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9: Solid Figures
9.25: Surface Area and Volume of Spheres

9.25: Surface Area and Volume of Spheres

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Jun 15, 2022
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- 9.24: Volume of Cones
- 9.26: Surface Area of Spheres

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Derive and use the formula: volume equals $\dfrac{4}{3}$ times pi times the radius cubed.

Spheres

Consider a solid figure consisting of the set of all points in three-dimensional space that are equidistant from a single point. How could you determine how much two-dimensional and three-dimensional space that figure occupies?

A sphere is the set of all points in three-dimensional space that are equidistant from a single point. The radius of a sphere has one endpoint on the sphere surface and the other endpoint at the center of that sphere. The diameter of a sphere must contain the center.

f-d_d6901d55b629c386c0641d1a93b432f521a443f41b631161a635b57c+IMAGE_TINY+IMAGE_TINY.png — Figure $\PageIndex{1}$

A great circle is the largest circular cross-section in a sphere. The circumference of a sphere is the circumference of a great circle. Every great circle divides a sphere into two congruent hemispheres.

f-d_e8e831771893961ae1e76f94c46db5cb4c08a8d5804f710dea4ddf93+IMAGE_TINY+IMAGE_TINY.png — Figure $\PageIndex{2}$

Surface Area

Surface area is a two-dimensional measurement that is the total area of all surfaces that bound a solid. The basic unit of area is the square unit.

Surface Area of a Sphere: $SA=4\pi r^{2}$

f-d_0dcbef245b10b2260b202a695b3958fdbe04cd19c27c9eb245876cd6+IMAGE_TINY+IMAGE_TINY.png — Figure $\PageIndex{3}$

Volume

To find the volume of any solid you must figure out how much space it occupies. The basic unit of volume is the cubic unit.

Volume of a Sphere: $V=\dfrac{4}{3}\pi r^{3}$

Example $\PageIndex{1}$

Find the surface area of the figure below, a hemisphere with a circular base.

f-d_1c8a850f4b14b8e2d4c602187936454a1578ba198d1fed823ebd9a20+IMAGE_TINY+IMAGE_TINY.png — Figure $\PageIndex{5}$

Solution

Use the formula for surface area:

$\begin{aligned} SA&=\pi r^{2} +124\pi r^{2} \\ &=\pi (6^{2})+2\pi (6^{2}) \\&=36\pi +72\pi =108\pi \text{ cm}^{2}\end{aligned}$

Example $\PageIndex{2}$

A sphere has a volume of 14,137.167 ft3. What is the radius?

Solution

Use the formula for volume, plug in the given volume and solve for the radius, r:

$\begin{aligned} V&=\dfrac{4}{3}\pi r^{3} \\ 14,137.167&=\dfrac{4}{3}\pi r^{3} \\ \dfrac{3}{4\pi} \cdot 14,137.167&=r^{3} \\ 3375&\approx r^{3}\end{aligned}$

At this point, you will need to take the cubed root of 3375. Your calculator might have a button that looks like \sqrt[3]{ }, or you can use $3375^{\dfrac{1}{3}}$ .

$\sqrt[3]{3375}=15\approx r$

Example $\PageIndex{3}$

The circumference of a sphere is 26\pi feet. What is the radius of the sphere?

Solution

The circumference is referring to the circumference of a great circle.

Use $C=2\pi r$ :

$\begin{aligned} 2\pi r&=26\pi \\ r&=13\text{ ft}\end{aligned}$

Example $\PageIndex{4}$

Find the surface area of a sphere with a radius of 14 feet.

Solution

Use the formula for surface area:

$SA=4\pi (14)^{2}$

$=784\pi \text{ ft}^{2}$ \)

Example $\PageIndex{5}$

Find the volume of a sphere with a radius of 6 m.

Solution

Use the formula for volume:

$\begin{aligned} V&=\dfrac{4}{3}\pi 63 \\ &=\dfrac{4}{3}\pi (216) \\ &=288\pi \text{ m}^{3}\end{aligned}$

Review

Are there any cross-sections of a sphere that are not a circle? Explain your answer.
List all the parts of a sphere that are the same as a circle.
List any parts of a sphere that a circle does not have.

For 4 - 11, find the surface area and volume of a sphere with the given dimension. Leave your answer in terms of $\pi$ .

a radius of 8 in.
a diameter of 18 cm.
a radius of 20 ft.
a diameter of 4 m.
a radius of 15 ft.
a diameter of 32 in.
a circumference of $26\pi \text{ cm}$ .
a circumference of $50\pi\text{ yds}$ .
The surface area of a sphere is $121\pi \text{ in}^{2}$ . What is the radius?
The volume of a sphere is $47916\pi \text{ m}^{3}$ . What is the radius?
The surface area of a sphere is $4\pi \text{ ft}^{2}$ . What is the volume?
The volume of a sphere is $36\pi \text{ mi}^{3}$ . What is the surface area?
Find the radius of the sphere that has a volume of $335 \text{ cm}^{3}$ . Round your answer to the nearest hundredth.
Find the radius of the sphere that has a surface area $225\pi \text{ ft}^{2}$ .

Find the surface area and volume of the following shape. Leave your answers in terms of \pi .

f-d_1a4017f7ca67016ee13cb88aaf6770f13677b6279e728394fe4c59da+IMAGE_TINY+IMAGE_TINY.png — Figure $\PageIndex{6}$

Review (Answers)

To see the Review answers, open this PDF file and look for section 11.7.

Vocabulary

Term	Definition
diameter	A chord that passes through the center of the circle. The length of a diameter is two times the length of a radius.
Sphere	A sphere is a round, three-dimensional solid. All points on the surface of a sphere are equidistant from the center of the sphere.
Radius	The radius of a circle is the distance from the center of the circle to the edge of the circle.
Volume	Volume is the amount of space inside the bounds of a three-dimensional object.
Cavalieri's Principle	States that if two solids have the same height and the same cross-sectional area at every level, then they will have the same volume.

Additional Resources

Interactive Element

Video: Spheres Principles - Basic

Activities: Spheres Discussion Questions

Study Aids: Spheres Study Guide

Practice: Surface Area and Volume of Spheres

Real World: Where We Live!

Spheres

Surface Area

Volume

Review

Review (Answers)

Vocabulary

Additional Resources

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