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<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3ODkxODk0MzA4NzQteTF2anl0Zmo0My5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg5NjI3MTk4LCJleHAiOjE3ODk2Mjc0OTh9.p-Ed456BpA0cZI7XBb1k4nq2HuqngV5J7Phc76oUFn8" 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>The figure below shows a tilted cuboid water tank with dimensions <strong>25 cm</strong> by <strong>20 cm</strong> by <strong>80 cm</strong>, such that the water in the tank forms a triangular prism with base triangle $BCM$. Given that $M$ and $N$ are midpoints of sides $CD$ and $EF$ of the cuboid tank, respectively.</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/1767771757170-8qtgpx3ubge.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3NzE3NTcxNzAtOHF0Z3B4M3ViZ2UucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYyNzE5OCwiZXhwIjoxNzg5NjI3NDk4fQ.Ms1IViN84xMHdwRhxvNavbkMAS5_hmHe2X9wbYYgfOM" 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 the water in the tank.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the surface area of the tank that is in contact with water.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the depth of water when the tank is placed in such a way that face $ABCD$ is the base.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the depth of water when the tank is placed in such a way that face $CDFE$ is the base.</span></div></li></ol>

The figure below shows a tilted cuboid water tank with dimensions **25 cm** by **20 cm** by **80 cm**, such that the water in the tank forms a triangular prism with base triangle $BCM$. Given that $M$ and $N$ are midpoints of sides $CD$ and $EF$ of the cuboid tank, respectively.

Find the volume of the water in the tank.

Find the surface area of the tank that is in contact with water.

Find the depth of water when the tank is placed in such a way that face $ABCD$ is the base.

Find the depth of water when the tank is placed in such a way that face $CDFE$ is the base.

<p>The figure below shows a container that is a trapezoidal prism. The container is empty at first, and then is filled with water by a pipe with diameter <strong>10 cm</strong> at a constant speed of <strong>6 m/s</strong>.</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/1767771591862-2iqlco8wksk.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3NzE1OTE4NjItMmlxbGNvOHdrc2sucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYyNzE5OCwiZXhwIjoxNzg5NjI3NDk4fQ.VU3x4nLij6xP7MMLQipRFfaSj_3P2DaHfSoh4p5GTUY" style="max-width: min(350px, 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 volume of water filled into the container per second in litres/s, leaving your answer in terms of $\pi$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence, find the total time needed for the container to be completely filled, correcting your answer to <strong>2</strong> decimal places.</span></div></li></ol>
<p>After $t$ seconds, the depth of the water in the container is <strong>1.4 m</strong> and the breadth of the water surface in the container is <strong>2.1 m</strong>.</p>
<ol class="abc-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Find the value of $t$, correcting your answer to <strong>2</strong> decimal places.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the volume of empty space in the container at this moment.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the area of the water surface at this moment.</span></div></li></ol>

The figure below shows a container that is a trapezoidal prism. The container is empty at first, and then is filled with water by a pipe with diameter **10 cm** at a constant speed of **6 m/s**.

Find volume of water filled into the container per second in litres/s, leaving your answer in terms of $\pi$.

Hence, find the total time needed for the container to be completely filled, correcting your answer to **2** decimal places.

After $t$ seconds, the depth of the water in the container is **1.4 m** and the breadth of the water surface in the container is **2.1 m**.

Find the value of $t$, correcting your answer to **2** decimal places.

Find the volume of empty space in the container at this moment.

Find the area of the water surface at this moment.

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3NzEzMzAxMzMtazdleGR2cGJ5MGsucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYyNzE5OCwiZXhwIjoxNzg5NjI3NDk4fQ.EqkaPIl-DcIuoHXD1Mnewd-Ky-yz7EOMpk7ML_Fa16g" 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>An open cylindrical can with diameter <strong>6 cm</strong> and height <strong>10 cm</strong> is used to fill water into an empty cuboid tank with dimensions <strong>48 cm</strong> by <strong>24 cm</strong> by $h$ <strong>cm</strong>. 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/1767771162674-ejcfdib5s1o.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3NzExNjI2NzQtZWpjZmRpYjVzMW8ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYyNzE5OCwiZXhwIjoxNzg5NjI3NDk4fQ.Ep91NFi_UU6YJQd_pwTQ9KWUdkdPyOihIfSc6q0u4b4" 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">Find the volume of the cylindrical can, leaving your answer in terms of $\pi$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that exactly <strong>128</strong> cans of water are needed to completely fill the empty tank, find the value of $h$, leaving your answer in terms of $\pi$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that $h =$ <strong>40 cm</strong>, find the minimum number of cans of water needed to completely fill the empty tank, correcting your answer to the nearest integer.</span></div></li></ol>

An open cylindrical can with diameter **6 cm** and height **10 cm** is used to fill water into an empty cuboid tank with dimensions **48 cm** by **24 cm** by $h$ **cm**. Assume the thickness of all containers is negligible.

Find the volume of the cylindrical can, leaving your answer in terms of $\pi$.

Given that exactly **128** cans of water are needed to completely fill the empty tank, find the value of $h$, leaving your answer in terms of $\pi$.

Given that $h =$ **40 cm**, find the minimum number of cans of water needed to completely fill the empty tank, correcting your answer to the nearest integer.

<p>The figure below shows a solid that is the remaining part of a cuboid, where a smaller cuboid is carved out to make a container.</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/1767770885707-dca69v6b5r.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3NzA4ODU3MDctZGNhNjl2NmI1ci5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg5NjI3MTk4LCJleHAiOjE3ODk2Mjc0OTh9.bdSFnZiIfewtN-7pYpmL_ORkmYhVRtunah4gI52KqEo" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="i-list"><li><div class="list-item-paragraph">Find</div><ol class="abc-list"><li><div class="list-item-paragraph">the total surface area of the solid, </div></li><li><div class="list-item-paragraph">the volume that the solid contains, and </div></li><li><div class="list-item-paragraph">the volume of the solid itself.</div></li></ol></li></ol>
<p>Another cuboid has the same dimensions of base of the smaller cuboid that is carved out of the solid above, as shown in the figure below. Suppose the cuboid below is fitted into the solid above to form a new solid.</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/1767770941804-ogg08qtkngl.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3NzA5NDE4MDQtb2dnMDhxdGtuZ2wucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYyNzE5OCwiZXhwIjoxNzg5NjI3NDk4fQ.GoD9tIujDNjMEPKtrvTO7nwRy_3UcoN4STMVNrHJr5s" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Find </div><ol class="abc-list"><li><div class="list-item-paragraph">the part of the surface area of the new smaller cuboid that is exposed as a part of the surface of the new solid,</div></li><li><div class="list-item-paragraph">the total surface area of the new solid, and </div></li><li><div class="list-item-paragraph">the volume of the new solid.</div></li></ol></li></ol>

The figure below shows a solid that is the remaining part of a cuboid, where a smaller cuboid is carved out to make a container.

Another cuboid has the same dimensions of base of the smaller cuboid that is carved out of the solid above, as shown in the figure below. Suppose the cuboid below is fitted into the solid above to form a new solid.

<p>The figure below shows a solid that is the remaining part of a larger triangular prism with a smaller triangular prism carved out of it.</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/1767770715447-jfu9gqyq22n.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3NzA3MTU0NDctamZ1OWdxeXEyMm4ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYyNzE5OCwiZXhwIjoxNzg5NjI3NDk4fQ.Az6zRw3oen5-32mW6oOuX82OT4JzHtue4IJYlgAG3kA" style="max-width: min(350px, 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 cross section in <strong>cm²</strong>.</div></li><li><div class="list-item-paragraph">the total surface area of the solid in <strong>cm²</strong>.</div></li><li><div class="list-item-paragraph">the volume of the solid in <strong>cm³</strong>, </div></li></ol>
<p>correcting your answers to the nearest integer.</p>

The figure below shows a solid that is the remaining part of a larger triangular prism with a smaller triangular prism carved out of it.

correcting your answers to the nearest integer.

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3NzA1NDgyNjYtemYyc3h4MHp4Yi5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg5NjI3MTk4LCJleHAiOjE3ODk2Mjc0OTh9.VVQfaKa6OsyloaNNj9MC9rV9oRknpPzsUjwqopw94Ao" 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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3NzA0MTE1NTUta3A3YTNsZHBmMy5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg5NjI3MTk4LCJleHAiOjE3ODk2Mjc0OTh9.uuVL5T2hMgDCkr62Oh_xjEeRCtnmrIB1GuIwGMKQYx0" 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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3NzAyMzA1NTctZXA2dHk1czRoMXYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYyNzE5OCwiZXhwIjoxNzg5NjI3NDk4fQ.TrBHg1HqLcxfVHhfSC5CFosiuMKF-LmxPBP9ETP2wNE" 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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3NzAwMTQzNzAtOHVlcmVzMnEweGwucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYyNzE5OCwiZXhwIjoxNzg5NjI3NDk4fQ.WZUalQateK-yA-OTw2t2UQ6hAGraza2m4AZgj7g1dUA" 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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3Njk2ODMwMTUtcWUyNXpqNDVlbGkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYyNzE5OCwiZXhwIjoxNzg5NjI3NDk4fQ.-rlVSK2P-G1tHRLUE7R4wN9FhuKTX5BYZ3iV5S_0xuQ" 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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3Njk1MjMyNDctcDUxMnZydmM0My5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg5NjI3MTk4LCJleHAiOjE3ODk2Mjc0OTh9.bu69MtaeagCH34hf-Yozi5319bpkdm_2inykbOQwC2s" 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 prism-shaped water drainage system is such that the cross section consists of a trapezium and a small rectangle as shown in the figures below.</p>
<ol class="i-list"><li><div class="list-item-paragraph">On a sunny day, only part of the rectangular part is submerged by water. Given that the water flows at a constant speed of $5$<strong>m/s</strong>, find:</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/migrated/2024/05/Question-25-i-diagram.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvbWlncmF0ZWQvMjAyNC8wNS9RdWVzdGlvbi0yNS1pLWRpYWdyYW0ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYyNzE5OCwiZXhwIjoxNzg5NjI3NDk4fQ.AbVBueQhrhkbTLS23FqhgPiaQ0mqeiAE2hj3oVXK5Og" alt="List item image" class="aligncenter custom-width" style="max-width: min(350px, 100%); height: auto;" /></div><ol class="abc-list"><li><div class="list-item-paragraph">the water-draining rate, which is given by the volume of water that flows through the cross section per unit of time in <strong>m$^3$/s</strong>, and</div></li><li><div class="list-item-paragraph">the volume of water drained in $30$ minutes, assuming that the area of the cross section of water and the speed at which water flows do not change.</div></li><li><div class="list-item-paragraph">the portion of the drainage capacity of the system used on this day, which is given by the area of the cross section of water as a percentage of the area of the cross section of the system, correcting your answer to $2$ decimal places.</div></li></ol></li><li><div class="list-item-paragraph">On a rainy day, the rectangular part is completely submerged by water and the area of the trapezoidal part of the cross section of water is <strong>$26$ m$^2$</strong>. Given that the water flows at a constant speed of <strong>$15$m/s</strong>, find:</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/migrated/2024/05/Question-25-ii-diagram.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvbWlncmF0ZWQvMjAyNC8wNS9RdWVzdGlvbi0yNS1paS1kaWFncmFtLnBuZyIsInNjb3BlIjoiZG93bmxvYWQiLCJpYXQiOjE3ODk2MjcxOTgsImV4cCI6MTc4OTYyNzQ5OH0.-L3kJxZbc6IgNAJ06fwqRHdc0mZnycDoLrwaTkPlZdg" alt="List item image" class="aligncenter custom-width" style="max-width: min(350px, 100%); height: auto;" /></div><ol class="abc-list"><li><div class="list-item-paragraph">the water-draining rate in terms of the volume of water that flows through the cross section per unit of time in <strong>m$^3$/s</strong>, and</div></li><li><div class="list-item-paragraph">the volume of water drained in $30$ minutes, assuming that the area of the cross section of water and the speed at which water flows do not change.</div></li><li><div class="list-item-paragraph">the portion of the drainage capacity of the system used on this day, which is given by the area of the cross section of water as a percentage of the area of the cross section of the system, correcting your answer to $2$ decimal places.</div></li></ol></li></ol>

A prism-shaped water drainage system is such that the cross section consists of a trapezium and a small rectangle as shown in the figures below.

<p>The figure below shows a container that is a triangular prism with a right-angled triangular cross section. The container is empty at first, and then is filled with water by a pipe with diameter $5$cm at a constant speed of $3$m/s.</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/1767695624847-xzbktg02uog.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTU2MjQ4NDcteHpia3RnMDJ1b2cucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYyNzE5OCwiZXhwIjoxNzg5NjI3NDk4fQ.a-rOoHtpNP-QxZ5EzVqNL2QoVhrfddbEKLqdSGOmqmg" style="max-width: min(250px, 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 volume of water filled into the container per second in cm$^3$/s, leaving your answer in terms of $\pi$.</span></div></li><li><div class="list-item-paragraph">Hence, find the total time needed, in seconds, for the container to be completely filled, correcting your answer to $2$ decimal places.</div></li></ol>
<p>After $t$ seconds, the depth of the water in the container is $1$m and the length of the water surface in the container is $0.8$m.</p>
<ol class="abc-list" start="3"><li><div class="list-item-paragraph">Find the value of $t$, correcting your answer to $2$ decimal places.</div></li><li><div class="list-item-content"><span class="list-item-text">Find the volume of empty space in the container at this moment.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the surface area of the container that is in contact with water at this moment.</span></div></li></ol>

The figure below shows a container that is a triangular prism with a right-angled triangular cross section. The container is empty at first, and then is filled with water by a pipe with diameter $5$cm at a constant speed of $3$m/s.

Find volume of water filled into the container per second in cm$^3$/s, leaving your answer in terms of $\pi$.

After $t$ seconds, the depth of the water in the container is $1$m and the length of the water surface in the container is $0.8$m.

Find the surface area of the container that is in contact with water at this moment.

<p>The two figures below show two solids that are the remaining parts of two identical cuboids, where two identical triangular prisms with right-angled triangular base is carved out in different ways.

</p>
<p>Given that the dimensions of the base triangle of the triangular prisms are $12$
 cm by $16$
 cm by $20$
 cm, and the height of the cuboids and triangular prisms are $x$
 cm, show that the difference between the total surface area of the two solids is $40$
cm$^2$.</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/1767695440403-zmw2r5zfjna.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTU0NDA0MDMtem13MnI1emZqbmEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYyNzE5OCwiZXhwIjoxNzg5NjI3NDk4fQ.2WC6NU1aw1YBxWI3PpejSezk5--7bJ6wzRU9iYNe02g" style="max-width: min(350px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>

The two figures below show two solids that are the remaining parts of two identical cuboids, where two identical triangular prisms with right-angled triangular base is carved out in different ways.



Given that the dimensions of the base triangle of the triangular prisms are $12$
 cm by $16$
 cm by $20$
 cm, and the height of the cuboids and triangular prisms are $x$
 cm, show that the difference between the total surface area of the two solids is $40$
cm$^2$.

<p>The figure below shows a solid that is the remaining part of a cuboid, where two square holes that cut through the cuboid are at the centre of two pairs of opposite faces of the cuboid. 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/1767695264981-mbhmdstfgdc.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTUyNjQ5ODEtbWJobWRzdGZnZGMucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYyNzE5OCwiZXhwIjoxNzg5NjI3NDk4fQ.XYbioqjlqbTIZQp5b2OWlcRyxgtTwZLO89LM7F3-jN8" style="max-width: min(350px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-paragraph">the total surface area of the solid, and</div></li><li><div class="list-item-paragraph">the volume of the solid.</div></li><li><div class="list-item-paragraph">Determine the type of the solid formed by the common part of the two square holes.</div></li></ol>

The figure below shows a solid that is the remaining part of a cuboid, where two square holes that cut through the cuboid are at the centre of two pairs of opposite faces of the cuboid. Find

 Volume and Surface Area of Composite Solids

<p>A cake of three layers can be deemed as three cylinders that stack together, such that the dimeter of each layer from top to bottom is $9$
 cm, $12$
 cm and $15$
 cm, respectively, and the height of each layer from top to bottom is $5$
 cm, $5$
 cm and $8$
 cm, respectively. Find</p>
<ol class="abc-list"><li><div class="list-item-paragraph">the total surface area of the cake, and</div></li><li><div class="list-item-paragraph">the volume of the cake.</div></li><li><div class="list-item-paragraph">Given that $9$
 people are to share the cake equally, find the volume of cake each person gets.</div></li></ol>

A cake of three layers can be deemed as three cylinders that stack together, such that the dimeter of each layer from top to bottom is $9$
 cm, $12$
 cm and $15$
 cm, respectively, and the height of each layer from top to bottom is $5$
 cm, $5$
 cm and $8$
 cm, respectively. Find

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTQ5MjU3MjktZTNzZ3JnZndrOWYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYyNzE5OCwiZXhwIjoxNzg5NjI3NDk4fQ.HC5amCRK4IJesR8i2xgQEEP5MdTh6gQDG_amR_sEO_k" 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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTQ4MDE1NjctOWx3MnQ5YzhsaDQucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYyNzE5OCwiZXhwIjoxNzg5NjI3NDk4fQ.iQRbK822aC96dLcwPcDBCJw326ceyhlEeQl1p_ZHqDw" 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

Math Question Bank