In this question the units are metres.

The diagram shows a circle, centre \(O\) and radius 2 .
The chord \(A B\) has length \(2 \sqrt{3}\).
The point \(Q\) lies on the circle such that \(A Q=B Q\).
The \(\operatorname{arc} A P B\) is part of a circle, centre \(Q\).
(a) Find the exact value of angle \(A Q B\) in radians.

(b) Hence find the area of the shaded region. Give your answer in terms of \(\pi\).

The diagram shows the shaded region \(ABCD\). The lines \(AC\) and \(BD\) each have length \(12\text{ cm}\) and bisect each other at \(O\). The lines \(AD\) and \(BC\) are parallel and each has length \(4\text{ cm}\). The arcs \(AB\) and \(DC\) are part of a circle with centre \(O\).

(a) Find the obtuse angle \(AOB\), giving your answer in radians.

(b) Use your answer to part (a) to find

(i) the perimeter of the shaded region,

(ii) the area of the shaded region.

(a) The diagram shows an isosceles triangle. Find the value of \(\theta\) in radians.

(b) The diagram shows a shape made of two arcs. Each arc is part of a circle with radius \(5\text{ cm}\). The height of the shape is \(8\text{ cm}\).

Use your answer to part (a) to find (i) the perimeter of the shape and (ii) the area of the shape.

The diagram shows a circle with centre \(O\) and radius \(5\text{ cm}\).

The point \(A\) lies on the circle. The point \(B\) is such that the line \(AB\) is a tangent to the circle. \(OB\) has length \(13\text{ cm}\).

(a) Find angle \(AOB\), giving your answer in radians.

(b) Find the perimeter of the shaded region.

(c) Find the area of the shaded region.

In this question all lengths are in metres.

The diagram shows a shape \(A B C D E F\). \(A B, B D\) and \(D E\) are three sides of a rectangle. \(O\) is the mid-point of \(B D\). \(A F E\) is an arc of a circle whose centre is \(O\). \(A B=\sqrt{3}, B C=C D=5\) and \(B D=6\). (a) Find the exact value of the perimeter of the shape, giving your answer in terms of \(\pi\).

(b) Find the exact value of the area of the shape, giving your answer in terms of \(\pi\).

The diagram shows a design for a logo. The logo is a sector of a circle, radius \(r \mathrm{~cm}\), with angle \(\alpha\) radians.

The area of the logo is \(9 \mathrm{~cm}^{2}\). (a) Show that the perimeter, \(P \mathrm{~cm}\), of the logo is given by \(P=2 r+\frac{18}{r} .\) (b) Given that \(r\) can vary, find the stationary value of \(P\) and determine its nature.

In this question all lengths are in centimetres and all angles are in radians.

The diagram shows a company logo. Each part of the logo is a sector of a circle with centre \(O\). Sector \(A O B\) has radius \(x\). Sector \(C O D\) has radius \(x+2\). Sector \(E O F\) has radius \(y\). The shaded region has area \(A \mathrm{~cm}^{2}\) and perimeter 24 . It is given that \(x\) and \(y\) can vary. (a) Show that \(A=\frac{91}{8} x^{2}-68 x+132\).

(b) Use differentiation to find the minimum possible area of the logo.

In this question, all lengths are in metres and all angles are in radians.

The diagram shows a circle with centre \(O\) and radius 5. The points \(A, B, C\) and \(D\) lie on the circumference of the circle. Angle \(D O C=\theta\). Angle \(A O D=\) angle \(C O B=0.5\). The length of the minor \(\operatorname{arc} D C\) is 3.75 . (a) Show that \(\theta=0.75\).

(b) Find the perimeter of the shaded region.

(c) Find the area of the shaded region.

The diagram shows a circle centre \(O\) with radius \(6\). The line \(AB\) is a tangent to the circle at the point \(B\). The point \(C\) lies on the circle such that \(AOC\) is a straight line. \(AB=8\).

(a) Find the perimeter of the shaded region.

(b) Find the area of the shaded region.

In the diagram, \(A D\) and \(B C\) are arcs of circles with common centre \(O\). \(O D C\) and \(O A B\) are straight lines with \(O A=5 \mathrm{~cm}\) and \(A B=4 \mathrm{~cm}\). Angle \(B O C=\theta\) radians. The area of the shaded region \(A B C D\) is \(4 \pi \mathrm{~cm}^{2}\). (a) Find \(\theta\).

(b)

The straight line \(A C\) is added to the diagram and the region \(A C D\) is now shaded. Find the perimeter of the shaded region \(A C D\).

In this question all lengths are in centimetres and all angles are in radians.

The diagram shows a circle with centre \(O\) and radius \(r\). The points \(A\) and \(B\) lie on the circumference of the circle such that the angle \(AOB\) is \(\theta\) and the length of the minor arc \(AB\) is 12. The area of the minor sector \(AOB\) is \(57.6\text{ cm}^2\). The point \(C\) lies on the tangent to the circle at \(A\) such that \(OBC\) is a straight line.

(a) Find the values of \(r\) and \(\theta\).

(b) Find the area of the shaded region. Give your answer correct to 1 decimal place.

In this question lengths are in centimetres and angles are in radians.

The diagram shows a circle with centre \(O\) and radius \(r\). The points \(A\) and \(B\) lie on the circumference of the circle. The area of the minor sector \(OAB\) is \(25\text{ cm}^2\). The angle \(AOB\) is \(\theta\).

(a) Find an expression for the perimeter, \(P\), of the minor sector \(OAB\), in terms of \(r\).

(b) Given that \(r\) can vary, show that \(P\) has a minimum value and find this minimum value.

In this question all lengths are in centimetres and all angles are in radians.

(a) The area of a sector of a circle of radius \(24\) is \(432\text{ cm}^2\). Find the length of the arc of the sector.

(b) The diagram shows an isosceles triangle \(OAB\), with \(AO=AB=y\) and height \(AD\). \(OCD\) is a sector of the circle with centre \(O\). Angle \(AOB\) is \(\alpha\).

(i) Find an expression for \(OB\) in terms of \(y\) and \(\alpha\).

(ii) Hence show that the area of the shaded region can be written as

\(\frac{y^2}{2}\cos\alpha(2\sin\alpha-\alpha\cos\alpha).\)

In this question all lengths are in centimetres and all angles are in radians.

The diagram shows a badge which consists of a minor sector \(OAB\), of the circle with centre \(O\) and radius 12, and a kite \(OBCD\), where \(OB=OD\) and \(CD=CB\). The arc \(AB\) has length 27. The line \(OB\) is perpendicular to the line \(CB\), and \(COA\) is a straight line.

(a) Find the perimeter of the badge.

(b) Find the area of the badge.

In this question, all lengths are in centimetres and all angles are in radians.

The diagram shows the sector \(OAB\) of a circle with centre \(O\) and radius \(20\). The perimeter of this sector is \(65\). The lines \(CA\) and \(CB\) are both tangents to the circle at the points \(A\) and \(B\), so that the triangle \(ABC\) is isosceles, with \(AC=CB\). The angle \(AOB\) is equal to \(\theta\).

Find the area of the shaded region.

The diagram shows the sector \(AOB\) of a circle, centre \(O\) and radius \(15\) cm. Angle \(AOB\) is \(\frac{\pi}{6}\) radians. Point \(C\) lies on \(OB\) such that \(CB\) is \(a\) cm. \(AC\) is a straight line.

(a) Find the exact value of \(a\) such that the area of triangle \(AOC\) is equal to the area of the shaded region \(ACB\).

(b) For the value of \(a\) found in part (a), find the perimeter of the shaded region. Give your answer correct to \(1\) decimal place.

In this question, all lengths are in centimetres and all angles are in radians.

(a) The diagram shows two sectors, \(AOB\) and \(COD\), with common centre \(O\). The angle \(AOB\) is \(\frac{3\pi}{8}\), \(OC=6.5\), and \(OA:OC=4:5\). Find the perimeter of the shaded region.

(b) The diagram shows a circle with centre \(O\) and radius \(a\). The sector \(PQR\) is part of a circle with centre \(R\) and radius \(y\). The angle \(OPR\) is \(\phi\). Find the total area of the three shaded regions in terms of \(a\) and \(\phi\), giving your answer in its simplest form.

In this question, all lengths are in centimetres and all angles are in radians.

The diagram shows a circle, centre \(O\), radius \(8\). The points \(A\), \(B\) and \(C\) lie on the circumference of the circle. The chord \(AB\) has length \(10\).

(a) Show that angle \(BOA\) is \(1.35\) correct to \(2\) decimal places.

(b) Given that the minor arc \(BC\) has a length of \(18\), find angle \(BOC\).

(c) Find the area of the minor sector \(AOC\).

In this question all lengths are in centimetres.

The diagram shows a circle, centre \(O\), radius \(a\). The lines \(PT\) and \(QT\) are tangents to the circle at \(P\) and \(Q\) respectively. Angle \(POQ\) is \(2\phi\) radians.

(a) In the case when the area of the sector \(OPQ\) is equal to the area of the shaded region, show that \(\tan\phi=2\phi\).

(b) In the case when the perimeter of the sector \(OPQ\) is equal to half the perimeter of the shaded region, find an expression for \(\tan\phi\) in terms of \(\phi\).

The diagram shows a circle with centre \(O\) and radius \(r\). \(OAB\) and \(OCD\) are sectors of a circle with centre \(O\) and radius \(x\), where \(0\lt x\lt r\). Angle \(AOB=\) angle \(COD=\theta\) radians, where \(0\lt \theta\lt \pi\).

(a) Find, in terms of \(r\), \(x\) and \(\theta\), the perimeter of the shaded region.

(b) Find, in terms of \(r\), \(x\) and \(\theta\), the area of the shaded region.

It is given that \(x\) can vary and that \(r\) and \(\theta\) are constant.

(c) Write down the least possible area of the shaded region in terms of \(r\) and \(\theta\).

The diagram shows a circle, centre \(O\), radius \(10\) cm. The points \(A\) and \(B\) lie on the circumference of the circle. The tangent at \(A\) and the tangent at \(B\) meet at the point \(C\). The angle \(AOB\) is \(\theta\) radians. The length of the minor arc \(AB\) is \(28\) cm.

(a) Find the value of \(\theta\).

(b) Find the perimeter of the shaded region.

(c) Find the area of the shaded region.

In this question all lengths are in metres.

The diagram shows a circle, centre \(O\), radius \(7\). The points \(A\) and \(B\) lie on the circumference of the circle. The line \(BC\) is a tangent to the circle at the point \(B\) such that the length of \(BC\) is \(24\). The length of the minor arc \(AB\) is \(12.25\).

(a) Find the obtuse angle \(AOB\), giving your answer in radians.

(b) Find the perimeter of the shaded region.

(c) Find the area of the shaded region.

\(AOB\) is a sector of a circle with centre \(O\) and radius \(16\text{ cm}\). Angle \(AOB\) is \(\dfrac{2\pi}{7}\) radians. The point \(C\) lies on \(OB\) such that \(OC\) is \(7.5\text{ cm}\), and \(AC\) is a straight line.

(a) Find the perimeter of the shaded region.

(b) Find the area of the shaded region.

In this question all lengths are in centimetres.

The diagram shows a shaded shape. The arc \(AB\) is the major arc of a circle, centre \(O\), radius \(10\). The line \(AB\) is of length \(15\), the line \(OC\) is of length \(25\) and the lengths of \(AC\) and \(BC\) are equal.

(a) Show that the angle \(AOB\) is \(1.70\) radians correct to 2 decimal places.

(b) Find the perimeter of the shaded shape.

(c) Find the area of the shaded shape.

The diagram shows a circle, centre \(O\), radius \(2a\). The points \(A\) and \(B\) lie on the circumference of the circle. The points \(C\) and \(D\) are the mid-points of the lines \(OB\) and \(OA\) respectively. The arc \(DC\) is part of a circle centre \(O\). The chord \(AB\) is of length \(2a\).

(a) Find angle \(AOB\), giving your answer in radians in terms of \(\pi\).

(b) Find, in terms of \(a\) and \(\pi\), the perimeter of the shaded region \(ABCD\).

(c) Find, in terms of \(a\) and \(\pi\), the area of the shaded region \(ABCD\).

\(DAB\) is a sector of a circle, centre \(A\), radius \(18\) cm. The lines \(CB\) and \(CD\) are tangents to the circle. Angle \(DAB\) is \(\frac{7\pi}{9}\) radians.

(a) Find the perimeter of the shaded region.

(b) Find the area of the shaded region.

The diagram shows a circle, centre \(O\), radius \(5\text{ cm}\). The lines \(AOB\) and \(COD\) are diameters of this circle. The line \(AC\) has length \(6\text{ cm}\).

(a) Show that angle \(AOC=1.287\) radians, correct to 3 decimal places.

(b) Find the perimeter of the shaded region.

(c) Find the area of the shaded region.

The diagram shows a circle, centre \(O\), radius \(12\text{ cm}\), and a rectangle \(ABCD\). The diagonals \(AC\) and \(BD\) intersect at \(O\). The sides \(AB\) and \(AD\) of the rectangle have lengths \(6\text{ cm}\) and \(4\text{ cm}\) respectively. The points \(M\) and \(N\) lie on the circumference of the circle such that \(MAC\) and \(NDB\) are straight lines.

(a) Show that angle \(AOD\) is \(1.176\) radians correct to 3 decimal places.

(b) Find the perimeter of the shaded region.

(c) Find the area of the shaded region.

The diagram shows a circle, centre \(O\), radius \(10\text{ cm}\). The points \(A\), \(B\) and \(P\) lie on the circumference of the circle. The chord \(AB\) is of length \(14\text{ cm}\). The point \(Q\) lies on \(AB\) and the line \(POQ\) is perpendicular to \(AB\).

(a) Show that angle \(POA\) is \(2.366\) radians, correct to 3 decimal places.

(b) Find the area of the shaded region.

(c) Find the perimeter of the shaded region.

(a) A sector of a circle has radius \(6\) cm and perimeter \(2(6+5\pi)\) cm. Find the area of the sector.

(b) The diagram shows a sector \(AOB\) of a circle with centre \(O\), radius \(7\) cm and angle \(\angle AOB=\frac14\pi\). Find the perimeter of the shaded region.

The diagram shows an isosceles triangle \(OAB\) such that \(OA=OB\) and angle \(AOB=\theta\) radians. The points \(C\) and \(D\) lie on \(OA\) and \(OB\) respectively. \(CD\) is an arc of length \(9.6\) cm of the circle, centre \(O\), radius \(12\) cm. The arc \(CD\) touches the line \(AB\) at the point \(M\).

(a) Find the value of \(\theta\).

(b) Find the total area of the shaded regions.

(c) Find the total perimeter of the shaded regions.

The diagram shows an isosceles triangle \(OAB\) such that \(OA=OB=12\) cm and angle \(AOB=\theta\) radians. Points \(C\) and \(D\) lie on \(OA\) and \(OB\) respectively such that \(CD\) is an arc of the circle, centre \(O\), radius \(10\) cm. The area of the sector \(OCD\) is \(35\text{ cm}^2\).

(a) Show that \(\theta=0.7\).

(b) Find the perimeter of the shaded region.

(c) Find the area of the shaded region.

The circles with centres \(C_1\) and \(C_2\) have equal radii of length \(r\) cm. The line \(C_1C_2\) is a radius of both circles. The two circles intersect at \(A\) and \(B\).

(a) Given that the perimeter of the shaded region is \(4\pi\) cm, find the value of \(r\).

(b) Find the exact area of the shaded region.

The diagram shows a shaded region in a rectangle. The arcs \(AB\) and \(CD\) are parts of circles with centres \(O\), and \(BC\) is an arc of a circle with centre \(O\). The radius is \(r\), and the length of arc \(BC\) is \(1.5r\).

(a) Find an expression for the perimeter of the shaded region in terms of \(r\).

(b) Find an expression for the area of the shaded region in the form \(kr^2\), where \(k\) is a constant to be found correct to 3 significant figures.

In this question all lengths are in centimetres.

The diagram shows the figure \(ABC\). The arc \(AB\) is part of a circle, centre \(O\), radius \(r\), and is of length \(1.45r\). The point \(O\) lies on the straight line \(CB\) such that \(CO=0.5r\).

(a) Find, in radians, the angle \(AOB\).

(b) Find the area of \(ABC\), giving your answer in the form \(kr^2\), where \(k\) is a constant.

(c) Given that the perimeter of \(ABC\) is \(12\) cm, find the value of \(r\).

The diagram shows a shape consisting of two circles of radii \(3\) cm and \(4\) cm with centres \(A\) and \(B\), which are \(5\) cm apart. The circles intersect at \(C\) and \(D\) as shown. The lines \(AC\) and \(BC\) are tangents to the circles with centres \(B\) and \(A\), respectively.

Find

(a) the angle \(CAB\), in radians,

(b) the perimeter of the whole shape,

(c) the area of the whole shape.

The diagram shows the right-angled triangle \(OAB\). The point \(C\) lies on \(OB\). Angle \(OAB=\frac{\pi}{2}\) radians and angle \(AOB=\theta\) radians. \(AC\) is an arc of the circle, centre \(O\), radius \(12\) cm and \(AC\) has length \(9.6\) cm.

(i) Find the value of \(\theta\).

(ii) Find the area of the shaded region.

The diagram shows a right-angled triangle \(ABC\) with \(AB=8\text{ cm}\) and angle \(ABC=\frac{\pi}{2}\) radians. The points \(D\) and \(E\) lie on \(AC\) and \(BC\) respectively. \(BAD\) and \(ECD\) are sectors of circles with centres \(A\) and \(C\) respectively. Angle \(BAD=\frac{2\pi}{9}\) radians.

(i) Find the area of the shaded region.

(ii) Find the perimeter of the shaded region.

The diagram shows a company logo, \(ABCD\). The logo is part of a sector, \(AOB\), of a circle, centre \(O\) and radius \(50\) cm. The points \(C\) and \(D\) lie on \(OB\) and \(OA\) respectively. The lengths \(AD\) and \(BC\) are equal and \(AD:AO=7:10\). The angle \(AOB\) is \(\dfrac{4\pi}{9}\) radians.

(i) Find the perimeter of \(ABCD\).

(ii) Find the area of \(ABCD\).

The diagram shows a circle with centre \(O\) and radius \(10\) cm. The points \(A\), \(B\), \(C\) and \(D\) lie on the circle such that the chord \(AB=15\) cm and the chord \(CD=10\) cm. The chord \(AB\) is parallel to the chord \(DC\).

(i) Show that the angle \(AOB\) is \(1.70\) radians correct to 2 decimal places.

(ii) Find the perimeter of the shaded region.

(iii) Find the area of the shaded region.

The diagram shows a circle centre \(O\), radius \(10\) cm. The points \(A\), \(B\) and \(C\) lie on the circumference of the circle such that \(AB=BC=18\) cm.

(i) Show that angle \(AOB=2.24\) radians correct to 2 decimal places.

(ii) Find the perimeter of the shaded region.

(iii) Find the area of the shaded region.

The diagram shows a sector \(OPQ\) of the circle centre \(O\), radius \(3r\) cm. The points \(S\) and \(R\) lie on \(OP\) and \(OQ\) respectively such that \(ORS\) is a sector of the circle centre \(O\), radius \(2r\) cm. The angle \(POQ=\theta\) radians. The perimeter of the shaded region \(PQRS\) is \(100\) cm.

(i) Find \(\theta\) in terms of \(r\).

(ii) Hence show that the area, \(A\text{ cm}^2\), of the shaded region \(PQRS\) is given by \(A=50r-r^2\).

(iii) Given that \(r\) can vary and that \(A\) has a maximum value, find this value of \(A\).

(iv) Given that \(A\) is increasing at the rate of \(3\text{ cm}^2\text{s}^{-1}\) when \(r=10\), find the corresponding rate of change of \(r\).

(v) Find the corresponding rate of change of \(\theta\) when \(r=10\).

The diagram shows a circle with centre \(O\) and radius \(8\) cm. The points \(A\), \(B\), \(C\), and \(D\) lie on the circumference of the circle. Angle \(AOB=\theta\) radians and angle \(COD=1.4\) radians. The area of sector \(AOB\) is \(20\text{ cm}^2\).

(i) Find angle \(\theta\).

(ii) Find the length of the arc \(AB\).

(iii) Find the area of the shaded segment.

In the diagram \(AOB\) and \(DOC\) are sectors of a circle centre \(O\). The angle \(AOB\) is \(x\) radians. The length of the arc \(AB\) is \(40\text{ cm}\) and the radius \(OB\) is \(16\text{ cm}\).

(i) Find the value of \(x\).

(ii) Find the area of sector \(AOB\).

(iii) Given that the area of the shaded region \(ABCD\) is \(140\text{ cm}^2\), find the length of \(OC\).

The diagram shows two shaded regions. \(ABC\) is an arc of a circle with centre \(O\), radius \(5\text{ cm}\), and angle \(AOC=1.5\) radians. \(AD\) and \(CE\) are diameters of the circle and \(DE\) is a straight line.

(i) Find the total perimeter of the shaded regions.

(ii) Find the total area of the shaded regions.

In the diagram \(AOB\) and \(DOC\) are sectors of a circle centre \(O\). The angle \(AOB\) is \(x\) radians. The length of the arc \(AB\) is \(40\text{ cm}\) and the radius \(OB\) is \(16\text{ cm}\).

(i) Find the value of \(x\).

(ii) Find the area of sector \(AOB\).

(iii) Given that the area of the shaded region \(ABCD\) is \(140\text{ cm}^2\), find the length of \(OC\).

The diagram shows a circle with centre \(O\) and radius \(r\) cm. The minor arc \(AB\) is such that angle \(AOB\) is \(\theta\) radians. The area of the minor sector \(AOB\) is \(48\text{ cm}^2\).

(i) Show that \(\theta=\dfrac{96}{r^2}\).

(ii) Given that the minor arc \(AB\) has length \(12\) cm, find the value of \(r\) and of \(\theta\).

(iii) Using your values of \(r\) and \(\theta\), find the area of the shaded region.

The diagram shows the sector \(OPQ\) of a circle, centre \(O\), radius \(r\) cm, where angle \(POQ=\theta\) radians. The perimeter of the sector is \(10\) cm.

(i) Show that the area, \(A\text{ cm}^2\), of the sector is given by

\(A=\frac{50\theta}{(2+\theta)^2}.\)

It is given that \(\theta\) can vary and \(A\) has a maximum value.

(ii) Find the maximum value of \(A\).

The diagram shows a sector \(POQ\) of a circle, centre \(O\), radius \(r\text{ cm}\), where angle \(POQ=\theta\) radians. The perimeter of the sector is \(20\text{ cm}\).

(i) Show that the area, \(A\text{ cm}^2\), of the sector is given by \(A=10r-r^2\).

It is given that \(r\) can vary and that \(A\) has a maximum value.

(ii) Find the value of \(\theta\) for which \(A\) has a maximum value.

The diagram shows a circle, centre \(O\), radius \(12\text{ cm}\). The points \(A\) and \(B\) lie on the circumference of the circle and form a rectangle with points \(C\) and \(D\). The length of \(AD\) is \(8\text{ cm}\), and the area of the minor sector \(AOB\) is \(150\text{ cm}^2\).

(i) Show that angle \(AOB\) is \(2.08\) radians, correct to 2 decimal places.

(ii) Find the area of the shaded region \(ADCB\).

(iii) Find the perimeter of the shaded region \(ADCB\).

The diagram shows a circle, centre \(O\), radius \(8\text{ cm}\). Points \(A,B,C,D\) lie on the circumference such that \(AB\) is parallel to \(DC\). The length of the arc \(AD\) is \(4\text{ cm}\), and the length of the chord \(AB\) is \(15\text{ cm}\).

(i) Find, in radians, \(\angle AOD\).

(ii) Hence show that \(\angle DOC=1.43\) radians, correct to 2 decimal places.

(iii) Find the perimeter of the shaded region.

(iv) Find the area of the shaded region.

The diagram shows a circle, centre \(O\) of radius \(r\text{ cm}\), and a chord \(AB\). Angle \(AOB=\theta\) radians.

The length of the major arc \(AB\) is \(5\) times the length of the minor arc \(AB\). The minor arc \(AB\) has length \(2\pi\text{ cm}\).

(i) Find the value of \(\theta\) and of \(r\).

(ii) Calculate the exact perimeter of the shaded segment.

(iii) Calculate the exact area of the shaded segment.

The diagram shows a circle, centre \(A\), radius \(10\) cm, intersecting a circle, centre \(B\), radius \(24\) cm. The two circles intersect at the points \(P\) and \(Q\). The radii \(AP\) and \(AQ\) are tangents to the circle with centre \(B\). The radii \(BP\) and \(BQ\) are tangents to the circle with centre \(A\).

(i) Show that angle \(PAQ\) is \(2.35\) radians, correct to 3 significant figures.

(ii) Find angle \(PBQ\) in radians.

(iii) Find the perimeter of the shaded region.

(iv) Find the area of the shaded region.

The diagram shows an isosceles triangle \(ABC\), where \(AB=AC=5\) cm. The arc \(BEC\) is part of the circle centre \(A\) and has length \(6.2\) cm. The point \(D\) is the midpoint of the line \(BC\). The arc \(BFC\) is a semi-circle centre \(D\).

(i) Show that angle \(BAC\) is \(1.24\) radians.

(ii) Find the perimeter of the shaded region.

(iii) Find the area of the shaded region.

The diagram shows a circle, centre \(O\), radius \(10\text{ cm}\). The points \(A\), \(B\), \(C\), and \(D\) lie on the circumference of the circle such that \(AB\) is parallel to \(DC\). The length of the minor arc \(AB\) is \(14.8\text{ cm}\). The area of the minor sector \(ODC\) is \(21.8\text{ cm}^2\).

(i) Write down, in radians, angle \(AOB\).

(ii) Find, in radians, angle \(DOC\).

(iii) Find the perimeter of the shaded region.

(iv) Find the area of the shaded region.