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9.25: Surface Area and Volume of Spheres

  • Page ID
    6748
  • 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: \(V=\dfrac{4}{3}\pi r^{3}\)

    f-d_0dcbef245b10b2260b202a695b3958fdbe04cd19c27c9eb245876cd6+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{4}\)

    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

    1. Are there any cross-sections of a sphere that are not a circle? Explain your answer.
    2. List all the parts of a sphere that are the same as a circle.
    3. 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 \(\pi\).

    1. a radius of 8 in.
    2. a diameter of 18 cm.
    3. a radius of 20 ft.
    4. a diameter of 4 m.
    5. a radius of 15 ft.
    6. a diameter of 32 in.
    7. a circumference of \(26\pi \text{ cm}\).
    8. a circumference of \(50\pi\text{ yds}\).
    9. The surface area of a sphere is \(121\pi \text{ in}^{2}\). What is the radius?
    10. The volume of a sphere is \(47916\pi \text{ m}^{3}\). What is the radius?
    11. The surface area of a sphere is \(4\pi \text{ ft}^{2}\). What is the volume?
    12. The volume of a sphere is \(36\pi \text{ mi}^{3}\). What is the surface area?
    13. Find the radius of the sphere that has a volume of \(335 \text{ cm}^{3}\). Round your answer to the nearest hundredth.
    14. 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!