Nets & surface area

Scale factor lab

Volume & displacement

<p class="paragraph-header">Mass, covering and capacity</p>
<p>Mass \(=\text{density}\times\text{volume}\). Equal-density solid objects therefore have the same mass and volume ratios. If densities differ, include the density ratio separately. Coating cost follows corresponding coated area only when the same price per unit area applies.</p>
<p>For a joined solid, add component volumes but count only exposed faces for surface area. A face glued to another face is internal and is not coated.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="width: 100%;"><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Solid</strong></th><th style="text-align: center; --cell-justify: center"><strong>Volume</strong></th><th style="text-align: center; --cell-justify: center"><strong>Curved area</strong></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">Cylinder</td><td style="text-align: center; --cell-justify: center">\(\pi r^2h\)</td><td style="text-align: center; --cell-justify: center">\(2\pi rh\)</td></tr><tr><td style="text-align: center; --cell-justify: center">Hemisphere</td><td style="text-align: center; --cell-justify: center">\(\dfrac23\pi r^3\)</td><td style="text-align: center; --cell-justify: center">\(2\pi r^2\)</td></tr></tbody></table></div></div>
<p>A hemisphere’s circular flat face has area \(\pi r^2\); add it only if exposed. Use matching length units before cubing, and retain exact \(\pi\) expressions until final rounding.</p>

Mass \(=\text{density}\times\text{volume}\). Equal-density solid objects therefore have the same mass and volume ratios. If densities differ, include the density ratio separately. Coating cost follows corresponding coated area only when the same price per unit area applies.

For a joined solid, add component volumes but count only exposed faces for surface area. A face glued to another face is internal and is not coated.

A hemisphere’s circular flat face has area \(\pi r^2\); add it only if exposed. Use matching length units before cubing, and retain exact \(\pi\) expressions until final rounding.

Physical assumptions and joined solids

Physical assumptions and joined solids

Solid	Volume	Curved area
Cylinder	\(\pi r^2h\)	\(2\pi rh\)
Hemisphere	\(\dfrac23\pi r^3\)	\(2\pi r^2\)