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Spheres

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What if you were given a solid figure consisting of the set of all points, in three-dimensional space, that are equidistant from a point? How could you determine how much two-dimensional and three-dimensional space that figure occupies? After completing this Concept, you'll be able to find the surface area and volume of a sphere.

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Spheres CK-12

Guidance

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

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.

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. The best way to understand the surface area of a sphere is to watch the link by Russell Knightley, http://www.rkm.com.au/ANIMATIONS/animation-Sphere-Surface-Area-Derivation.html

Surface Area of a Sphere: SA=4 \pi r^2 .

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. To see an animation of the volume of a sphere, see http://www.rkm.com.au/ANIMATIONS/animation-Sphere-Volume-Derivation.html by Russell Knightley.

Volume of a Sphere: V=\frac{4}{3} \pi r^3 .

Example A

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

The circumference is referring to the circumference of a great circle. Use C = 2 \pi r .

2 \pi r &= 26\pi\\r &= 13 \ ft.

Example B

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

Use the formula.

SA &= 4\pi (14)^2\\&= 784 \pi \ ft^2

Example C

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

Use the formula for volume:

V &= \frac{4}{3} \pi 6^3\\&= \frac{4}{3} \pi (216)\\&= 288 \pi \ m^3

Spheres CK-12

Guided Practice

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

2. The surface area of a sphere is 100 \pi \ in^2 . What is the radius?

3. A sphere has a volume of 14,137.167 \ ft^3 . What is the radius?

Answers:

1. Use the formula for surface area.

SA &= \pi r^2+\frac{1}{2} 4 \pi r^2\\&= \pi (6^2 )+2 \pi (6^2)\\&= 36 \pi +72 \pi =108 \pi \ cm^2

2. Use the formula for surface area.

SA &= 4 \pi r^2\\100 \pi &= 4 \pi r^2\\25 &= r^2\\5 &= r

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

V &= \frac{4}{3} \pi r^3\\14,137.167 &= \frac{4}{3} \pi r^3\\\frac{3}{4 \pi} \cdot 14,137.167 &= r^3\\3375 \approx  r^3

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 do 3375^{\frac{1}{3}} .

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

Practice

  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.

Find the surface area and volume of a sphere with: (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 \ cm .
  8. a circumference of 50 \pi \ yds .
  9. The surface area of a sphere is 121 \pi \ in^2 . What is the radius?
  10. The volume of a sphere is 47916 \pi \ m^3 . What is the radius?
  11. The surface area of a sphere is 4 \pi \ ft^2 . What is the volume?
  12. The volume of a sphere is 36 \pi \ mi^3 . What is the surface area?
  13. Find the radius of the sphere that has a volume of 335 \ cm^3 . Round your answer to the nearest hundredth.
  14. Find the radius of the sphere that has a surface area 225 \pi \ ft^2 .

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

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