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PSLE

PSLE Foundation Mathematics

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3ODkxODk0MzA4NzQteTF2anl0Zmo0My5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyODUxMDgxLCJleHAiOjE3OTI4NTEzODF9.8_r_yUP8dUUyGeH8qqKdfAQvGL1uj18hofvhodC1KsU" 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 Cylinders

Applications of Pythagoras’ Theorem

Volume and Surface Area of Spheres

Volume and Surface Area of 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="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" 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 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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3NzEzMzAxMzMtazdleGR2cGJ5MGsucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg1MTA4MSwiZXhwIjoxNzkyODUxMzgxfQ.zelZwgalItBowQ9FaV71HPFozDcO56m3A435BoPhnv8" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li data-list-item-id="item-1767771333313-g1qo5jp1j" data-list-item-path="item-1767771333313-g1qo5jp1j"><div class="list-item-paragraph">the area of the cross section in $\text{m}^2$,</div></li><li data-list-item-id="item-1767771333897-jrgt8arj2" data-list-item-path="item-1767771333897-jrgt8arj2"><div class="list-item-paragraph">the total surface area of the prism in $\text{m}^2$, and</div></li><li data-list-item-id="item-1767771334921-8yysn2035" data-list-item-path="item-1767771334921-8yysn2035"><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

Conversion between Cubic Units

Conversions between Square Units

Perimeter and Area of Plane Figures

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3NzA4ODU3MDctZGNhNjl2NmI1ci5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyODUxMDgxLCJleHAiOjE3OTI4NTEzODF9.YjHCqdyLVb4cn1Y-Ybz6hk1vHd5Mkejbn-eST69Jn54" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Find</div><ol class="abc-list"><li data-list-item-id="subitem-1767770890972-wj6mn5mki" data-list-item-path="item-i/subitem-1767770890972-wj6mn5mki"><div class="list-item-paragraph">the total surface area of the solid, </div></li><li data-list-item-id="subitem-1767770892924-r1xfmmbdt" data-list-item-path="item-i/subitem-1767770892924-r1xfmmbdt"><div class="list-item-paragraph">the volume that the solid contains, and </div></li><li data-list-item-id="subitem-1767770893692-egzn73x9g" data-list-item-path="item-i/subitem-1767770893692-egzn73x9g"><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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3NzA5NDE4MDQtb2dnMDhxdGtuZ2wucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg1MTA4MSwiZXhwIjoxNzkyODUxMzgxfQ.8G6Kzeq6WOZSLjW9tOookNPF77gztD__xdMtOAIzbrA" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="i-list" start="2"><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Find </div><ol class="abc-list"><li data-list-item-id="subitem-1767770948325-qofkgfhdk" data-list-item-path="item-ii/subitem-1767770948325-qofkgfhdk"><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 data-list-item-id="subitem-1767770949788-zzh1os73w" data-list-item-path="item-ii/subitem-1767770949788-zzh1os73w"><div class="list-item-paragraph">the total surface area of the new solid, and </div></li><li data-list-item-id="subitem-1767770950356-1leirjo5s" data-list-item-path="item-ii/subitem-1767770950356-1leirjo5s"><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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3NzA3MTU0NDctamZ1OWdxeXEyMm4ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg1MTA4MSwiZXhwIjoxNzkyODUxMzgxfQ.Japn305CTv15qCmNMbc6nokGu5hX0oN_jc47AimDuI0" 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>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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvbWlncmF0ZWQvMjAyNC8wNS9RdWVzdGlvbi0yNS1pLWRpYWdyYW0ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg1MTA4MSwiZXhwIjoxNzkyODUxMzgxfQ.Nwsju5OqPJ_4mxrZgy9hhkBe9vQTnW34-5iRcGi0qxw" 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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvbWlncmF0ZWQvMjAyNC8wNS9RdWVzdGlvbi0yNS1paS1kaWFncmFtLnBuZyIsInNjb3BlIjoiZG93bmxvYWQiLCJpYXQiOjE3OTI4NTEwODEsImV4cCI6MTc5Mjg1MTM4MX0._IEP7i1UW6gjw-7Z15bur2Vcd79IpBVtYzW1qJKLfl4" 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.

Expressing One Quantity as a Percentage of Another

Calculating Rates and Quantities

Percentages

Ratio, Rate and Speed

<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.<br><br></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 $40x$ 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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTU0NDA0MDMtem13MnI1emZqbmEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg1MTA4MSwiZXhwIjoxNzkyODUxMzgxfQ.7HZG2PujTDEegbFLueU4int7IFkvLAORrgZiRc3i9Qk" style="max-width: min(350px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Correction: the source claim omits the factor $x$. The difference is $40x\text{ cm}^2$, as the solution shows.</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.



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 $40x$ cm$^2$.

Correction: the source claim omits the factor $x$. The difference is $40x\text{ cm}^2$, as the solution shows.

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTUyNjQ5ODEtbWJobWRzdGZnZGMucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg1MTA4MSwiZXhwIjoxNzkyODUxMzgxfQ.wxwdAzG3dx0c_PQPge7ZLtdAqjAuqb_O9Ytp7nX8nNU" 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 diameter 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 data-list-item-id="item-1767695201257-2mvyo60jp" data-list-item-path="item-1767695201257-2mvyo60jp"><div class="list-item-paragraph">the total surface area of the cake, and</div></li><li data-list-item-id="item-1767695201841-qyufs3gmt" data-list-item-path="item-1767695201841-qyufs3gmt"><div class="list-item-paragraph">the volume of the cake.</div></li><li data-list-item-id="item-1767695202393-kg1ojpdo6" data-list-item-path="item-1767695202393-kg1ojpdo6"><div class="list-item-paragraph">Given that&nbsp;$9$ 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 diameter 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

Volume and Surface Area of Cylinder

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTQ4MDE1NjctOWx3MnQ5YzhsaDQucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg1MTA4MSwiZXhwIjoxNzkyODUxMzgxfQ.o9JSfTkY0q9vm8zh-MJDG0mCtohns8F9LwaOj_xun_M" 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, where the circular portion is a quadrant, and an isosceles right-angled triangular prism is removed from the solid. 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/1767694543950-zsch4vo2if.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTQ1NDM5NTAtenNjaDR2bzJpZi5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyODUxMDgxLCJleHAiOjE3OTI4NTEzODF9.-BV758j_mv7OVledWwrx4rrXJvLufltDKR6iGSq_DXE" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">the area of the cross section in cm$^2$,</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">the total surface area of the solid in cm$^2$, and</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">the volume of the solid in cm$^3$.</div></li></ol>
<p>The source labels the sloping cut as 240 mm. This is rounded: its two perpendicular sides are 170 mm, so the exact sloping length is $170\sqrt2\text{ mm}$. Use the exact sloping length before rounding the requested answers.</p>

The figure below shows a solid of a uniform cross section, where the circular portion is a quadrant, and an isosceles right-angled triangular prism is removed from the solid. Find, correcting your answers to the nearest integer,

the area of the cross section in cm$^2$,

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

The source labels the sloping cut as 240 mm. This is rounded: its two perpendicular sides are 170 mm, so the exact sloping length is $170\sqrt2\text{ mm}$. Use the exact sloping length before rounding the requested answers.

<p>The figure below shows a solid that is the remaining part of a cuboid, where a half-cylinder and two identical triangular prisms with isosceles right-angled triangular base are removed from the prism. 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/1767694445113-yzs28ilx1a9.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTQ0NDUxMTMteXpzMjhpbHgxYTkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg1MTA4MSwiZXhwIjoxNzkyODUxMzgxfQ.Bi-vmgLuZtUJRNf6rOWdms1oQjuxK34ArVuCgzv3iyk" 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 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 that is the remaining part of a cuboid, where a half-cylinder and two identical triangular prisms with isosceles right-angled triangular base are removed from the prism. 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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTQzMzA5MzMtcXF2endlMmw0cGsucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg1MTA4MSwiZXhwIjoxNzkyODUxMzgxfQ.toC-RMTp610eYUzwf4ZaJDAiDY26MOR4GjeXe8xqiTI" 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 prism, where the cross section consists of a right-angled triangle, a rectangle and an isosceles trapezium.</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/1767694063028-xrit7n5zous.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTQwNjMwMjgteHJpdDduNXpvdXMucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg1MTA4MSwiZXhwIjoxNzkyODUxMzgxfQ.g46CSonx8qOfFTpYc9RL6TXn8QTCpwDnJfYrGHRREmA" 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,</div></li><li><div class="list-item-paragraph">
the total surface area of the prism, and</div></li><li><div class="list-item-paragraph">the volume of the prism.</div></li></ol>

The figure below shows a prism, where the cross section consists of a right-angled triangle, a rectangle and an isosceles trapezium.

<p>The figure below shows a prism, where the cross section consists of a rectangle and two semicircles.</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/1767693945438-993s50xi8sf.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTM5NDU0MzgtOTkzczUweGk4c2YucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg1MTA4MSwiZXhwIjoxNzkyODUxMzgxfQ.LO8ebcduUVNiYGgyUM4lD-a1qwszdeRQtSOWV_VzgxM" 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,</div></li><li><div class="list-item-paragraph">the total surface area of the prism, and</div></li><li><div class="list-item-paragraph">the volume of the prism.</div></li></ol>

The figure below shows a prism, where the cross section consists of a rectangle and two semicircles.

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjUyNjI0MzU3OTAtanE1b3hmeWZqeC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyODUxMDgxLCJleHAiOjE3OTI4NTEzODF9.bN2JwdHPCuVSIPEYr8oK7b-slSmb9xKqPqL-JHcawlI" 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.

Pythagoras’ Theorem in Three Dimensions

Hemisphere Minus Cone: Surface Area, Volume & Mass Application

<p>In a restaurant, the fruit punch is mixed in a container with no lid as shown in Figure 1. The container is a prism whose cross-section is a trapezium. The lengths of the parallel sides of the trapezium are $10$ cm and $15$ cm and the distance between the parallel sides is $9$ cm. It is given that the capacity of the container is $1.8$ litres.</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/1765261978122-37qq3cy9096.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjUyNjE5NzgxMjItMzdxcTNjeTkwOTYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg1MTA4MSwiZXhwIjoxNzkyODUxMzgxfQ.6AkIpcDWtwdswSdbe1C2uQ5GRwhbMmFVldtjErsc8XM" style="max-width: min(370px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that the height of the container is $16$ cm.</div></li></ol>
<p>All the fruit punch in one full container is transferred to a jar as shown in Figure 2.</p>
<p>The jar is made by joining a hemisphere of radius $8$ cm to a part of a right circular cone of height, $h$ cm.</p>
<ol class="abc-list" start="2"><li><div class="list-item-paragraph">Given that the slant height of the cone is $17$ cm, show that&nbsp;$h=15$ cm.</div></li><li><div class="list-item-paragraph">Find the depth of the fruit punch in this jar after all the fruit punch has been transferred 	from the container in Figure 1.</div></li></ol>

In a restaurant, the fruit punch is mixed in a container with no lid as shown in Figure 1. The container is a prism whose cross-section is a trapezium. The lengths of the parallel sides of the trapezium are $10$ cm and $15$ cm and the distance between the parallel sides is $9$ cm. It is given that the capacity of the container is $1.8$ litres.

All the fruit punch in one full container is transferred to a jar as shown in Figure 2.

The jar is made by joining a hemisphere of radius $8$ cm to a part of a right circular cone of height, $h$ cm.

Conservation of Volume and Displacement

Volume Transfer Between Composite Solids

<p>A metal structure formed by a cone of radius $15$ cm and height $34$ cm and a cylindrical base of the same radius and height $h$ cm is shown below.</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="abc-list"><li><div class="list-item-paragraph">Given that the volume of the structure is $15080$ cm$^{3}$, find the height, $h$ cm of the cylindrical base.</div></li><li><div class="list-item-paragraph">The solid is melted and recast into $30$ identical hemispherical shapes. Find the radius of each hemisphere.</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/1765260959599-okf6hwhyp1h.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjUyNjA5NTk1OTktb2tmNmh3aHlwMWgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg1MTA4MSwiZXhwIjoxNzkyODUxMzgxfQ.1Z-bMOJfSSQHIiWFOfRz91N2gm27t4KF_vTLn6glnWM" style="max-width: min(200px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

A metal structure formed by a cone of radius $15$ cm and height $34$ cm and a cylindrical base of the same radius and height $h$ cm is shown below.

Volume and Melting Application of a Composite Solid

<p>A hollow hemispheric container has an external radius of $15$ cm and an internal radius $12$ cm.</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>Given that&nbsp;$\pi =3.142$,</p><ol class="abc-list"><li><div class="list-item-paragraph">find the total surface area of the container.</div></li><li><div class="list-item-paragraph">find the volume of the container.</div></li><li><div class="list-item-paragraph">If the hemispheric container is melted and recast into small cubes of length $2$ cm on each side. Find the maximum number of cubes which can be made.</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/1765260560487-k69ebacwypa.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjUyNjA1NjA0ODctazY5ZWJhY3d5cGEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg1MTA4MSwiZXhwIjoxNzkyODUxMzgxfQ.XIVXsMjQcnjqar2duenTviC7DjyIt5zI4AW1zOH_HuY" style="max-width: min(220px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

A hollow hemispheric container has an external radius of $15$ cm and an internal radius $12$ cm.

Hollow Hemisphere: Surface Area and Volume

<p>The iron solid is made up of a hemisphere joined to a cylinder with a radius of $6$ cm and a height of $10$ cm as shown.</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="abc-list"><li><div class="list-item-paragraph">Calculate</div><ol class="i-list"><li><div class="list-item-paragraph">the total volume of the solid, </div></li><li><div class="list-item-paragraph">the total surface area of the solid.</div></li></ol></li><li><div class="list-item-paragraph">If the solid is melted to form a pyramid of a base area of $250$ cm$^{2}$, find the height of the pyramid.</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/1765260054310-ofpdiqfhk2i.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjUyNjAwNTQzMTAtb2ZwZGlxZmhrMmkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg1MTA4MSwiZXhwIjoxNzkyODUxMzgxfQ.tQBIqiHJHkm1UslnYij4ktoqHYwO34qmme32ElHsG3k" style="max-width: min(200px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

The iron solid is made up of a hemisphere joined to a cylinder with a radius of $6$ cm and a height of $10$ cm as shown.

Math Question Bank