Circles, Arcs & Sectors (OCR GCSE Maths: Higher): Exam Questions

Jon has a flower garden in the shape of a circle.
The diameter of the garden is 5 metres.

Jon wants to put fencing around the edge of the garden.
The fencing costs £1.80 per metre.

Work out the total cost of the fencing.

£ .................................

The diagram shows a circle drawn inside a square.

The circle has a radius of 6 cm.
The square has a side of length 12 cm.

Work out the shaded area.
Give your answer in terms of $\pi$.

The diagram shows a circle inside a square.

$ABCD$ is a square of side 10cm.
Each side of the square is a tangent to the circle.

Work out the total area of the shaded regions in terms of $\pi$.
Give your answer in its simplest form.

The diagram shows the top of Levi's birthday cake.

The top of the cake is in the shape of a circle.
The diameter of the circle is 7 inches.

A ribbon is going to be put around the side of the cake.
Ribbons are sold in 50 cm lengths.

1 inch is 2.54 cm.

Work out if one length of ribbon is long enough to go all the way around the cake.
You must show your working.

Hasmeet walks once round a circle with diameter 80 metres.

There are 8 points equally spaced on the circumference of the circle.

Find the distance Hasmeet walks between one point and the next point.

Four of the points are moved, as shown in the diagram below.

Hasmeet walks once round the circle again.

Has the mean distance that Hasmeet walks between one point and the next point changed?
You must give a reason for your answer.

$OAB$ is a sector of a circle with centre $O$ and radius 7 cm.

The area of the sector is 40 cm².

Calculate the perimeter of the sector.
Give your answer correct to 3 significant figures.

The shape below is formed from two semicircles and a straight line.

The radius of the large semicircle is 8cm.
The radius of the small semicircle is $t$cm.

Find an expression, in terms of $t$, for the exact perimeter of the shaded shape.

........................ cm

The design below is made from two sectors of circles, centre O.

Calculate the perimeter of the shaded part.
Give your answer correct to 3 significant figures.

......................... cm

Points $P$, $Q$ and $R$ lie on the circumference of a circle, centre $O$.

Angle $PQR=90°$, $PQ=12$ cm and $QR=9$ cm.

Work out the area of the circle.

You must show your working.

The diagram shows a circle with centre $O$

$A,B$ and $C$are points on the circle so that the length of the arc $ABC$ is 5 cm.

Given that angle $AOC$ = 55°

work out the area of the circle.
Give your answer correct to one decimal place.

A circle has an area of 50 cm².

Show that the circumference of the circle is 25.1 cm, correct to 3 significant figures.

The region, shown shaded in the diagram, is a path.

The boundary of the path is formed by two semicircles, with the same centre $O$, and two straight lines.

The inner semicircle has a radius of 7 metres.
The path has a width of 2 metres.

Work out the perimeter of the path.
Give your answer correct to one decimal place.

...................................................... m

A circle centre $O$ has radius 9 cm.

Calculate the perimeter of the shaded sector of the circle.
Give your answer correct to 3 significant figures.

...................................................... cm

The diagram shows Yuen’s garden.

The garden is in the shape of a semicircle of radius $7.2\mathrm{m}$.
Yuen is going to cover his garden with grass seed.

Yuen has 12 boxes of grass seed.
Each box of grass seed contains enough seed to cover $6{\mathrm{m}}^2$ of the garden.

Has Yuen enough grass seed for his garden?
Show your working clearly.

Two wire shapes make an earring.
The shapes are

a circle with radius 21 mm

and

a quarter circle.

Show that the radius of the quarter circle is 6 mm.

Work out the total length of the wire in the earring.

Give your answer in the form $a\pi +b$ where $a$ and $b$ are integers.

...............................mm

A shaded semicircle is inside a circle as shown.

The radius of the circle is 10 cm
The diameter of the semicircle is 8 cm

How many times bigger is the unshaded area than the shaded area?

Here are two sectors from different circles.

Which sector has the bigger area?

Tick a box.

$\square$ Sector A            $\square$ Sector B

Show working to support your answer.

Two identical quarter circles are cut from a rectangle as shown.

Work out the shaded area.

..........................cm²

Here is a circle touching a square.

The area of the square is 64 cm²

Work out the area of the circle.

Give your answer in terms of $\pi$.

........................................ cm²

The diagram shows a plan of Brian's lawn.

The edge of the lawn consists of two semicircles and two straight lines.
Each semicircle has centre $O$.
The diameters of the semicircles are 9 m and 5 m.

Brian is going to put lawn edging around the edge of the lawn. Lawn edging is sold in 2.4 metre rolls.

Brian has £35

Has Brian got enough money to buy all the rolls of lawn edging he needs?
You must show all your working.

Saphia is organising a conference.

People at the conference will sit at circular tables.

Each table has a diameter of 140 cm.
Each person needs 60 cm around the circumference of the table.

There are 12 of these tables in the conference room.
A total of 90 people will be at the conference.

Are there enough tables in the conference room?

$ONQ$ is a sector of a circle with centre $O$ and radius 11 cm.

$A$ is the point on $ON$ and $B$ is the point on $OQ$ such that $AOB$ is an equilateral triangle of side 7 cm.

Calculate the area of the shaded region as a percentage of the area of the sector $ONQ$.
Give your answer correct to 1 decimal place.

The diagram shows a square $ABCD$ inside a circle.

The points $A,B,C$ and $D$ lie on the circle.

The radius of the circle is 6 cm.

Work out the total area of the shaded regions.
Give your answer correct to 3 significant figures.

The diagram shows a logo made from three circles.

Each circle has centre $O$.
Daisy says that exactly $\frac{1}{3}$ of the logo is shaded.
Is Daisy correct?
You must show all your working.

The arc $ABC$ is a quarter of a circle with centre $O$ and radius 4.8 cm.
$AC$ is a chord of the circle.

Work out the area of the shaded segment.
Give your answer correct to 3 significant figures.

$OAB$ is a sector of a circle, centre $O$.
$OCD$ is a sector of a circle, centre $O$.
$OCA$ and $ODB$ are straight lines.

Angle $AOB=75°$

$$\begin{gathered} OD=6\mathrm{cm} \\ DB=4\mathrm{cm} \end{gathered}$$

Calculate the perimeter of the shaded region.
Give your answer correct to 3 significant figures.

$A,B$ and $C$ are points on a circle of radius 5 cm, centre $O$.
$DA$ and $DC$ are tangents to the circle.
$DO=9\mathrm{cm}$

Work out the length of arc $ABC$.
Give your answer correct to 3 significant figures.

$OAC$ is a sector of a circle, centre $O$, radius 10 m.

$BA$ is the tangent to the circle at point $A$.
$BC$ is the tangent to the circle at point $C$.

Angle $AOC=120°$

Calculate the area of the shaded region.
Give your answer correct to 3 significant figures.

AOB is a sector of a circle, centre O and radius 6cm.
The length of arc AB is $5\mathrm{\pi }$cm.

Find the area of the sector.
Give your answer in terms of $\mathrm{\pi }$.

......................... cm²

The diagram shows a semi-circle inside a rectangle of length 120 m.
The semi-circle touches the rectangle at A, B and C.

Calculate the perimeter of the shaded region.
Give your answer correct to 3 significant figures.

...................................................... m

Four pencils are held together with a band.
The figure below shows the bottom end of the pencils and the band.

Each of the pencils has diameter 9 mm.
Find the length of the band in this position.

............................ mm

The diagram shows a shaded shape $ABCD$made from a semicircle $ABC$and a right-angled triangle $ACD$.

$AC$is the diameter of the semicircle $ABC$.

Work out the perimeter of the shaded shape.
Give your answer correct to 3 significant figures.

....................$\mathrm{cm}$

The diagram shows a shaded shape made by removing sector $OAB$ from sector $OCD$.

Both sectors have an angle of 45°.

The radius, $OA$, of the smaller sector is $r\text{ cm}$.

The ratio of radius $OA$ to radius $OC$ is $1:3$.

Work out, in terms of $\pi$ and $r$, the perimeter of the shaded shape $ACDB$.

Give your answer in its simplest form.

You must show your working.

The diagram shows a shape ABCDE.
The shape is made from a rectangle, a right-angled triangle and a quarter of a circle.

F is the mid-point of BD.
AE = 18 m and the perpendicular distance from C to AE is 41 m.

Work out the perimeter of the shape ABCDE.

....................................................... m

OAB is a sector of a circle, centre O.

OA = 6 cm and AX is perpendicular to OB.

The area of sector OAB is 6$\pi$cm².
Show that AX = 3 $\sqrt{3}$ cm.

A, B, C and D are points on the circumference of a circle, centre O.

Angle CAD = 28° and CD = 6.4cm.
BD is a diameter of the circle.

Calculate the area of the circle.

...................... cm²

Here are two watch faces, A and B.

Both watch faces are circular with radius 2cm.

The materials used to make both watch faces have the same thickness.

A is made entirely of plastic.
B has a 20° sector of metal and a 340° sector of plastic.

The ratio of the cost per cm² of the metal to the cost per cm² of the plastic is 3 : 2

Work out the ratio of the cost of the materials for A to the cost of the materials for B.
Give your answer in its simplest form.
You must show all your working.

Three identical small circles are drawn inside one large circle, as shown in the diagram.
The centres of the small circles lie on the diameter of the large circle.

Find the fraction of the large circle that is shaded.

In the diagram, $ABC$ is a right-angled triangle and $DEF$ is a semicircular arc.

In triangle $ABC$

The points $D$ and $F$ lie on $AC$ so that $DF$ is the diameter of the semicircular arc $DEF$
The radius of the semicircular arc is 3cm.

Work out the length of $AFEDC$
Give your answer correct to 2 significant figures.

The diagram shows a sector $OBC$ of a circle with centre $O$ and radius $(6+x)\mathrm{cm}$.

$A$ is the point on $OB$ and $D$ is the point on $OC$ such that $OA=OD=6\mathrm{cm}$
Angle $BOC=50°$

Given that

the perimeter of sector $OBC=2\times$the perimeter of triangle $OAD$

find the value of $x$.
Give your answer correct to 3 significant figures.

$A,B$ and $C$ are points on a circle with centre $O$.

Angle $ABC=75°$

The area of the shaded segment is 200cm²

Calculate the radius of the circle.
Give your answer correct to 3 significant figures.

...................................................... cm

The diagram shows two circles such that the region $\mathbf{R}$, shown shaded in the diagram, is the region common to both circles.

One of the circles has centre $O$ and radius 5cm.
The other circle has centre $P$ and radius 4 cm.
Angle $AOB=50°$

Calculate the area of region $\mathbf{R}$.
Give your answer correct to 3 significant figures.

...................................................... cm²

Here is a sector, $AOB$ , of a circle with centre $O$and angle $AOB$= $x°$

The sector can form the curved surface of a cone by joining $OA$to $OB$ .

The height of the cone is $25\mathrm{cm}$.
The volume of the cone is $1600{\mathrm{cm}}^3$

Work out the value of $x.$
Give your answer correct to the nearest whole number.

$x=$.......................

$OPQ$ is a sector of a circle, centre $O$
$OAB$ is a sector of a circle, centre $O$

$$\begin{gathered} A\mathrm{is}\mathrm{the}\mathrm{point}\mathrm{on}OP\mathrm{such}\mathrm{that}OA:AP=3:2 \\ B\mathrm{is}\mathrm{the}\mathrm{point}\mathrm{on}OQ\mathrm{such}\mathrm{that}OB:BQ=3:2 \end{gathered}$$

Angle $POQ=45°$

The area of the shaded region is $\frac{81}{2}\pi {\mathrm{cm}}^2$
Work out the perimeter of the shaded region.
Give your answer in terms of $\pi$.

The diagram shows a sector $OAPB$ of a circle, centre $O$

$AB$ is a chord of the circle.

Angle$AOB=80°$

The area of sector $OAPB$ is $\frac{25}{2}\mathrm{\pi }{\mathrm{cm}}^2$

Work out the perimeter of the shaded segment.
Give your answer correct to 3 significant figures.

......................$\mathrm{cm}$

The vertices of a regular hexagon lie on a circle with centre $O$ and radius 5 cm

Work out the shaded area.
Give your answer in the form $\frac{a\pi -b\sqrt{c}}{12}$ where $a,b$ and $c$ are integers.

Here are a circle and a sector of the circle.

They each have radius $r$.

circumference of circle = perimeter of sector

Work out the size of angle $x$.

Give your answer in terms of $\mathrm{\pi }$.

...................................degrees

A motor racing circuit consists of

   two parallel straight sections, each of length 0.75 km
   a semicircle of diameter 0.9 km
   three equal, smaller semicircles.

The length of a motor race must be greater than 305 km

What is the lowest number of full laps needed at this circuit?
You must show your working.

Here is a tunnel for a toy train.

The diagram below shows the cross section of the tunnel.

$AD$ is a semicircular arc of radius 10 cm
$BC$ is a semicircular arc of radius 7 cm
The length of the tunnel is 30 cm

Work out the total area of all six faces of the tunnel.
Give your answer in terms of $\pi$.

....................cm²

The cross section of an earring is a semicircle, centre $C$, radius 25 mm
The earring is black and white.

The shaded area is black.

Sector $BCD$ is white and has radius 12 mm

Is more than 20% of the semicircle white?

You must show your working.

The diagram shows a logo.

$ABE$ and $DCE$ are congruent triangles.
$BCE$ is a sector of a circle, centre $E$.

Show that the area of the logo is 510 cm² to 2 significant figures.