Density of regular solids means determining how much mass is packed into a known geometric volume (cube, rectangular prism, cylinder, sphere, cone, etc.).
Core definition
Density is mass per unit volume: \[ \rho = \frac{m}{V}. \]
Common units: kg/m3, g/cm3. Consistency of units for \(m\) and \(V\) is essential.
Step-by-step method for any regular solid
1) Identify the solid and measure its dimensions. Examples: cube side \(a\), cylinder radius \(r\) and height \(h\), sphere radius \(r\), rectangular prism sides \(L,W,H\).
2) Compute the volume \(V\) using the correct geometric formula. (A reference table is provided below.)
3) Measure or obtain the mass \(m\). Use a balance/scale; keep units consistent (e.g., grams with cm3, kilograms with m3).
4) Compute density.
\[ \rho = \frac{m}{V}. \]
Volume formulas for common regular solids
| Solid | Typical dimensions | Volume \(V\) | Density expression \( \rho = \frac{m}{V} \) |
|---|---|---|---|
| Cube | Side \(a\) | \(V = a^3\) | \(\rho = \frac{m}{a^3}\) |
| Rectangular prism | \(L, W, H\) | \(V = LWH\) | \(\rho = \frac{m}{LWH}\) |
| Cylinder | Radius \(r\), height \(h\) | \(V = \pi r^2 h\) | \(\rho = \frac{m}{\pi r^2 h}\) |
| Sphere | Radius \(r\) | \(V = \frac{4}{3}\pi r^3\) | \(\rho = \frac{m}{\frac{4}{3}\pi r^3}\) |
| Cone | Base radius \(r\), height \(h\) | \(V = \frac{1}{3}\pi r^2 h\) | \(\rho = \frac{m}{\frac{1}{3}\pi r^2 h}\) |
| Right circular pyramid (square base) | Base side \(a\), height \(h\) | \(V = \frac{1}{3}a^2 h\) | \(\rho = \frac{m}{\frac{1}{3}a^2 h}\) |
| General prism | Base area \(A_{\text{base}}\), height \(h\) | \(V = A_{\text{base}} h\) | \(\rho = \frac{m}{A_{\text{base}} h}\) |
Worked example (cylinder): density from measurements
Consider a right circular cylinder with measured radius \(r = 2.50 \text{ cm}\), height \(h = 10.0 \text{ cm}\), and mass \(m = 420 \text{ g}\).
Compute the volume.
\[ V = \pi r^2 h = \pi (2.50\ \text{cm})^2 (10.0\ \text{cm}) = \pi \cdot 6.25\ \text{cm}^2 \cdot 10.0\ \text{cm} = 62.5\pi\ \text{cm}^3. \]
\[ V \approx 62.5 \cdot 3.1416\ \text{cm}^3 \approx 196.35\ \text{cm}^3. \]
Compute the density.
\[ \rho = \frac{m}{V} = \frac{420\ \text{g}}{196.35\ \text{cm}^3} \approx 2.14\ \text{g/cm}^3. \]
Unit check
Using grams and cubic centimeters produces g/cm3. Converting to kg/m3: \[ 2.14\ \text{g/cm}^3 = 2140\ \text{kg/m}^3. \]
Composite solids and average density
For an object made of multiple regular solids (or pieces) with total mass equal to the sum of the masses and total volume equal to the sum of the volumes, the average density is:
\[ \rho_{\text{avg}} = \frac{m_1 + m_2 + \cdots + m_n}{V_1 + V_2 + \cdots + V_n}. \]
This form is also used for an object with internal voids (subtract the void volume from the outer volume) as long as the mass corresponds to the material actually present.
Visualization: mass–volume relationship and slope as density
Density as the slope of a mass–volume graph
Common pitfalls (quick checks)
- Mixed units: convert all lengths to the same unit before computing volume; keep mass units consistent with the desired density unit.
- Radius vs diameter: for circles, \(r = \frac{d}{2}\) must be used in \( \pi r^2 \).
- Hollow objects: use the material volume (outer volume minus inner void volume).
- Significant figures: report density with precision consistent with the measurements of \(m\) and the dimensions.