Geometry - How to find the diameter of a sphere

Find the diameter of a sphere if it has a volume of $24\pi$ .

Explanation

Recall how to find the volume of a sphere:

$\text{Volume of Sphere}=\frac{4}{3}\pi r^3$ , where is the radius of the sphere.

Now, since the radius is half the diameter, the equation for the volume of a sphere can be rewritten as thus:

$\text{Volume of Sphere}=\frac{4}{3}\pi(\frac{d}{2})^3=\frac{1}{6}\pi d^3$ , where is the diameter of the sphere.

Rewrite the equation to solve for .

$d^3=\frac{6(\text{Volume of Sphere})}{\pi}$

$d=\sqrt[3]{\frac{6(\text{Volume of Sphere})}{\pi}}$

Now, plug in the volume of the sphere to find the diameter.

$d=\sqrt[3]{\frac{6(24\pi)}{\pi}}=5.24$

Find the diameter of the sphere if it has a volume of $39\pi$ .

Explanation

Recall how to find the volume of a sphere:

$\text{Volume of Sphere}=\frac{4}{3}\pi r^3$ , where is the radius of the sphere.

Now, since the radius is half the diameter, the equation for the volume of a sphere can be rewritten as thus:

$\text{Volume of Sphere}=\frac{4}{3}\pi(\frac{d}{2})^3=\frac{1}{6}\pi d^3$ , where is the diameter of the sphere.

Rewrite the equation to solve for .

$d^3=\frac{6(\text{Volume of Sphere})}{\pi}$

$d=\sqrt[3]{\frac{6(\text{Volume of Sphere})}{\pi}}$

Now, plug in the volume of the sphere to find the diameter.

$d=\sqrt[3]{\frac{6(39\pi)}{\pi}}=6.16$

Find the diameter of a sphere if the volume is $\pi$ .

$2\sqrt[3]{\frac{3}{4}}$

$\sqrt[3]{\frac{3}{4}}$

$\frac{1}{2}$

Explanation

Write the formula for the volume of a sphere.

$V=\frac{4}{3}\pi r^3$

Plug in the given volume and solve for the radius.

$\pi=\frac{4}{3}\pi r^3$

$\frac{3}{4}= r^3$

$r=\sqrt[3]{\frac{3}{4}}$

The diameter is double the radius.

$d=2\sqrt[3]{\frac{3}{4}}$

A company wants to construct an advertising balloon spherical in shape. It can afford to buy 28,000 square meters of material to make the balloon. What is the largest possible diameter of this balloon (nearest whole meter)?

Explanation

This is equivalent to asking the diameter of a balloon with surface area 28,000 square meters.

The relationship between the surface area and the radius is:

$A=4\pi r^{2}$

To find the radius, substitute for the surface area, then solve:

$28,000=4\pi r^{2}$

$r^{2}=\frac{28,000}{4\pi }$

$r=\sqrt{\frac{28,000}{4\pi }}\approx47$

To find the diameter , double the radius—this is 94.

Find the diameter of a sphere if it has a volume of $106\pi$ .

Explanation

Recall how to find the volume of a sphere:

$\text{Volume of Sphere}=\frac{4}{3}\pi r^3$ , where is the radius of the sphere.

Now, since the radius is half the diameter, the equation for the volume of a sphere can be rewritten as thus:

$\text{Volume of Sphere}=\frac{4}{3}\pi(\frac{d}{2})^3=\frac{1}{6}\pi d^3$ , where is the diameter of the sphere.

Rewrite the equation to solve for .

$d^3=\frac{6(\text{Volume of Sphere})}{\pi}$

$d=\sqrt[3]{\frac{6(\text{Volume of Sphere})}{\pi}}$

Now, plug in the volume of the sphere to find the diameter.

$d=\sqrt[3]{\frac{6(106\pi)}{\pi}}=8.60$

Make sure to round to places after the decimal.

Find the diameter of a sphere if it has a volume of $650\pi$ .

Explanation

Recall how to find the volume of a sphere:

$\text{Volume of Sphere}=\frac{4}{3}\pi r^3$ , where is the radius of the sphere.

Now, since the radius is half the diameter, the equation for the volume of a sphere can be rewritten as thus:

$\text{Volume of Sphere}=\frac{4}{3}\pi(\frac{d}{2})^3=\frac{1}{6}\pi d^3$ , where is the diameter of the sphere.

Rewrite the equation to solve for .

$d^3=\frac{6(\text{Volume of Sphere})}{\pi}$

$d=\sqrt[3]{\frac{6(\text{Volume of Sphere})}{\pi}}$

Now, plug in the volume of the sphere to find the diameter.

$d=\sqrt[3]{\frac{6(650\pi)}{\pi}}=15.74$

Make sure to round to places after the decimal.

If the volume of a sphere is , what is the sphere's diameter?

$2\sqrt[3]{\frac{3}{2\pi}}$

$2\sqrt[3]{\frac{2\pi}{3}}$

$\sqrt[3]{\frac{4\pi}{3}}$

$\sqrt[3]{\frac{3}{2\pi}}$

$\sqrt[3]{\frac{2\pi}{3}}$

Explanation

Write the formula for the volume of a sphere:

$V=\frac{4}{3}\pi r^3$

Plug in the volume and find the radius by solving for :

$2=\frac{4}{3}\pi r^3$

Start solving for by multiplying both sides of the equation by :

$2\cdot3=\frac{4}{3}\pi r^3\cdot3$

$6=4\pi r^3$

Now, divide each side of the equation by $4\pi$ :

$\frac{6}{4\pi}=\frac{4\pi r^3}{4\pi}$

$\frac{6}{4\pi}=r^3$

Reduce the left side of the equation:

$\frac{3}{2\pi}=r^3$

Finally, take the cubed root of both sides of the equation:

$\sqrt[3]{\frac{3}{2\pi}}=r$

Keep in mind that you've solved for the radius, not the diameter. The diameter is double the radius, which is: $2\sqrt[3]{\frac{3}{2\pi}}$ .

The surface area of a sphere is $100\pi ; in^{2}$ . What is the diameter of the sphere?

Explanation

The surface area of a sphere is given by $SA=4\pi r^{2}$

So the equation to sovle becomes $4\pi r^{2}=100\pi$ or $r^{2}=25$ so

To answer the question we need to find the diameter:

$d=2r=2\cdot 5=10; in$

Find the diameter of a sphere if it has a volume of .

Explanation

Recall how to find the volume of a sphere:

$\text{Volume of Sphere}=\frac{4}{3}\pi r^3$ , where is the radius of the sphere.

Now, since the radius is half the diameter, the equation for the volume of a sphere can be rewritten as thus:

$\text{Volume of Sphere}=\frac{4}{3}\pi(\frac{d}{2})^3=\frac{1}{6}\pi d^3$ , where is the diameter of the sphere.

Rewrite the equation to solve for .

$d^3=\frac{6(\text{Volume of Sphere})}{\pi}$

$d=\sqrt[3]{\frac{6(\text{Volume of Sphere})}{\pi}}$

Now, plug in the volume of the sphere to find the diameter.

$d=\sqrt[3]{\frac{6(100)}{\pi}}=5.76$

Find the diameter of a sphere that has a volume of $69\pi$ .

Explanation

Recall how to find the volume of a sphere:

$\text{Volume of Sphere}=\frac{4}{3}\pi r^3$ , where is the radius of the sphere.

Now, since the radius is half the diameter, the equation for the volume of a sphere can be rewritten as thus:

$\text{Volume of Sphere}=\frac{4}{3}\pi(\frac{d}{2})^3=\frac{1}{6}\pi d^3$ , where is the diameter of the sphere.

Rewrite the equation to solve for .

$d^3=\frac{6(\text{Volume of Sphere})}{\pi}$

$d=\sqrt[3]{\frac{6(\text{Volume of Sphere})}{\pi}}$

Now, plug in the volume of the sphere to find the diameter.

$d=\sqrt[3]{\frac{6(69\pi)}{\pi}}=7.45$

Make sure to round to places after the decimal.

How to find the diameter of a sphere

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