Key takeaway: The volume of a sphere is four thirds times pi times the radius cubed, and it measures all the space inside a perfectly round ball.
Grade 88.G.C.9Geometry and Measurement
Definition of Volume of a Sphere
The volume of a sphere tells how much a ball, a bubble or a spherical tank can hold. A sphere is the set of all points in space at one fixed distance, the radius, from a center point.
A sphere fits snugly inside a cylinder whose height equals its width. The ball takes up two thirds of that cylinder, and this relationship gives the formula. Because the radius is cubed, small changes in size matter a lot: a ball with twice the radius has eight times the volume.
Formula and notation
The volume of a sphere with radius \(r\):
\[V = \frac{4}{3}\pi r^{3}\]
Cube the radius first, then multiply by \(\pi\), then by \(\frac{4}{3}\). If you know the diameter \(d\), halve it to get \(r = \frac{d}{2}\). The result is in cubic units.
Key facts
- Only the radius is needed for the volume of a sphere; it has no faces or edges.
- Doubling the radius multiplies the volume by 2 cubed, which is 8.
- The sphere’s volume is two thirds of its tightest cylinder.
- A hemisphere holds exactly half the volume of the full sphere.
Volume of a Sphere example
A decorative globe has a diameter of 15 inches. Find the volume of air inside it, to the nearest tenth of a cubic inch.
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Radius: \(15 : 2 = 7.5\) inches.
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Cube it: \(7.5^{3} = 421.875\).
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Multiply: \(V = \frac{4}{3}\pi (421.875) = 562.5\pi\).
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Approximate: \(562.5 \times 3.14159 \approx 1767.1\).
The globe holds about 1,767.1 cubic inches, exactly \(562.5\pi\) cubic inches.
Common mistake
Watch out
Using the 15-inch diameter as the radius, Cal gets \(V = \frac{4}{3}\pi (15)^{3} = 4500\pi \approx 14{,}137\) cubic inches.
Why it is wrong
The formula needs the radius, which is half the diameter. Because the radius is cubed, the mistake makes the answer \(2^{3} = 8\) times too big: \(4500\pi : 8 = 562.5\pi\). Write \(r\) next to the number before computing.
Related terms and lessons
Browse the glossary: A–Z list · Grade 8 vocabulary · Geometry and Measurement