since is a distance, only the principal root is considered. Therefore, the area of the circular cross-section can be found using the formula for area of a circle. For consistency, will be used in place of in the standard formula.
This equation gives the area of the cross-section of the hemisphere at altitude

Finding the Cylinder's Cross-Sectional Area

The area of the cross-section of the cylinder can be found similarly. The cross-section's area is equal to the area between two circles. Since the height and the radius of the cylinder are equal, an isosceles right triangle can be formed inside the cylinder. Therefore, the radius of the smaller circle is also
Cross Section of a Cylinder cut by a cone
Now that the radii of the circles are known, the area of the cross-section can be calculated. It is the difference between the area of the greater circle and the area of the smaller circle.
The area of the cross-section of the cylinder at altitude can be found by using the above equation.

Conclusion

It can be stated that both solids have the same cross-sectional area at every altitude. Moreover, they have the same height. By Cavalieri's Principle, the hemisphere and the cylinder with a cone removed from its interior have the same volume.

Hemisphere volume and cylinder section with same volume
If the volume of the cone is subtracted from the volume of the cylinder, the volume of the hemisphere can be found.
Simplify right-hand side
Finally, by multiplying the volume of the hemisphere by the formula for the volume of a sphere will be obtained.

The formula for volume of a sphere has been developed and proven. It can now be put to use to solve a real-world problem!

Example

Volume of a Water Tank

Last weekend, Tearrik installed a spherical water tank for his house.

Spherical Tank of Water

If the tank has a radius of feet, what is the maximum amount of water it can hold? Round the answer to two decimal places.

Hint

Find the volume of the tank.

Solution

Finding the largest amount of water the tank can hold is the same as the finding its volume. Since it has a spherical shape, its volume is four-thirds of pi multiplied by the radius cubed. It is given that the radius of the tank is feet. To find the volume, all that is needed is to substitute this value into the volume formula and solve.
Simplify
Tearrik's tank can hold about cubic feet of water.

Example

Earth's Radius

The volume of the Earth is approximately cubic miles.

Planet Earth

Assuming the Earth is spherical, what is the Earth's radius? Round the answer to one decimal place.

Hint

Solve the volume formula of the sphere for the radius and substitute the given volume.

Solution

Begin by writing the formula to find the volume of a sphere. Since the volume is given and the radius is needed, solve the formula for
Solve for
Next, to find the Earth's radius, substitute the given volume into the derived formula and simplify.
Consequently, the Earth has a radius of approximately miles.
Knowing how to find the volume of different geometric solids helps to model some scenarios in the real world. Although geometric solids could not perfectly model real-world objects, these geometric solids can provide an approximation that yields valuable information about the scenario.

Example

School Competition

Davontay needs to fill a cylindrical bucket in a school competition by throwing water balloons from a distance of feet. Each balloon has a spherical shape with a radius of inches. The radius of the bucket is inches, with a height of inches.
Man throwing balloons
What is the minimum number of balloons needed to fill up the bucket?

Hint

Find the volume of the bucket and the volume of each balloon. Then, divide the volume of the bucket by the volume of a balloon.

Solution

To determine the minimum number of balloons needed to fill up the bucket, first solve for the volume of the bucket and the volume of each balloon. To do so, recall the formulas to find the volume of a cylinder and a sphere.

Volume of a Cylinder Volume of a Sphere
Using this information, to find the volume of the bucket, and can be substituted into the formula for the volume of a cylinder.
Simplify right-hand side
Next, substitute into the formula for the volume of a sphere to find the volume of each balloon.
Simplify right-hand side
Now, to determine how many water balloons will fill up the bucket, divide the volume of the bucket by the volume of a balloon. Simplifying the right-hand side will give the number of balloons needed.
Simplify right-hand side
By multiplying the last equation by the equation is obtained. This indicates that water balloons will fill up the bucket. Remember, what has been determined is the minimum number of balloons needed, but there could be a need for more if, for instance, Davontay misses some shots.

Example

Volume of a Pencil

Ramsha bought a standard pencil whose radius is millimeters and the length, not including the eraser, is millimeters. After a good sharpening, the tip turned into a millimeter high cone.

Pencil

Assuming the eraser is half of a sphere, what is the volume of the pencil? Round the answer to two decimal places.

Hint

The pencil is made of a cone, a cylinder, and a hemisphere, all with the same radius. Therefore, the volume of the pencil equals the sum of the volumes of each solid.

Solution

The given pencil can be seen to consist of a cone, a cylinder, and half of a sphere — all with the same radius.
Parts of the Pencil
Consequently, the volume of the pencil equals the sum of the volumes of each of these solids.
Volume of a Cone Volume of a Cylinder Volume of a Hemisphere

Volume of the Pencil's Tip

The tip of the pencil has a radius of millimeters and a height of millimeters. Substituting these values into the first formula will give the volume of the tip.
Simplify right-hand side
The pencil's tip has a volume of cubed millimeters.

Volume of the Pencil's Body

The body of the pencil has a radius of millimeters. To find the height of this cylinder, subtract the height of the pencil's tip from the original length of the pencil. Next, substitute and into the formula for a cylinder's volume.
Simplify right-hand side
It has been found that the pencil's body has a volume of cubed millimeters.

Volume of the Pencil's Eraser

The eraser is a hemisphere with a radius of Therefore, to find its volume, substitute into the hemisphere volume formula.
Simplify right-hand side
Consequently, the pencil's eraser has a volume of cubed millimeters.

Pencil's Volume

Finally, the volume of the pencil is equal to the sum of the volume of its parts.
Substitute values
Rewrite
Simplify
In conclusion, the volume of the Ramsha's pencil is approximately cubed millimeters.

Discussion

Formula for the Surface Area of a Sphere

Consider the fact that baseballs commonly have a radius of inches. With this information, the volume of a baseball can be calculated, right? What about the surface area? How much leather is needed to make the lining of a baseball?

Baseball

To answer these question, the following formula can be used.

Rule

Surface Area of a Sphere

The surface area of a sphere with radius is four times pi multiplied by the radius squared.

Surface area of a Sphere

Proof

Informal Justification

The aim of this proof is to consider areas and volumes of known figures to approximate the ratio of the surface area of the sphere to the volume of the sphere. Suppose that a sphere is filled with congruent pyramids. Consider that the area of the base of one of those pyramids is

Pyramid inside the sphere
There is a formula for the volume of a pyramid using its height and the area of the base. Notice that since the base of the pyramid lies on the surface of the sphere, the height of the pyramid is the radius Now the ratio of the area of the base of the pyramid to its volume can be obtained by dividing over the formula for the volume.
Evaluate
It should be noted that this ratio is equal to the ratio of the areas of the bases of congruent pyramids to their volumes.
Evaluate
Since the sphere is filled with these pyramids, the sum of the areas of their bases is the same as the surface area of the sphere. Also, the sum of the volumes of all pyramids is approximately equal to the volume of the sphere. Due to those relationships, the ratio of the surface area of the sphere to its volume is the same as the ratio of the areas of the bases of the pyramids to their volumes. Since the formula for the volume of a sphere is known, it can be substituted in this ratio to find the formula for the surface area of the sphere.
Evaluate
It should be noted that this is an informal justification for this formula and not a formal proof.

Example

Surface Area of a Baseball

Find the amount of leather needed to make the lining of a baseball that has a radius of inches. Round the answer to one decimal place.

Baseball

Hint

Use the formula to find the surface area of a sphere.

Solution

The amount of leather needed to make the lining of a baseball is the same as the surface area of the baseball. Therefore, the following formula can be used. Next, substitute into the previous formula.
Substitute for and evaluate
Consequently, to make the lining of a baseball, about square inches of leather are needed.

Covering of the Baseball

Notice that the surface area of a sphere with radius is four times the area of a circle with the same radius as the sphere. This relationship can be roughly seen in the covering of a baseball.

Covering of a Baseball

Pop Quiz

Finding the Volume or Surface Area of Different Spheres

In the following applet, calculate either the volume or the surface area of the given sphere and round the answer to two decimal places.

Random Spheres

Illustration

Napkin Ring Problem

In talking aboutspheres, there is a fascinating case called the Napkin Ring Problem. Consider two solids that have spherical shapes, for example, a soccer ball and the planet Earth. Each is represented in the diagram, along with its respective diameter.

Soccer ball and the Earth
Imagine each of these objects is cut with a horizontal plane centimeters above the center of the sphere. Then, imagine making a second cut, this time centimeters below the center of each sphere. Both the soccer ball and the Earth will be left with their own respective solid with a circular base. Additionally, the heights of each solid are the same.
Soccer ball and the Earth being cut
Next, bore a cylindrical hole through the center of each solid so that the radius of each cylinder is the radius of the corresponding circular base. The resulting solids are called napkin rings. Note that both cylinders have the same height.
Two napkin rings with the same height

The interesting fact about these two napkin rings is that, since they have the same height, they have exactly the same volume.

Napkin rings from another view

Why

Showing the Volumes are Equal
To show that the two napkin rings have the same volume, the Cavalieri's Principle will come into action. Start by considering a cross-section of each napkin ring at the same height.
Cross-Section of two Napkin Rings with the same height
The area of each cross-section is equal to the area of the outer circle minus the area of the inner circle. For a moment, focus all the attention on only one of the cross-sections. Let be the height of the cylinder, be the radius of the sphere, and be the height above the center at which the cross-section was made.
Top and Front View of the Cross-Section

From the diagram, the radius of the inner circle is which is equal to Since is a right triangle, by the Pythagorean Theorem, can be found.

Also, is the radius of the outer circle and applying the Pythagorean Theorem to the right triangle an expression for it can be deducted.

Having the two radii, the area of the cross-section can be found.
Substitute values and simplify
As might be seen, the area of the cross-section does not depend on the radius of the sphere, it depends only on the height of the napkin ring and the height at which the cross-section was made. Therefore, finding the area of the second cross-section will produce the same expression.
Cross-Section of two Napkin Rings with the same height
Consequently, since both napkin rings have the same height and the same cross-sectional area at every altitude, by Cavalieri's Principle, the two napkin rings have the same volume.

Closure

Relationship Between Circles and Spheres

Briefly describe a circle and a sphere. Write the definitions side by side.

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.

As can be seen, the two definitions are identical except for the fact that a circle is a two-dimensional figure while a sphere is three-dimensional. But this is not all. There is one more gnarly relationship. To see it, consider a circle and draw a line containing one diameter.

A Circle and Its diameter
Next, rotate the circle about this line to see what figure is formed.
Rotating a Circle about a Diameter
As can be seen, when a circle is rotated about a diameter, it generates a sphere.
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