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The base and the height of a triangle are each reduced by 40% followed by an increase by 50%. Find the percentage reduction in the area of the triangle.

The base and the height of a triangle are each reduced by $40\%$ followed by an increase by $50\%$. Find the percentage reduction in the area of the triangle.

Percentage Change and Reverse Percentage

Perimeter and Area of Common Plane Figures

2025 CCHY PRELIMS P1 Q10

Percentages

Perimeter and Area of Plane Figures

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3NzA1NDgyNjYtemYyc3h4MHp4Yi5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg5NjEyODI4LCJleHAiOjE3ODk2MTMxMjh9.HoPsbIIveAapT-25ne6QWDFLuDUGFpSVZEm4Ih62JlA" 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.

Volume and Surface Area of Prisms

Mensuration (Pyramid, Cones and Spheres)

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3NzA0MTE1NTUta3A3YTNsZHBmMy5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg5NjEyODI4LCJleHAiOjE3ODk2MTMxMjh9.gYtCOyD39IlYyxkv-_Nms4tKHZb0xZxnu6n7AYVdgf8" 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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3NzAyMzA1NTctZXA2dHk1czRoMXYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYxMjgyOCwiZXhwIjoxNzg5NjEzMTI4fQ.mh2DZbTIxeQ_Wo77HiqQalM1C9CirP98-BD6MNaGB60" 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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3NzAwMTQzNzAtOHVlcmVzMnEweGwucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYxMjgyOCwiZXhwIjoxNzg5NjEzMTI4fQ.NNxlWOksb_ywoM2txOCEoK03Sj1tFRC3nzIPXth91o8" 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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3Njk2ODMwMTUtcWUyNXpqNDVlbGkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYxMjgyOCwiZXhwIjoxNzg5NjEzMTI4fQ.zYasSpAac_3lP4zNoaWlba-OE6yZzJ3jicnLxiKe5tg" 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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3Njk1MjMyNDctcDUxMnZydmM0My5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg5NjEyODI4LCJleHAiOjE3ODk2MTMxMjh9.Ihlz1yczFDMLTpykWzKfhKryeQ7iMgl5A7dIlVhn78I" 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 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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTE0MjY1MzAtcTR3eHBoampmbjkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYxMjgyOCwiZXhwIjoxNzg5NjEzMTI4fQ.bX_M0il60q-2fGzA7qYkIWVnc4oEg5iwmNID5-bxWIU" 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.

<p>In the figure below, $ABCD$ and $PQRS$ are identical squares, where $P$ is the centre of $ABCD$. Given that $C$ is the centre of $PQRS$, such that the shaded region is also bounded by a square. Find the area of the shaded region where the two squares overlap.</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/1767691181895-jy8apto1qmq.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTExODE4OTUtank4YXB0bzFxbXEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYxMjgyOCwiZXhwIjoxNzg5NjEzMTI4fQ.g0qrlAXuONv_Nmm37qMQx-E9BKO64eY5WqBmVly_HKw" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>

In the figure below, $ABCD$ and $PQRS$ are identical squares, where $P$ is the centre of $ABCD$. Given that $C$ is the centre of $PQRS$, such that the shaded region is also bounded by a square. Find the area of the shaded region where the two squares overlap.

<p>In the figure below, arcs $BAC$, $DAE$ are semicircles and arcs $BD$, $CE$ are quadrants that are fitted in a grid consisting of $8$ identical squares of length of sides of $2$ cm. Find the perimeter and the area of the shaded region. Leave your answer in terms of $\pi$.</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/1767691079044-v81etk5i5g.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTEwNzkwNDQtdjgxZXRrNWk1Zy5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg5NjEyODI4LCJleHAiOjE3ODk2MTMxMjh9.berNh6swqkG3EtiUMENrmojPaIjeHkIG6nwAVIA5Vh0" style="max-width: min(350px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>

In the figure below, arcs $BAC$, $DAE$ are semicircles and arcs $BD$, $CE$ are quadrants that are fitted in a grid consisting of $8$ identical squares of length of sides of $2$ cm. Find the perimeter and the area of the shaded region. Leave your answer in terms of $\pi$.

<p>In the figure below, $ABCD$ is a square and the circular portions are two quadrants. Find the perimeter and the area of the shaded region. Leave your answer in terms of $\pi$.</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/1767690931081-tndtp9hf4vo.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTA5MzEwODEtdG5kdHA5aGY0dm8ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYxMjgyOCwiZXhwIjoxNzg5NjEzMTI4fQ.SbPBpAxDLSltkNEjOKOf7oVQOTlHbZnMKWnrHOvZk_o" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>

In the figure below, $ABCD$ is a square and the circular portions are two quadrants. Find the perimeter and the area of the shaded region. Leave your answer in terms of $\pi$.

<p>A roller brush used for wall-painting has a circular cross section of diameter $8$ cm, and the breadth of the roller is $21$ cm. A painter uses this roller brush to paint a wall of breadth $4.2$ m and height $5$ m. He paints in vertical rectangular paint columns of breadth $21$ cm and in a speed of $10$ cm/s in the vertical direction. </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/1767690677788-jn9mox3202r.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTA2Nzc3ODgtam45bW94MzIwMnIucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYxMjgyOCwiZXhwIjoxNzg5NjEzMTI4fQ._flkrSo0mzlfRp3ubIrdsfU1Aj44re8CU4Gg5BGEN8g" style="max-width: min(340px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Assuming there are no overlapping paint columns, find</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the number of paint columns needed to finish painting the whole area of the wall,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the number of rotations of the roller to finish painting one column, correcting your answer to the nearest integer,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the least possible time needed for the painter to fully paint the wall.</span></div></li></ol>

A roller brush used for wall-painting has a circular cross section of diameter $8$ cm, and the breadth of the roller is $21$ cm. A painter uses this roller brush to paint a wall of breadth $4.2$ m and height $5$ m. He paints in vertical rectangular paint columns of breadth $21$ cm and in a speed of $10$ cm/s in the vertical direction. 

Assuming there are no overlapping paint columns, find

the number of paint columns needed to finish painting the whole area of the wall,

the number of rotations of the roller to finish painting one column, correcting your answer to the nearest integer,

the least possible time needed for the painter to fully paint the wall.

<p>The figure below shows the diagram of a rectangle-outlined community, which has four rectangular blocks that are by the sides of two perpendicular straight pavements.</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/1767690521067-gew9wpqcxvt.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTA1MjEwNjctZ2V3OXdwcWN4dnQucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYxMjgyOCwiZXhwIjoxNzg5NjEzMTI4fQ.bPcJ9vIcuQoI7ApNe847x0TJeRosAXK7H4Dw3z-omJI" style="max-width: min(375px, 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 total area of the pavements,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the ratio of the area of Block C to the area of Block B,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the total area of four blocks as a percentage of the area of the whole community.</span></div></li></ol>

The figure below shows the diagram of a rectangle-outlined community, which has four rectangular blocks that are by the sides of two perpendicular straight pavements.

the ratio of the area of Block C to the area of Block B,

the total area of four blocks as a percentage of the area of the whole community.

<p>In the figure below, $ABCD$ is a parallelogram and $E$ is a point on $AD$, $F$ is a point on $BC$ such that $\frac{BF}{BC} = \frac{3}{4}$.</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/1767690388269-n0k9lnh3l5.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTAzODgyNjktbjBrOWxuaDNsNS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg5NjEyODI4LCJleHAiOjE3ODk2MTMxMjh9.ZmmEx1mvkBYDk5ebYtLbx3FEmL9leWCfo9K-eX3QIvg" 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-content"><span class="list-item-text">the areas of parallelogram $ABCD$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the area of the shaded region,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the ratio of the area of the shaded region to the area of parallelogram $ABCD$.</span></div></li></ol>

In the figure below, $ABCD$ is a parallelogram and $E$ is a point on $AD$, $F$ is a point on $BC$ such that $\frac{BF}{BC} = \frac{3}{4}$.

the ratio of the area of the shaded region to the area of parallelogram $ABCD$.

<p>In the figure below, three smaller circles are fitted in a larger circle, which share a common diameter.</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/1767690256715-uhfbcr6uno.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTAyNTY3MTUtdWhmYmNyNnVuby5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg5NjEyODI4LCJleHAiOjE3ODk2MTMxMjh9.4L_g2pvepaLQfRhCnvWk2zvd13fJnvswXc1n_0SJEDM" 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 perimeter and the area of the shaded region,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the ratio of the area of the shaded region to the area of the largest circle.</span></div></li></ol>

In the figure below, three smaller circles are fitted in a larger circle, which share a common diameter.

the perimeter and the area of the shaded region,

the ratio of the area of the shaded region to the area of the largest circle.

<p>In the figure below, the circular portions are a quadrant and a semicircle. 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/1767690177389-meme0e9w09h.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTAxNzczODktbWVtZTBlOXcwOWgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYxMjgyOCwiZXhwIjoxNzg5NjEzMTI4fQ.rsvEqc0zuhWnMHqwi4h4bXCyh-FGRYyngJpG1bBOieY" style="max-width: min(200px, 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 perimeter and the area of the shaded region,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the ratio of the area of the shaded region to the area of the larger quadrant.</span></div></li></ol>

In the figure below, the circular portions are a quadrant and a semicircle. Find

the ratio of the area of the shaded region to the area of the larger quadrant.

<p>In the figure below, four identical sectors are removed from a square, where the circular portions are quadrants. Find the perimeter and the area of the shaded region.</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/1767689969999-mfwyy26nwt.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2ODk5Njk5OTktbWZ3eXkyNm53dC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg5NjEyODI4LCJleHAiOjE3ODk2MTMxMjh9.q_wTIW5WNU6WNNrV_DBscStYHRZTRq0Yk0oW6AFyUJ0" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>

In the figure below, four identical sectors are removed from a square, where the circular portions are quadrants. Find the perimeter and the area of the shaded region.

<p>In the figure below, $ABCD$ is an isosceles trapezium, where circle $O$ touches the two parallel sides of trapezium $ABCD$. 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/1767689858177-qcmttf4crl.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2ODk4NTgxNzctcWNtdHRmNGNybC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg5NjEyODI4LCJleHAiOjE3ODk2MTMxMjh9.PQWK6RHxf4nmKxCyePCIaCzvtGZUiwe4ut7XuVow340" 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 area of the shaded region,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the ratio of the area of the shaded region to the area of trapezium $ABCD$. Leave your answer in terms of $\pi$.</span></div></li></ol>

In the figure below, $ABCD$ is an isosceles trapezium, where circle $O$ touches the two parallel sides of trapezium $ABCD$. Find

the ratio of the area of the shaded region to the area of trapezium $ABCD$. Leave your answer in terms of $\pi$.

<ol class="i-list"><li><div class="list-item-paragraph">The figure below shows a ring that consists of two concentric circles. Find the area of the shaded region.</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-14-i-diagram-.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvbWlncmF0ZWQvMjAyNC8wNS9RdWVzdGlvbi0xNC1pLWRpYWdyYW0tLnBuZyIsInNjb3BlIjoiZG93bmxvYWQiLCJpYXQiOjE3ODk2MTI4MjgsImV4cCI6MTc4OTYxMzEyOH0.x80sJi_h7PZcyRGJdRW2-zjxNCytjECRYAowpRDBlGE" alt="List item image" class="aligncenter custom-width" style="max-width: min(300px, 100%); height: auto;" /></div></li></ol>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">The figure below shows a half of a ring that consists of two concentric circles. Find the perimeter and the area of the shaded region.</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-14-ii-diagram-.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvbWlncmF0ZWQvMjAyNC8wNS9RdWVzdGlvbi0xNC1paS1kaWFncmFtLS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg5NjEyODI4LCJleHAiOjE3ODk2MTMxMjh9.VW1ngtSimZVKK8Tiiwvh-4EEmrjLYiXZbpzNcy8e7HE" alt="List item image" class="aligncenter custom-width" style="max-width: min(300px, 100%); height: auto;" /></div></li></ol>

The figure below shows a ring that consists of two concentric circles. Find the area of the shaded region.

The figure below shows a half of a ring that consists of two concentric circles. Find the perimeter and the area of the shaded region.

<ol class="i-list"><li><div class="list-item-paragraph">In the figure below, $ABCD$
 is a square. Find the perimeter and the area of the shaded region.</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-13-i-diagram-.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvbWlncmF0ZWQvMjAyNC8wNS9RdWVzdGlvbi0xMy1pLWRpYWdyYW0tLnBuZyIsInNjb3BlIjoiZG93bmxvYWQiLCJpYXQiOjE3ODk2MTI4MjgsImV4cCI6MTc4OTYxMzEyOH0.CzNyPxx56iYcIwjUdwEa4mjh9a2hXtN1eYHEbQ4WffY" alt="List item image" class="aligncenter custom-width" style="max-width: min(300px, 100%); height: auto;" /></div></li><li><div class="list-item-paragraph">In the figure below, $ABCD$
 is a square. Find the perimeter and the area of the shaded region.</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-13-ii-diagram-.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvbWlncmF0ZWQvMjAyNC8wNS9RdWVzdGlvbi0xMy1paS1kaWFncmFtLS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg5NjEyODI4LCJleHAiOjE3ODk2MTMxMjh9.uzn-lrPKbnFzFhVwmH2v3ouhVx29j9RXL0dWa2JAgew" alt="List item image" class="aligncenter custom-width" style="max-width: min(300px, 100%); height: auto;" /></div></li></ol>

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