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<p>The diagram shows sphere with centre $O$ and a total surface area of $\frac{192\pi}{3}$ $\text{cm}^2$. Vertices of an equilateral triangle $ABC$ are on the surface of the sphere. Given that&nbsp;$AB = 2$ $\text{cm}$, find</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="i-list"><li><div class="list-item-paragraph">the total surface area of pyramid $O-ABC$,</div></li><li><div class="list-item-paragraph">the volume of pyramid $O-ABC$, leaving your answers in the exact form.</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1770522016003-tj1ptowyk1h.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA1MjIwMTYwMDMtdGoxcHRvd3lrMWgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU0NjYyNiwiZXhwIjoxNzkzNTQ2OTI2fQ.Xep86xCngbTQoZuvgJ3cjNHX8dMchh9A1qV2-Ap9iRA" style="max-width: min(200px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

The diagram shows sphere with centre $O$ and a total surface area of $\frac{192\pi}{3}$ $\text{cm}^2$. Vertices of an equilateral triangle $ABC$ are on the surface of the sphere. Given that $AB = 2$ $\text{cm}$, find

Volume and Surface Area of Pyramids

Volume and Surface Area of Spheres

Pythagoras’ Theorem in Three Dimensions

IP2U11 HOT Q4

Mensuration (Pyramid, Cones and Spheres)

Pythagoras’ Theorem

<p>A cube fits inside a sphere with all eight vertices on the sphere. The greatest circular cross-section of the sphere, through its centre, has area $\pi\text{ cm}^2$. Find the side length of the cube.</p>
<div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1768920383306-39p3v12cqq9.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njg5MjAzODMzMDYtMzlwM3YxMmNxcTkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU0NjYyNiwiZXhwIjoxNzkzNTQ2OTI2fQ.FlrqRXMNIvzFd0F9Rp1LpWOv6laeZJOxpRBw0M7iafE" style="max-width: min(150px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div>

A cube fits inside a sphere with all eight vertices on the sphere. The greatest circular cross-section of the sphere, through its centre, has area $\pi\text{ cm}^2$. Find the side length of the cube.

IP1U13 SWII (Pythagoras) Q16

<p>A cylinder is three times as high as it is wide.</p>
<p>The diagonal of the axial rectangular cross-section shown is $20\text{ cm}$; its width is the cylinder’s diameter.</p>
<p>Find the <strong>EXACT</strong> surface area of the top of the cylinder (as shaded in the image).</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><p style="text-align: left">(Note: Your solution will have the symbol $\pi $ in it.)</p></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1768919168947-al2ixqh65g.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njg5MTkxNjg5NDctYWwyaXhxaDY1Zy5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTQ2NjI2LCJleHAiOjE3OTM1NDY5MjZ9.DxGfow4Re4uonDsoLiAYo5588LiDYMfz4WyKnrbHU88" style="max-width: min(160px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

A cylinder is three times as high as it is wide.

The diagonal of the axial rectangular cross-section shown is $20\text{ cm}$; its width is the cylinder’s diameter.

Find the **EXACT** surface area of the top of the cylinder (as shaded in the image).

Circumference and Area of Circles

IP1U13 SWII (Pythagoras) Q14

Perimeter and Area of Plane Figures

<p>A pencil box, in the shape of a rectangular prism, measures $16\text{ cm}\times 12\text{ cm}\times 8\text{ cm}$.<br></p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><p>Find the length of the longest pencil that would fit inside the box.<br></p></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1768912850367-tljf1lbpp1.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njg5MTI4NTAzNjctdGxqZjFsYnBwMS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTQ2NjI2LCJleHAiOjE3OTM1NDY5MjZ9.YilWQ_wKMX2ALlJMUlcNJRQO1I0ZGLzCkjtifEBfGEQ" style="max-width: min(200px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

A pencil box, in the shape of a rectangular prism, measures $16\text{ cm}\times 12\text{ cm}\times 8\text{ cm}$.


IP1U13 SWII (Pythagoras) Q7

<p>A container in the form of a hemisphere of radius $10.8$ cm has a conical part removed as shown in the diagram.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1765262435790-jq5oxfyfjx.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjUyNjI0MzU3OTAtanE1b3hmeWZqeC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTQ2NjI2LCJleHAiOjE3OTM1NDY5MjZ9.NM9hnMhFcvIcOnYjhPr-ckTUWvAO6Sua0nn9OP72fTU" style="max-width: min(220px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-paragraph">Calculate the</div><ol class="i-list"><li><div class="list-item-paragraph">slant height of the cone,</div></li><li><div class="list-item-paragraph">total surface area of the container.</div></li></ol></li><li><div class="list-item-paragraph">Show that the volume of the container is $1319$ cm$^{3}$, correct to the nearest cm$^{3}$.</div></li><li><div class="list-item-paragraph">The container is made of a metal alloy. Given that the density of the alloy is $5$g/cm$^{3}$, 	find the mass of the container correct to the nearest kg.</div></li></ol>

A container in the form of a hemisphere of radius $10.8$ cm has a conical part removed as shown in the diagram.

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