Volume of a Sphere

A sphere is a perfectly round geometrical object in 3-dimensional space. It can be characterized as the set of all points located distance $r$ (radius) away from a given point (center). It is perfectly symmetrical and has no edges or vertices.

Note: The earth is not a sphere! As mentioned above, a sphere has no edges or vertices. But the earth is slightly flattened on the poles, which makes its shape un-sphere-ish. Its shape is given a special name: the geoid.

A sphere with radius $r$ has volume $\frac{4}{3} \pi r^3$ and surface area $4 \pi r^2$.

Interesting fact: Of all shapes with the same surface area, the sphere has the largest volume.

What is the volume of a sphere of radius $5$?

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ANSWER

The volume of a sphere of radius 5 is $\frac{4}{3} \pi \times 5^3 = \frac{ 500 } { 3} \pi = 166 \frac{2}{3} \pi$. $_\square$

If the surface area of a sphere is $144\pi,$ what is the volume of the sphere?

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ANSWER

Observe that the surface area of the sphere can be rewritten as

$$144\pi=4\pi \times 6^2.$$

Then, since the surface area of a sphere with radius $r$ is $4 \pi r^2 ,$ it follows that the radius of the sphere in this problem is $r=6.$ Hence, its volume is

$$\frac{4}{3} \pi r^3 =\frac{4}{3} \pi \times 6^3 =288\pi. \ _\square$$

You have a gold sphere whose volume is $\frac{4\pi}{3} \text{ cm}^3.$ If you want the gold sphere to be twice as large in size, how much more gold do you have to bring to a jeweler in addition to the gold sphere you currently have?

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ANSWER

From the formula $V=\frac{4}{3} \pi r^3$ for the volume of a sphere with radius $r,$ you know that the radius of your gold sphere is $r=1 \text{ cm}.$ Since you want the radius to be $2 \text{ cm},$ the amount of gold required to make the new, bigger sphere is $\frac{4}{3} \pi \times 2^3 =\frac{32\pi }{3}\left(\text{cm}^3\right).$ Hence, the additional amount of gold required is

$$\frac{32 \pi}{3}\text{ cm}^3-\frac{4\pi}{3}\pi \text{ cm}^3=\frac{28\pi}{3} \text{ cm}^3. \ _\square$$

The volume of sphere $a$ is $\frac{1}{27}$ times that of sphere $b.$ The surface area of $a$ is how many times the surface area of $b?$

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ANSWER

Let $R_a$ and $R_b$ denote the radii of spheres $a$ and $b,$ respectively. Then from the formula $V=\frac{4}{3} \pi r^3$ for the volume of a sphere with radius $r,$ we have

$$\frac{4}{3} \pi {R_a}^3=\frac{1}{27}\cdot \frac{4}{3} \pi {R_b}^3,$$

which implies $R_a=\frac{1}{3}R_b.$

Therefore, from the formula $S=4 \pi r^2$ for the surface area of a sphere with radius $r,$ we conclude that the surface area of sphere $a$ is $\left(\frac{1}{3}\right)^2=\frac{1}{9}$ times the surface area of sphere $b.$ $_\square$