mirror of
https://github.com/HananoshikaYomaru/openfreebooks.git
synced 2026-11-12 20:45:52 +00:00
77 lines
2.2 KiB
HTML
77 lines
2.2 KiB
HTML
<div class="book-prose">
|
|
<p>
|
|
Mensuration in 3D focuses on volume (space inside solids) and surface area (total outer area).
|
|
</p>
|
|
|
|
<h2 id="units" class="book-prose__heading">Units and interpretation</h2>
|
|
<ul>
|
|
<li>Length: cm, m</li>
|
|
<li>Area: cm\(^2\), m\(^2\)</li>
|
|
<li>Volume: cm\(^3\), m\(^3\)</li>
|
|
</ul>
|
|
<p>
|
|
Always keep units consistent before substituting values.
|
|
</p>
|
|
|
|
<h2 id="prisms-cylinders" class="book-prose__heading">Prisms and cylinders</h2>
|
|
<p>
|
|
For any prism/cylinder:
|
|
</p>
|
|
<p class="book-formula">\[
|
|
\text{Volume}=\text{base area}\times\text{height}.
|
|
\]</p>
|
|
<p>
|
|
Cylinder formulas:
|
|
</p>
|
|
<p class="book-formula">\[
|
|
V=\pi r^2h,\qquad
|
|
\text{TSA}=2\pi r^2+2\pi rh.
|
|
\]</p>
|
|
|
|
<h2 id="pyramids-cones" class="book-prose__heading">Pyramids and cones</h2>
|
|
<p class="book-formula">\[
|
|
V_{\text{pyramid}}=\frac13(\text{base area})h,\qquad
|
|
V_{\text{cone}}=\frac13\pi r^2h.
|
|
\]</p>
|
|
<p>
|
|
Cone curved surface area:
|
|
</p>
|
|
<p class="book-formula">\[
|
|
\text{CSA}_{\text{cone}}=\pi rl
|
|
\]</p>
|
|
<p>
|
|
where \(l\) is slant height. Total surface area adds base area \(\pi r^2\).
|
|
</p>
|
|
|
|
<h2 id="spheres" class="book-prose__heading">Spheres</h2>
|
|
<p class="book-formula">\[
|
|
V=\frac43\pi r^3,\qquad
|
|
\text{Surface area}=4\pi r^2.
|
|
\]</p>
|
|
|
|
<h2 id="composite-solids" class="book-prose__heading">Composite solids</h2>
|
|
<p>
|
|
Split complex shapes into known solids, then add/subtract volumes or areas appropriately.
|
|
</p>
|
|
<ul>
|
|
<li>Add volumes when parts are joined.</li>
|
|
<li>Subtract volumes for holes/cut-outs.</li>
|
|
<li>For surface area, remove hidden contact faces.</li>
|
|
</ul>
|
|
|
|
<h2 id="density-context" class="book-prose__heading">Volume in context</h2>
|
|
<p>
|
|
Common contexts include capacity, material usage, and mass:
|
|
</p>
|
|
<p class="book-formula">\[
|
|
\text{mass}=\text{density}\times\text{volume}.
|
|
\]</p>
|
|
|
|
<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
|
|
<ul>
|
|
<li>Sketch and label all dimensions before formula use.</li>
|
|
<li>Choose exact \(\pi\) form first; round at final step if required.</li>
|
|
<li>Convert units early (e.g., cm to m).</li>
|
|
<li>For composite shapes, explain add/subtract logic briefly.</li>
|
|
</ul>
|
|
</div>
|