How to Calculate Density of Sphere

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Density is a useful characteristic. Every material has a characteristic density, and none are the same, so you can use density as an identification method. That's how Archimedes managed to determine whether a crown the king had given him was made of gold or not.

Density is defined as mass per unit volume, which means if you want to calculate the density of anything, you have to measure it's mass, then calculate its volume. The density formula is

\rho = \frac{m}{V}

where ρ is the density, m is the mass and V is the volume of the material.

The volume calculation is easy for regular figures, such as cubes, rectangular boxes and pyramids, because all you need to do is measure the dimensions and use a formula. That's also true for spheres.

How to Calculate the Volume of a Sphere

The formula for the volume of a sphere is 4/3 × π_r_3, where r is the sphere's radius. That's pretty straightforward, except in practice, it can be difficult to measure the radius. Even if you have a scaled 2D projection of the sphere to work with, it can still be difficult to pinpoint the center.

It's usually easier to measure the diameter, which is equal to twice the radius. This means r = d/2, so after doing the arithmetic, you can rewrite the volume formula in terms of diameter this way:

V = \frac{1}{6} × πd^3

Mass of a Sphere vs. Weight

There's always a little confusion between mass and weight. Mass, which is the quantity you need to determine density, is a body's inherent inertial resistance to a change in motion, but weight is the force exerted by gravity on the body. Mass can be measured in kilograms, but weight is measured in newtons. In the imperial system, the unit for mass is slugs, while weight is measured in pounds.

The convention is to weigh objects in kilograms in the SI system, which are units of mass, and in pounds in the imperial system, which are units of weight. While performing measurements on the surface of the Earth, it's usually safe to ignore these distinctions, but not in space, where the force of gravity is different.

Calculating the Density of a Sphere

Once you weigh the sphere in question, you have a value for m. Now all you have to do is calculate its volume (V), which you can do if you measure its diameter, d. The density formula is ρ = m/V, and you can rearrange this volume formula to express the relationship in terms of d:

\begin{aligned} \rho &= \frac{m}{(1/6) × πd^3}\\ &=\frac{6m}{πd^3} \end{aligned}

Using Density to Calculate Mass or Volume of a Sphere

Suppose you have a cannonball made completely of iron. You can look up the density of iron in a table: 7.8 g/cm3. You weigh the cannonball and find it weighs 20 lbs. You now have enough information to calculate its volume, so just rearrange the density formula to solve for V: V = m/ρ.

There's just one problem. The density is in CGS metric units and the weight is in imperial units. Depending on whether you want the volume in metric or imperial units, you can either convert the weight to kilograms or you can look up the density in pounds per cubic inch. Use either of these conversions:

1 \;\text{lb} = 0.45359 \;\text{kg, so } 20 \;\text{lbs} = 9.07 \;\text{kg} \\ 7.8 \;\text{g/cm}^3 = 0.28 \;\text{lb/in}^3

Alternatively, you can calculate the weight (mass) of the cannonball if you can measure its diameter. Use this formula:

m = \frac{1}{6}\rhoπd^3