It doesn't look like an O'level explanation at all 

It doesn't looks like a O' level explanation at all 

<h2>Volume of Pyramids, Cones and Spheres</h2>
<div>
<table>
<tbody>
<tr>
<td style="width: 87px;">
<p><strong>Square pyramid</strong></p>
</td>
<td style="width: 246px;"><img alt="23da4349df92b529dc49e3a0bee65cf7.png" src="https://data.ulearngo.com/assets/3073c9d9-fa7a-4195-9d68-4fed296c6598"/></td>
<td style="width: 654px;">$\begin{array}{rl} \text{Volume}&amp; = \cfrac{1}{3}\times \text{area of base} \times \\ &amp; \text{height of pyramid}\\ &amp; = \cfrac{1}{3}\times {b}^{2}\times H \end{array}$</td>
</tr>
<tr>
<td style="width: 87px;">
<p><strong>Triangular pyramid</strong></p>
</td>
<td style="width: 246px;"><img alt="fc309fbc970e05e7a137139e655909d0.png" src="https://data.ulearngo.com/assets/45b43b76-efb3-4fc1-9652-167398024b18"/></td>
<td style="width: 654px;">$\begin{array}{rl} \text{Volume} &amp; = \cfrac{1}{3}\times \text{area of base} \times \\ &amp; \text{height of pyramid}\\ &amp; = \cfrac{1}{3} \times \cfrac{1}{2}bh \times H \end{array}$</td>
</tr>
<tr>
<td style="width: 87px;">
<p><strong>Right cone</strong></p>
</td>
<td style="width: 246px;"><img alt="48165c0ece92cd4f11f0a725c4db9bdd.png" src="https://data.ulearngo.com/assets/58cbeeb8-7e32-4bb8-b886-5cc4c07cc06f"/></td>
<td style="width: 654px;">$\begin{array}{rl} \text{Volume}&amp; = \cfrac{1}{3}\times \text{area of base} \times \\ &amp; \text{height of cone}\\ &amp; = \cfrac{1}{3}\times \pi {r}^{2}\times H \end{array}$</td>
</tr>
<tr>
<td style="width: 87px;">
<p><strong>Sphere</strong></p>
</td>
<td style="width: 246px;"><img alt="ad6f55dab33291d980ecfd5daea07d15.png" src="https://data.ulearngo.com/assets/f8c123fc-9c61-4954-ace7-35e6294eccd6"/></td>
<td style="width: 654px;">
<p>$\text{Volume}=\cfrac{4}{3}\pi {r}^{3}$</p>
</td>
</tr>
</tbody>
</table>
</div>
<div class="note visit">
<p>This video gives an example of calculating the volume of a sphere.</p>
<iframe allowfullscreen="allowfullscreen" frameborder="0" height="360" src="https://www.youtube.com/embed/IelS2vg7JO8?rel=0&amp;modestbranding=1&amp;cc_lang_pref=en&amp;cc_load_policy=1" width="640"></iframe></div>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<h3>Question</h3>
<div>
<p>Find the volume of a square pyramid with a height of 3 cm and a side length of 2 cm.</p>
</div>
<div>
<h3>Sketch and label the pyramid</h3>
<img alt="6718f60833199a452680f54daab7cc23.png" src="https://data.ulearngo.com/assets/91abfa42-eff9-4b75-af33-30c606aa1795"/></div>
<div>
<h3>Select the correct formula and substitute the given values</h3>

\[V = \cfrac{1}{3}\times {b}^{2}\times H\]

<p>We are given $b=2$ and $H=3$, therefore</p>

\begin{align*} V &amp; = \cfrac{1}{3}\times {2}^{2}\times 3 \\ &amp; = \cfrac{1}{3}\times 12 \\ &amp; = \text{4}\text{ cm$^{3}$} \end{align*}</div>
<div>
<h3>Write the final answer</h3>
<p>The volume of the square pyramid is $\text{4}$ $\text{cm$^{3}$}$.</p>
</div>
</div>
</div>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<h3>Question</h3>
<div>
<p>Find the volume of the following triangular pyramid (correct to 1 decimal place):</p>
<img alt="2283c5653898afeb38d1375cbe6a0991.png" src="https://data.ulearngo.com/assets/6133e64b-edfc-43c5-88cc-41f2e5049bd4"/></div>
<div>
<h3>Sketch the base triangle and calculate its area</h3>
<img alt="05f5fc91bd2b378a6a6f33501d62d3f6.png" src="https://data.ulearngo.com/assets/ae42ece1-d1e1-4e84-a8b3-c8649d1ab8ca"/>
<p>The height of the base triangle (${h}_{b}$) is:</p>

\begin{align*} {8}^{2}&amp; ={4}^{2}+{h}_{b}^{2} \\ \therefore {h}_{b}&amp; =\sqrt{{8}^{2}-{4}^{2}} \\ &amp; = 4\sqrt{3}~\text{cm} \end{align*}

<p>The area of the base triangle is:</p>

\begin{align*} \text{ area of base triangle}&amp; =\cfrac{1}{2}b\times {h}_{b} \\ &amp; =\cfrac{1}{2}\times 8\times 4\sqrt{3} \\ &amp; =16\sqrt{3} ~\text{cm$^{2}$} \end{align*}</div>
<div>
<h3>Sketch the side triangle and calculate pyramid height $H$</h3>
<img alt="0279bc7cf58b08c8a44405d103670aef.png" src="https://data.ulearngo.com/assets/9e9c5859-2239-4e3d-9228-abece29441f6"/></div>
<div>
<h3>Calculate the volume of the pyramid</h3>

\begin{align*} V &amp; = \cfrac{1}{3}\times \cfrac{1}{2}b{h}_{b} \times H \\ &amp; = \cfrac{1}{3} \times 16\sqrt{3}\times \sqrt{130} \\ &amp; = \text{105.3}\text{ cm$^{3}$} \end{align*}</div>
<div>
<h3>Write the final answer</h3>
<p>The volume of the triangular pyramid is $\text{105.3}$ $\text{cm$^{3}$}$.</p>
</div>
</div>
</div>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<h3>Question</h3>
<div>
<p>Find the volume of the following cone (correct to $\text{1}$ decimal place):</p>
<img alt="115ea932606ae315d3b9bce84977c9c6.png" src="https://data.ulearngo.com/assets/c8785ce7-df65-4b34-bde1-c08dce992b96"/></div>
<div>
<h3>Find the area of the base</h3>

\begin{align*} \text{area of circle}&amp; =\pi {r}^{2} \\ &amp; = \pi \times {3}^{2} \\ &amp; = 9\pi ~\text{cm$^{2}$} \end{align*}</div>
<div>
<h3>Calculate the volume</h3>

\begin{align*} V &amp; =\cfrac{1}{3}\times \pi {r}^{2}\times H \\ &amp; = \cfrac{1}{3}\times 9\pi \times 11 \\ &amp; = \text{103.7}\text{ cm$^{3}$} \end{align*}</div>
<div>
<h3>Write the final answer</h3>
<p>The volume of the cone is $\text{103.7}$ $\text{cm$^{3}$}$.</p>
</div>
</div>
</div>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<h3>Question</h3>
<div>
<p>Find the volume of the following sphere (correct to 1 decimal place):</p>
<img alt="6a180dbf1bf072d7296367996d5c3038.png" src="https://data.ulearngo.com/assets/72b82638-50e6-4bc8-9109-e2370f9a7194"/></div>
<div>
<h3>Use the formula to find the volume</h3>

\begin{align*} \text{volume}&amp; =\cfrac{4}{3}\pi {r}^{3} \\ &amp; =\cfrac{4}{3}\pi {(4)}^{3} \\ &amp; = \text{268.1}\text{ cm$^{3}$} \end{align*}</div>
<div>
<h3>Write the final answer</h3>
<p>The volume of the sphere is $\text{268.1}$ $\text{cm$^{3}$}$.</p>
</div>
</div>
</div>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<h3>Question</h3>
<div>
<p>A triangular pyramid is placed on top of a triangular prism, as shown below. The base of the prism is an equilateral triangle of side length $\text{20}$ $\text{cm}$ and the height of the prism is $\text{42}$ $\text{cm}$. The pyramid has a height of $\text{12}$ $\text{cm}$. Calculate the total volume of the object.</p>
<img alt="2b420a55c517408aa1c7c0154b446711.png" src="https://data.ulearngo.com/assets/e09c0fd8-483c-4c1b-9fb3-0d6f73caa57e"/></div>
<div>
<h3>Calculate the volume of the prism</h3>
<p>First find the height of the base triangle using the theorem of Pythagoras:</p>
<img alt="8910dd96d1c7aed337132c4f40b87a3a.png" src="https://data.ulearngo.com/assets/a6b73b80-4e8f-4f9d-99ae-05828e8b188b"/>
<p>Next find the area of the base triangle:</p>

\begin{align*} \text{area of base triangle} &amp; = \cfrac{1}{2}\times 20\times 10\sqrt{3} \\ &amp; = 100\sqrt{3} ~\text{cm$^{2}$} \end{align*}

<p>Now we can find the volume of the prism:</p>

\begin{align*} \therefore \text{volume of prism} &amp; = \text{area of base triangle}\times \text{height of prism} \\ &amp; = 100\sqrt{3}\times 42 \\ &amp; = \text{4 200}\sqrt{3} ~\text{cm$^{3}$} \end{align*}</div>
<div>
<h3>Calculate the volume of the pyramid</h3>
<p>The area of the base triangle is equal to the area of the base of the pyramid.</p>

\begin{align*} \therefore \text{volume of pyramid} &amp; = \cfrac{1}{3}(\text{area of base})\times H \\ &amp; = \cfrac{1}{3}\times 100\sqrt{3}\times 12 \\ &amp; = 400\sqrt{3} ~\text{cm$^{3}$} \end{align*}</div>
<div>
<h3>Calculate the total volume</h3>

\begin{align*} \text{total volume} &amp; = \text{4 200} \sqrt{3} + 400\sqrt{3} \\ &amp; = \text{4 600} \sqrt{3} \\ &amp; = \text{7 967.4}\text{ cm$^{3}$} \end{align*}

<p>Therefore the total volume of the object is $\text{7 967.4}$ $\text{cm$^{3}$}$.</p>
</div>
</div>
</div>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<h3>Question</h3>
<div>
<p>With the same complex object as in the previous example, you are given the additional information that the slant height ${h}_{s}$ = $\text{13.3}$ $\text{cm}$. Now calculate the total surface area of the object.</p>
</div>
<div>
<h3>Calculate the surface area of each exposed face of the pyramid</h3>

\begin{align*} \text{area of one pyramid face}&amp; = \cfrac{1}{2}b\times {h}_{s} \\ &amp; = \cfrac{1}{2}\times 20 \times \text{13.3} \\ &amp; = \text{133}\text{ cm$^{2}$} \end{align*}

<p>Because the base triangle is equilateral, each face has the same base, and therefore the same surface area. Therefore the surface area for each face of the pyramid is $\text{133}$ $\text{cm$^{2}$}$.</p>
</div>
<div>
<h3>Calculate the surface area of each side of the prism</h3>
<p>Each side of the prism is a rectangle with base $b = \text{20}\text{ cm}$ and height ${h}_{p} = \text{42}\text{ cm}$.</p>

\begin{align*} \text{area of one prism side}&amp; = b\times {h}_{p} \\ &amp; = 20\times 42 \\ &amp; = \text{840}\text{ cm$^{2}$} \end{align*}

<p>Because the base triangle is equilateral, each side of the prism has the same area. Therefore the surface area for each side of the prism is $\text{840}$ $\text{cm$^{2}$}$.</p>
</div>
<div>
<h3>Calculate the total surface area of the object</h3>

\begin{align*} \text{total surface area} =&amp; \text{area of base of prism} + \text{area of sides of prism} + \text{area of exposed faces of pyramid}\\ = &amp; (100\sqrt{3}) + 3(840) + 3(133)\\ =&amp; \text{3 092.2}\text{ cm$^{2}$} \end{align*}

<p>Therefore the total surface area (of the exposed faces) of the object is $\text{3 092.2}$ $\text{cm$^{2}$}$.</p>
</div>
</div>
</div>
<div class="note visit">
<p>This video shows an example of calculating the volume of a complex object.</p>
<iframe allowfullscreen="allowfullscreen" frameborder="0" height="360" src="https://www.youtube.com/embed/Q9vYvyvDzI8?rel=0&amp;modestbranding=1&amp;cc_lang_pref=en&amp;cc_load_policy=1" width="640"></iframe></div>
<section></section>

77e14597-4d58-4399-be3f-b1bb5c28bc54

Volume of Pyramids, Cones and Spheres

Mathematics studies measurement, relationships, and properties of quantities and sets, using numbers and symbols. Arithmetic, algebra, geometry, and calculus are popular topics in mathematics.

Mathematics

Explore how to calculate area, surface area, and volume of various geometric shapes including polygons, prisms, cones, pyramids, and spheres in this tutorial.


Volume of Pyramids, Cones and Spheres

Lesson video

Lesson video

Track Your Learning Progress

Earlier discussion