STEM Calculators

Slide presentation

Density of Regular Solids

General Chemistry • Matter, Its Properties, and Measurement

Open calculator Open questions
Slide 1 of 1

Matter, properties, and measurement

Regular solids let geometry become part of density measurement

For cubes, rectangular prisms, cylinders, and spheres, volume can be calculated from measured dimensions. Once volume is known, density connects the solid’s mass to the space it occupies.

Learning target

  • Calculate density from mass and geometric volume.
  • Choose the correct volume formula for a regular solid.
  • Use correct units and significant figures.
  • Compare solid materials using density values.
Regular solid density overview A balance measures mass and geometry measurements determine volume for regular solids. measure mass balance reading in grams length height width calculate volume mass and geometry

Why it matters

Density helps identify and compare solid materials

Regular solids are common in laboratory samples, engineering parts, crystal models, and classroom measurement activities. Their simple shapes make them useful for connecting geometry to chemical properties.

Identification

Compare unknown solids

A measured density can help distinguish between materials such as aluminum, copper, glass, and plastic.

Measurement

Connect tools to formulas

A balance gives mass, while a ruler or caliper gives dimensions for the volume formula.

Reasonableness

Check your answer

A dense metal should have a larger density than a low-density plastic for the same shape size.

Geometry
Dimensions give volume when the shape is regular.

then

Density
Mass divided by calculated volume gives the material ratio.

Core concept

For a regular solid, volume comes from shape measurements

Unlike irregular solids, regular solids do not require water displacement. Their volume can be calculated from dimensions because the shape follows a predictable geometric model.

Geometric volume model for a regular solid A rectangular prism shows length, width, and height measurements used to calculate volume. length height width measure dimensions volume is calculated before density is calculated

Two-step reasoning

First, calculate the volume using the correct geometry formula. Second, divide the measured mass by that calculated volume.

\[ d = \frac{m}{V} \]

The density value depends on both the balance measurement and the accuracy of the measured dimensions.

Vocabulary and units

Regular-solid density combines mass units and length-derived volume units

Because volume is calculated from dimensions, the length unit determines the final volume unit.

Quantity or term Meaning Common units Measurement note
Mass, \(m\) Amount of matter in the solid sample. g, kg Measured with a balance; usually reported in grams for introductory chemistry.
Length, width, height Linear dimensions used for rectangular prisms and cubes. cm, mm, m Measure carefully; small errors can affect volume strongly.
Radius, \(r\) Distance from the center of a circular face or sphere to its edge. cm, mm, m For cylinders and spheres, radius is half the diameter.
Volume, \(V\) Space occupied by the regular solid. cm3, mL, m3 If dimensions are in cm, volume is in cm3.
Density, \(d\) or \(\rho\) Mass per unit volume. g/cm3, g/mL, kg/m3 Use units that match the mass and volume measurements.

Unit habit

For solids measured in centimeters, density is commonly reported as g/cm3. Since \(1\ \text{cm}^{3} = 1\ \text{mL}\), g/cm3 and g/mL are numerically equivalent.

Main formulas

Choose the volume formula that matches the solid

The density formula stays the same, but the volume formula changes with the shape.

Density formula

\[ d = \frac{m}{V} \]
Rectangular prism

\(V = lwh\)

Use length, width, and height.

Cube

\(V = s^{3}\)

Use one side length.

Cylinder

\(V = \pi r^{2}h\)

Use radius and height.

Sphere

\(V = \frac{4}{3}\pi r^{3}\)

Use radius only.

Measurement accuracy

When a dimension is squared or cubed, measurement error is amplified. A small error in radius can create a much larger error in cylinder or sphere volume.

Interactive geometry model

Change shape, dimensions, and mass to calculate density

Select a regular solid and adjust the measurements. The model updates the volume formula, volume, density, and visual shape.

Volume 24.0 cm³
Density 4.00 g/cm³
Formula V = lwh

A rectangular prism with volume 24.0 cm³ and mass 96.0 g has density 4.00 g/cm³.

Interactive regular solid density simulation A regular solid shape, dimension labels, volume formula, and relative density meter update with the controls. length 4.0 cm width 3.0 cm height 2.0 cm rectangular prism V = length × width × height relative density high low 4.00 g/cm³

Dynamic relationship

For the same mass, larger volume means lower density

Regular solids make this relationship easy to visualize. If mass stays fixed while dimensions increase, the volume increases and the density decreases.

Choose the fixed mass

For a 90 g solid, density decreases as volume increases because the same mass is spread through more space.

Density versus volume graph for fixed mass Three inverse curves show density decreasing as calculated volume increases for different fixed masses. calculated volume density high low small large 150 g 90 g 30 g same volume comparison

Worked example

Calculate density of a rectangular metal block

A rectangular metal block has a mass of \(96.0\ \text{g}\), length \(4.00\ \text{cm}\), width \(3.00\ \text{cm}\), and height \(2.00\ \text{cm}\). Calculate its density.

  1. Identify the shape and formula.

    The solid is a rectangular prism, so \(V = lwh\).

  2. Calculate the volume.

    \(V = (4.00\ \text{cm})(3.00\ \text{cm})(2.00\ \text{cm}) = 24.0\ \text{cm}^{3}\).

  3. Use the density formula.

    \(d = m/V = 96.0\ \text{g} / 24.0\ \text{cm}^{3}\).

  4. Calculate density.

    \(d = 4.00\ \text{g/cm}^{3}\).

  5. Check significant figures.

    All given measurements have 3 significant figures, so the density is reported with 3 significant figures.

Final answer

The density of the regular solid is \(4.00\ \text{g/cm}^{3}\).

Common misconception

Do not add dimensions to find volume

Length, width, and height are one-dimensional measurements. Volume is three-dimensional, so the dimensions must be multiplied, not added.

Mistake

“The block is \(4.00 + 3.00 + 2.00 = 9.00\ \text{cm}\), so the volume is \(9.00\ \text{cm}\).”

Correction

Volume is \(4.00 \times 3.00 \times 2.00 = 24.0\ \text{cm}^{3}\). The unit is cubed because volume is three-dimensional.

Adding dimensions misconception A comparison shows the incorrect addition of dimensions and the correct multiplication of dimensions for volume. Incorrect 4 + 3 + 2 = 9 cm This is a length-like result. It is not volume. Correct 4 × 3 × 2 = 24 cm³ This is a three-dimensional result. Use it in density. Always check that the volume unit is cubic before calculating solid density.

Practice check

Calculate density from a cylinder’s dimensions

A cylindrical solid has a mass of \(62.8\ \text{g}\), radius \(1.50\ \text{cm}\), and height \(4.00\ \text{cm}\).

Question

Calculate the volume of the cylinder and then calculate its density. Report the density with correct significant figures.

Show answer

Volume: \(V = \pi r^{2}h = \pi(1.50\ \text{cm})^{2}(4.00\ \text{cm}) = 28.274...\ \text{cm}^{3}\).

Density: \(d = 62.8\ \text{g} / 28.274...\ \text{cm}^{3} = 2.221...\ \text{g/cm}^{3}\).

With 3 significant figures, the density is \(2.22\ \text{g/cm}^{3}\).

Reasoning check

The answer has units of g/cm3 because mass is in grams and volume is calculated from centimeter measurements.

Apply the topic

Use geometry before density

For regular solids, density problems often test both measurement reasoning and formula choice. A correct solution starts by identifying the solid shape.

Open calculator Density of Regular Solids Calculator

Calculate density using mass and geometric volume formulas for common regular solids.

 Practice questions Density of Regular Solids Questions

Practice volume formulas, units, significant figures, and density comparisons.

How to apply this topic

Identify the shape, select the volume formula, calculate volume with cubic units, divide mass by volume, then round using significant figures from the measurements.

Final summary

The essential takeaways

Density is mass per unit volume.

Use \(d = m/V\) after the solid’s volume is known.

Regular solids use geometry.

Choose the volume formula that matches the solid’s shape.

Volume units are cubic.

Length measurements in cm produce volume in cm3.

Shape formulas matter.

Rectangular prisms, cubes, cylinders, and spheres use different volume relationships.

Measurement accuracy affects density.

Errors in dimensions can strongly affect calculated volume and density.

Round from measured values.

Final density should reflect the limiting significant figures in the measurements.