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IP Math

<p>A cylindrical solid has a hemispherical top of the same radius $r\text{ cm}$. The cylinder has height $2r\text{ cm}$, and the volume of the entire solid is $576\pi\text{ cm}^3$.</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/1789189430874-y1vjytfj43.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3ODkxODk0MzA4NzQteTF2anl0Zmo0My5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg5NjMzNjM3LCJleHAiOjE3ODk2MzM5Mzd9.0ysE-7X26umqtzqPXsELzXiXS5oEvDAAJDGz3WiuTL0" style="max-width: min(150px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li data-list-item-id="sec4-em-2025-mgs-s-p2-q7-b3-i1" data-list-item-path="sec4-em-2025-mgs-s-p2-q7-b3-i1"><div class="list-item-paragraph">Show that&nbsp;$r=6$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="sec4-em-2025-mgs-s-p2-q7-b3-i2" data-list-item-path="sec4-em-2025-mgs-s-p2-q7-b3-i2"><div class="list-item-paragraph">The entire solid is melted to form a solid cone whose radius and height are equal. Find the total surface area of the cone.<span class="inline-marks">[4]</span></div></li></ol>

A cylindrical solid has a hemispherical top of the same radius $r\text{ cm}$. The cylinder has height $2r\text{ cm}$, and the volume of the entire solid is $576\pi\text{ cm}^3$.

The entire solid is melted to form a solid cone whose radius and height are equal. Find the total surface area of the cone.

Volume and Surface Area of Cylinder

Applications of Pythagoras’ Theorem

Volume and Surface Area of Spheres

Cones

Volume and Surface Area of Composite Solids

2025 MGS(S) PRELIMS P2 Q7

Volume and Surface Area of Prisms

Mensuration (Pyramid, Cones and Spheres)

Pythagoras’ Theorem

<p>A solid square pyramid has height 33 cm and base side 21 cm. It is melted and recast as a sphere. Using $\pi=22/7$, find the sphere radius.</p>
<div class="graph-block" data-graph-config="eyJheGlzIjpmYWxzZSwiZ3JpZCI6ZmFsc2UsIndpZHRoIjo1MjAsImhlaWdodCI6MzQwLCJlbGVtZW50cyI6W3siaWQiOiJ0MTEtcTA5LXMxIiwidHlwZSI6InNlZ21lbnQiLCJjb2xvciI6IiMzMzQxNTUiLCJmaXhlZCI6dHJ1ZSwicG9pbnQxIjpbMCwwXSwicG9pbnQyIjpbMTYsMF0sInN0cm9rZVdpZHRoIjoxLjh9LHsiaWQiOiJ0MTEtcTA5LXMyIiwidHlwZSI6InNlZ21lbnQiLCJjb2xvciI6IiMzMzQxNTUiLCJmaXhlZCI6dHJ1ZSwicG9pbnQxIjpbMTYsMF0sInBvaW50MiI6WzExLDRdLCJzdHJva2VXaWR0aCI6MS44fSx7ImlkIjoidDExLXEwOS1zMyIsInR5cGUiOiJzZWdtZW50IiwiY29sb3IiOiIjMzM0MTU1IiwiZml4ZWQiOnRydWUsInBvaW50MSI6WzExLDRdLCJwb2ludDIiOlszLDRdLCJzdHJva2VXaWR0aCI6MS44fSx7ImlkIjoidDExLXEwOS1zNCIsInR5cGUiOiJzZWdtZW50IiwiY29sb3IiOiIjMzM0MTU1IiwiZml4ZWQiOnRydWUsInBvaW50MSI6WzMsNF0sInBvaW50MiI6WzAsMF0sInN0cm9rZVdpZHRoIjoxLjh9LHsiaWQiOiJ0MTEtcTA5LXM1IiwidHlwZSI6InNlZ21lbnQiLCJjb2xvciI6IiMzMzQxNTUiLCJmaXhlZCI6dHJ1ZSwicG9pbnQxIjpbOCwxNl0sInBvaW50MiI6WzAsMF0sInN0cm9rZVdpZHRoIjoxLjh9LHsiaWQiOiJ0MTEtcTA5LXM2IiwidHlwZSI6InNlZ21lbnQiLCJjb2xvciI6IiMzMzQxNTUiLCJmaXhlZCI6dHJ1ZSwicG9pbnQxIjpbOCwxNl0sInBvaW50MiI6WzE2LDBdLCJzdHJva2VXaWR0aCI6MS44fSx7ImlkIjoidDExLXEwOS1zNyIsInR5cGUiOiJzZWdtZW50IiwiY29sb3IiOiIjMzM0MTU1IiwiZml4ZWQiOnRydWUsInBvaW50MSI6WzgsMTZdLCJwb2ludDIiOlsxMSw0XSwic3Ryb2tlV2lkdGgiOjEuOH0seyJpZCI6InQxMS1xMDktczgiLCJ0eXBlIjoic2VnbWVudCIsImNvbG9yIjoiIzMzNDE1NSIsImZpeGVkIjp0cnVlLCJwb2ludDEiOls4LDE2XSwicG9pbnQyIjpbMyw0XSwic3Ryb2tlV2lkdGgiOjEuOH0seyJpZCI6InQxMS1xMDktczkiLCJkYXNoIjoyLCJ0eXBlIjoic2VnbWVudCIsImNvbG9yIjoiIzMzNDE1NSIsImZpeGVkIjp0cnVlLCJwb2ludDEiOls4LDE2XSwicG9pbnQyIjpbOCwyXSwic3Ryb2tlV2lkdGgiOjEuOH0seyJ4Ijo5LCJ5IjoxNywiaWQiOiJ0MTEtcTA5LXQxMCIsInR5cGUiOiJ0ZXh0IiwiZml4ZWQiOnRydWUsImxhYmVsIjoiXFwoMzNcXCkifSx7IngiOjUsInkiOi0yLCJpZCI6InQxMS1xMDktdDExIiwidHlwZSI6InRleHQiLCJmaXhlZCI6dHJ1ZSwibGFiZWwiOiJcXCgyMVxcdGltZXMyMVxcKSJ9XSwic2hvd1hBeGlzIjpmYWxzZSwic2hvd1lBeGlzIjpmYWxzZSwieEF4aXNMYWJlbCI6IlxcKHhcXCkiLCJ5QXhpc0xhYmVsIjoiXFwoeVxcKSIsImJvdW5kaW5nQm94IjpbLTE0LDI0LDI4LC00XSwib3JpZ2luTGFiZWwiOiJcXChPXFwpIiwiZ3JpZFNldHRpbmdzIjp7Im1ham9yQ29sb3IiOiIjOTRhM2I4IiwibWFqb3JTdGVwWCI6MSwibWFqb3JTdGVwWSI6MSwibWlub3JDb2xvciI6IiNjYmQ1ZTEiLCJtYWpvck9wYWNpdHkiOjAuNSwibWlub3JPcGFjaXR5IjowLjMsIm1pbm9yRWxlbWVudHNYIjo0LCJtaW5vckVsZW1lbnRzWSI6NCwibWFqb3JTdHJva2VXaWR0aCI6MC44LCJtaW5vclN0cm9rZVdpZHRoIjowLjV9LCJzaG93WmVyb1RpY2siOmZhbHNlLCJzaG93QXhpc0xhYmVscyI6ZmFsc2UsInNob3dOYXZpZ2F0aW9uIjpmYWxzZSwia2VlcEFzcGVjdFJhdGlvIjp0cnVlLCJtYWpvclNxdWFyZVNpemVNbSI6MTB9" data-width="520" data-height="340" style="width:520px;height:340px;max-width:100%;display:block;margin:0 auto;"></div>

A solid square pyramid has height 33 cm and base side 21 cm. It is melted and recast as a sphere. Using $\pi=22/7$, find the sphere radius.

Square pyramid recast as a sphere

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA1MjIwMTYwMDMtdGoxcHRvd3lrMWgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYzMzYzNywiZXhwIjoxNzg5NjMzOTM3fQ.uW9hjwerDXMcb35_gXs7KWEU-TEAjNCEyYDp-9VuDvw" 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

IP2U11 HOT Q4

<p>The diagram shows a solid $ABCDEF$, where the base $ABCD$ is a square, $DE$ and $BF$ are perpendicular to the plane $ABCD$. Given that&nbsp;$AB = DE = 2BF = 2$, find the exact volume of the solid $ABCDEF$.</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/1770521325143-zw4jig8cse.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA1MjEzMjUxNDMtenc0amlnOGNzZS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg5NjMzNjM3LCJleHAiOjE3ODk2MzM5Mzd9.gzMu_TQw2EX7PE9pu_DH5bxNFeUJXg7Te3OjOpnZBEk" style="max-width: min(220px, 100%); height: auto; display: block;" alt="Solid ABCDEF with square base ABCD" class="alignright custom-width" /></figure></div>

The diagram shows a solid $ABCDEF$, where the base $ABCD$ is a square, $DE$ and $BF$ are perpendicular to the plane $ABCD$. Given that $AB = DE = 2BF = 2$, find the exact volume of the solid $ABCDEF$.

IP2U11 HOT Q3

<p>The diagram shows a solid $ABCDE$, where&nbsp;$AE$ and $CD$ are perpendicular to the plane $ABC$. Given that&nbsp;$AE = 2CD = 2$ cm, $CA = CB = 3$ cm and $AB = 2\sqrt{5}$ cm, find the exact volume of the solid $ABCDE$.</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/1770520793503-7156h958mrf.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA1MjA3OTM1MDMtNzE1Nmg5NThtcmYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYzMzYzNywiZXhwIjoxNzg5NjMzOTM3fQ.ncBuk-n7CVivv5_L1oMkyT-m-tJ2zQWUx6Q-Ig_q3G8" style="max-width: min(200px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div>

The diagram shows a solid $ABCDE$, where $AE$ and $CD$ are perpendicular to the plane $ABC$. Given that $AE = 2CD = 2$ cm, $CA = CB = 3$ cm and $AB = 2\sqrt{5}$ cm, find the exact volume of the solid $ABCDE$.

IP2U11 HOT Q2

<p>The figure below shows a staircase-shaped prism of a uniform cross section. Given that every step is identical and is $30$
 cm in breadth, $400$
 cm in length and $40$
 cm in height, find</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/1767771330133-k7exdvpby0k.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3NzEzMzAxMzMtazdleGR2cGJ5MGsucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYzMzYzNywiZXhwIjoxNzg5NjMzOTM3fQ._sSxfl84oTdaVvqVnI9bL_efpuMH71CkivwicCSlZ1g" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-paragraph">the area of the cross section in $m^3$,</div></li><li><div class="list-item-paragraph">the total surface area of the prism in $m^3$, and</div></li><li><div class="list-item-paragraph">the volume of the prism in $m^3$.</div></li></ol>

The figure below shows a staircase-shaped prism of a uniform cross section. Given that every step is identical and is $30$
 cm in breadth, $400$
 cm in length and $40$
 cm in height, find

<p>In the figure below, $ABCD$
 is a kite, where diagonals $AC$
 and $BD$
 are of length $a$
 cm. Find the area of $ABCD$
 in terms of $a$
 and $b$. Hence, state a general formula of the area of a kite in terms of the lengths of the diagonals of the kite.</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/1767770548266-zf2sxx0zxb.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3NzA1NDgyNjYtemYyc3h4MHp4Yi5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg5NjMzNjM3LCJleHAiOjE3ODk2MzM5Mzd9.zvsuM1yuOlfVDegpbC0iDxKp6FNqI4jJBLrRmdq8nCo" style="max-width: min(350px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>

In the figure below, $ABCD$
 is a kite, where diagonals $AC$
 and $BD$
 are of length $a$
 cm. Find the area of $ABCD$
 in terms of $a$
 and $b$. Hence, state a general formula of the area of a kite in terms of the lengths of the diagonals of the kite.

Perimeter and Area of Common Plane Figures

Perimeter and Area of Plane Figures

<p>In the figure below, $ABCD$
 is a rhombus, where diagonals $AC$
 and $BD$
 are of lengths $a$
 cm and $b$
 cm, respectively. Find the area of $ABCD$
 in terms of $a$
 and $b$
.</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/1767770411555-kp7a3ldpf3.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3NzA0MTE1NTUta3A3YTNsZHBmMy5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg5NjMzNjM3LCJleHAiOjE3ODk2MzM5Mzd9.MqXLbOgaQVn9AwW62qltbB5GEMfuOhnajkq-7e0p4a4" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>

In the figure below, $ABCD$
 is a rhombus, where diagonals $AC$
 and $BD$
 are of lengths $a$
 cm and $b$
 cm, respectively. Find the area of $ABCD$
 in terms of $a$
 and $b$
.

Perimeter and Area of Parallelogram

<p>A lawn mower has a blade of breadth $0.75$
 m, and its two front wheels are identical of diameter $20$
 cm, its two rear wheels are identical of dimeter $40$
 cm. A gardener uses this lawn mower to mow a square lawn of dimension of $18$
 m by $18$
 m. He alternates his mowing directions, as shown in the figure below, at a speed of $5$
m/min.</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/1767770230557-ep6ty5s4h1v.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3NzAyMzA1NTctZXA2dHk1czRoMXYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYzMzYzNywiZXhwIjoxNzg5NjMzOTM3fQ.Un3SGRie5i6umNh9fWonOFF1SQCFXAHGJ88Il5Cw5tE" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Assuming there is no overlapping of mowing areas, find</p>
<ol class="abc-list"><li><div class="list-item-paragraph">the number of times he needs to move from one side to the opposite side of the lawn to finish mowing the whole area of the lawn,</div></li><li><div class="list-item-paragraph">the number of rotations of the front and rear wheels, respectively, to finish mowing from one side to the opposite side, ignoring the length of the mower (correcting your answers to the nearest integer),</div></li><li><div class="list-item-paragraph">
the least possible time needed for the gardener to fully mow the lawn.</div></li></ol>

A lawn mower has a blade of breadth $0.75$
 m, and its two front wheels are identical of diameter $20$
 cm, its two rear wheels are identical of dimeter $40$
 cm. A gardener uses this lawn mower to mow a square lawn of dimension of $18$
 m by $18$
 m. He alternates his mowing directions, as shown in the figure below, at a speed of $5$
m/min.

Assuming there is no overlapping of mowing areas, find

<p>The figure below shows a piece of rectangular cardboard of dimension <strong>8 cm</strong> by <strong>4 cm</strong>. A square, a $\frac{3}{4}$ disc, a right-angled triangle and a parallelogram are removed from the cardboard. Find the perimeter and the area of the remaining part of the cardboard.</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/1767770014370-8ueres2q0xl.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3NzAwMTQzNzAtOHVlcmVzMnEweGwucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYzMzYzNywiZXhwIjoxNzg5NjMzOTM3fQ.9J98AGm-an8JZpKKzkL-TK4aO0H43gzDIVybcw0Z0gc" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>

The figure below shows a piece of rectangular cardboard of dimension **8 cm** by **4 cm**. A square, a $\frac{3}{4}$ disc, a right-angled triangle and a parallelogram are removed from the cardboard. Find the perimeter and the area of the remaining part of the cardboard.

Perimeter and Area of Composite Figures

<p>In the figure below, four smaller identical parallelograms are removed from a larger parallelogram.</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/1767769683015-qe25zj45eli.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3Njk2ODMwMTUtcWUyNXpqNDVlbGkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYzMzYzNywiZXhwIjoxNzg5NjMzOTM3fQ.XsIqxzm0RXWKjRnWM_Mqn3fbpROAxstJkq4hJojbYGg" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Find</p>
<ol class="abc-list"><li><div class="list-item-paragraph">the area of the shaded region,</div></li><li><div class="list-item-paragraph">the ratio of the area of the shaded region to the area of the larger parallelogram.</div></li></ol>

In the figure below, four smaller identical parallelograms are removed from a larger parallelogram.

<p>In the figure below, the circular portions are two quadrants which are fitted within rectangle $ABCD$.</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/1767769523247-p512vrvc43.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3Njk1MjMyNDctcDUxMnZydmM0My5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg5NjMzNjM3LCJleHAiOjE3ODk2MzM5Mzd9.rGcZ2sH0w8ZFY92XDVgCfreYuSvX6z3AibNqwRwoR0M" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Find</p>
<ol class="abc-list"><li><div class="list-item-paragraph">the perimeter and the area of the shaded region.</div></li><li><div class="list-item-paragraph">the ratio of the area of the shaded region to the area of rectangle $ABCD$.</div></li></ol>

In the figure below, the circular portions are two quadrants which are fitted within rectangle $ABCD$.

<p>A cuboid water tank with dimensions $12$ cm by $12$ cm by $32$ cm contains water to $\frac{3}{4}$ its height and is used to fill water to a number of identical open cylindrical cans with diameter $4$ cm and height $h$ cm. Assume the thickness of all containers is negligible.</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/1767694925729-e3sgrgfwk9f.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTQ5MjU3MjktZTNzZ3JnZndrOWYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYzMzYzNywiZXhwIjoxNzg5NjMzOTM3fQ.8O8Os9uII2ICf2j1lW--pXaepyODQTvPaKp023Q44nE" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the volume of water contained in the cuboid tank.</span></div></li><li><div class="list-item-paragraph">Given that the maximum number of cylinder cans that can be completely filled is $8$, find the integer value of $h$. Hence, find the volume of remaining water in the tank.</div></li><li><div class="list-item-content"><span class="list-item-text">Given that $h = 4$ cm, find the maximum number of cylinder cans that can be completely filled. Hence, find the volume of remaining water in the tank.</span></div></li></ol>

A cuboid water tank with dimensions $12$ cm by $12$ cm by $32$ cm contains water to $\frac{3}{4}$ its height and is used to fill water to a number of identical open cylindrical cans with diameter $4$ cm and height $h$ cm. Assume the thickness of all containers is negligible.

Find the volume of water contained in the cuboid tank.

Given that $h = 4$ cm, find the maximum number of cylinder cans that can be completely filled. Hence, find the volume of remaining water in the tank.

<p>The figure below shows a cup-shaped solid that is the remaining part of a cylinder, where a smaller cylinder is carved out. Given that the thickness of the lateral and the bottom of the cup are both $2$ cm, and the height of the solid is $15$ cm, find</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/1767694801567-9lw2t9c8lh4.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTQ4MDE1NjctOWx3MnQ5YzhsaDQucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYzMzYzNywiZXhwIjoxNzg5NjMzOTM3fQ.-GAwbU5wHRTi1qbvMIupxMho3r3dR9o_-UAQUf9Y3nQ" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the total surface area of the solid,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the volume that the solid contains, and</span></div></li><li><div class="list-item-content"><span class="list-item-text">the volume of the solid itself</span></div></li></ol>

The figure below shows a cup-shaped solid that is the remaining part of a cylinder, where a smaller cylinder is carved out. Given that the thickness of the lateral and the bottom of the cup are both $2$ cm, and the height of the solid is $15$ cm, find

<p>The figure below shows a solid of a uniform cross section, which consists of a quadrant and a right-angled triangle. </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/1767694330933-qqvzwe2l4pk.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTQzMzA5MzMtcXF2endlMmw0cGsucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYzMzYzNywiZXhwIjoxNzg5NjMzOTM3fQ.gEqDMz9DOZQ0JFrUzV5UHnzDsSqJ5OIBBFHvuDxXocY" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Find</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the area of the cross section,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the total surface area of the solid, and</span></div></li><li><div class="list-item-content"><span class="list-item-text">the volume of the solid.</span></div></li></ol>

The figure below shows a solid of a uniform cross section, which consists of a quadrant and a right-angled triangle. 

<p>The figure below shows a pentagonal prism. Given that the perimeter of its cross section is $489.4$mm, find, correcting your answers to the nearest integer,</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/1767694189025-w6omtkdpz0o.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTQxODkwMjUtdzZvbXRrZHB6MG8ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYzMzYzNywiZXhwIjoxNzg5NjMzOTM3fQ.NQlm2wk3_nlIcAUsIxyAoD1cldJJpQKfowfRbptIliY" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the value of $x$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the area of the cross section in cm$^2$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the total surface area of the prism in cm$^2$, and</span></div></li><li><div class="list-item-content"><span class="list-item-text">the volume of the prism in cm$^3$.</span></div></li></ol>

The figure below shows a pentagonal prism. Given that the perimeter of its cross section is $489.4$mm, find, correcting your answers to the nearest integer,

the total surface area of the prism in cm$^2$, and

<p>The figure below shows a net of a prism, which consists of $3$
 pairs of identical rectangles.</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/1767693836575-2pvpkqevsea.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTM4MzY1NzUtMnB2cGtxZXZzZWEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYzMzYzNywiZXhwIjoxNzg5NjMzOTM3fQ.vM45586m70oomFH-V3tJCd1SYIOdNZORW6XiXQyRf10" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-paragraph">State the type of the prism.</div></li><li><div class="list-item-paragraph">Find its total surface area, and its volume</div></li></ol>

The figure below shows a net of a prism, which consists of $3$
 pairs of identical rectangles.

<p>The figure below shows a cylinder that has a volume of $208\pi$ cm$^3$.</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/1767693559374-spfhsw9dnb9.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTM1NTkzNzQtc3BmaHN3OWRuYjkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYzMzYzNywiZXhwIjoxNzg5NjMzOTM3fQ.fXB2p9NQEQtwDSxSPLSj2l4jW0t6oCiybM3FRXocTOY" style="max-width: min(280px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Find</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the value of $r$, and</span></div></li><li><div class="list-item-content"><span class="list-item-text">its total surface area.</span></div></li></ol>

The figure below shows a cylinder that has a volume of $208\pi$ cm$^3$.

<p>Express</p>
<ol class="i-list"><li><div class="list-item-paragraph"><strong>$170\ 964$cm$^3$ in m$^3$</strong></div></li><li><div class="list-item-paragraph"><strong>$7.97$m$^3$ in litres</strong></div></li><li><div class="list-item-paragraph"><strong>$2\ 481$millilitres in m$^3$</strong></div></li><li><div class="list-item-paragraph"><strong>$729$litres in cm$^3$</strong></div></li></ol>

Conversion between Cubic Units

<p>A factory produces circular iron discs out of rectangular iron boards of dimension of $32$
 cm by $64$
 cm. For Type A of circular iron disc, at most $8$
 of which can be produced out of one rectangular iron board, with each disc being fitted exactly in the rectangular iron board as shown in the figure below.</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/1767691426530-q4wxphjjfn9.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTE0MjY1MzAtcTR3eHBoampmbjkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYzMzYzNywiZXhwIjoxNzg5NjMzOTM3fQ.tscTSM9QjANPJcFrSpUYDxMJ7HM464r5f_3XlYOfe34" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Find, for Type A of circular iron disc,</p>
<ol class="abc-list"><li><div class="list-item-paragraph">the radius, perimeter and area, respectively,</div></li><li><div class="list-item-paragraph">the area of the shaded region,</div></li><li><div class="list-item-paragraph">
the utilisation efficiency of the rectangular iron board, denoted by the total area of circular iron discs as a percentage of the area of the rectangular iron board. Correct your answer to $1$
 decimal place.</div></li></ol>
<p>Another two types of circular iron discs, Type B and Type C, have radii of $2$
 cm and $4$
 cm, respectively.</p>
<ol class="abc-list" start="4"><li><div class="list-item-paragraph">Find the utilisation efficiency of the rectangular iron board for Type B and Type C, respectively. Correct your answer to $1$
 decimal place.
Hence, state your finding(s) about the utilisation efficiency of Types A, B and C.</div></li></ol>

A factory produces circular iron discs out of rectangular iron boards of dimension of $32$
 cm by $64$
 cm. For Type A of circular iron disc, at most $8$
 of which can be produced out of one rectangular iron board, with each disc being fitted exactly in the rectangular iron board as shown in the figure below.

Another two types of circular iron discs, Type B and Type C, have radii of $2$
 cm and $4$
 cm, respectively.

Math Question Bank