Arc length and sector area

The shape ABCD, shown in Figure 1, consists of a triangle ABD containing a sector ABC of a circle with centre B. Given that

• AD = 6.4 cm

• BD = 13 cm

• BA = BC = 8 cm

show that angle ABC = 0.394 radians to 3 significant figures. The region R, shown shaded in Figure 1, is bounded by the line CD, the line DA and the arc AC.

Find the area of R, giving the answer in cm² to 3 significant figures. You must make your method clear.

Figure 4

Figure 4 shows the design of a badge.

The shape ABCOA is a semicircle with centre O and diameter 10 cm.

OB is the arc of a circle with centre A and radius 5 cm.

The region R, shown shaded in Figure 4, is bounded by the arc OB, the arc BC and the line OC.

Find the exact area of R.

FRONT OF STAGE

Figure 1 shows the plan view of a stage.

The plan view shows two congruent triangles ABO and GFO joined to a sector OCDEO of a circle, centre O, where

• angle COE = 2.3 radians

• arc length CDE = 27.6 m

• AOG is a straight line of length 15 m

Show that OC = 12 m.

Show that the size of angle AOB is 0.421 radians correct to 3 decimal places.

Given that the total length of the front of the stage, BCDEF, is 35 m,

find the total area of the stage, giving your answer to the nearest square metre.

The shape OABCDEFO shown in Figure 1 is a design for a logo.

In the design

• OAB is a sector of a circle centre O and radius r

• sector OFE is congruent to sector OAB

• ODC is a sector of a circle centre O and radius 2 r

• AOF is a straight line

Given that the size of angle COD is θ radians,

write down, in terms of θ, the size of angle AOB

Show that the area of the logo is

Find the perimeter of the logo, giving your answer in simplest form in terms of r, θ and π.

The shape OABCDEFO shown in Figure 1 is a design for a logo. In the design

• OAB is a sector of a circle centre O and radius r

• sector OFE is congruent to sector OAB

• ODC is a sector of a circle centre O and radius 2 r

• AOF is a straight line

Given that the size of angle COD is θ radians,

write down, in terms of θ, the size of angle AOB

Show that the area of the logo is

Find the perimeter of the logo, giving your answer in simplest form in terms of r, θ and π.

Circle C₁ has equation x² + y² = 100 Circle C₂ has equation (x − 15)² + y² = 40 The circles meet at points A and B as shown in Figure 3.

Show that angle AOB = 0.635 radians to 3 significant figures, where O is the origin. The region shown shaded in Figure 3 is bounded by C₁ and C₂

Find the perimeter of the shaded region, giving your answer to one decimal place.

Figure 1 shows a sector AOB of a circle with centre O, radius 5 cm and angle AOB = 40° The attempt of a student to find the area of the sector is shown below.

Explain the error made by this student.

Write out a correct solution.

Figure 1 shows a sector AOB of a circle with centre O and radius rcm.

The angle AOB is θ radians. The area of the sector AOB is 11 cm²

Given that the perimeter of the sector is 4 times the length of the arc AB, find the exact value of r.

Figure 2 shows a flag XYWZX. The flag consists of a triangle XYZ joined to a sector ZYW of a circle with radius 5 cm and centre Y. The angle of the sector, angle ZYW, is 0.7 radians. The points X, Y and W lie on a straight line with XY= 7 cm and YW= 5 cm. Find

the area of the sector ZYW in cm²,

the area of the flag, in cm², to 2 decimal places,

the length of the perimeter, XYWZX, of the flag, in cm to 2 decimal places.

Figure 1 is a sketch representing the cross-section of a large tent ABCDEF. AB and DE are line segments of equal length. Angle FAB and angle DEF are equal. F is the midpoint of the straight line AE and FC is perpendicular to AE. BCD is an arc of a circle of radius 3.5 m with centre at F. It is given that AF = FE = 3.7 m BF = FD = 3.5 m angle BFD = 1.77 radians Find

the length of the arc BCD in metres to 2 decimal places,

the area of the sector FBCD in m² to 2 decimal places,

the total area of the cross-section of the tent in m² to 2 decimal places.

Figure 4 shows a plan view of a sheep enclosure. The enclosure ABCDEA, as shown in Figure 4, consists of a rectangle BCDE joined to an equilateral triangle BFA and a sector FEA of a circle with radius x metres and centre F. The points B, F and E lie on a straight line with FE = x metres and 10 x ≤ 25

Find, in m², the exact area of the sector FEA, giving your answer in terms of x, in its simplest form.

Given that BC = y metres, where y 0, and the area of the enclosure is 1000 m²,

show that

Hence show that the perimeter P metres of the enclosure is given by

Use calculus to find the minimum value of P, giving your answer to the nearest metre.

Justify, by further differentiation, that the value of P you have found is a minimum.

The shape ABCDEA, as shown in Figure 2, consists of a right-angled triangle EAB and a triangle DBC joined to a sector BDE of a circle with radius 5 cm and centre B. The points A, B and C lie on a straight line with BC = 7.5 cm. Angle EAB = radians, angle EBD = 1.4 radians and CD = 6.1 cm.

Find, in cm, the area of the sector BDE.

Find the size of the angle DBC, giving your answer in radians to 3 decimal places.

Find, in cm², the area of the shape ABCDEA, giving your answer to 3 significant figures.

A 23m B C

Figure 2 shows a plan view of a garden. The plan of the garden ABCDEA consists of a triangle ABE joined to a sector BCDE of a circle with radius 12m and centre B. The points A, B and C lie on a straight line with AB = 23m and BC = 12m. Given that the size of angle ABE is exactly 0.64 radians, find

the area of the garden, giving your answer in m², to 1 decimal place,

the perimeter of the garden, giving your answer in metres, to 1 decimal place.

Figure 2 shows the design for a triangular garden ABC where AB = 7 m, AC = 13 m and BC = 10 m. Given that angle BAC = radians,

show that, to 3 decimal places, = 0.865

The point D lies on AC such that BD is an arc of the circle centre A, radius 7 m. The shaded region S is bounded by the arc BD and the lines BC and DC. The shaded region S will be sown with grass seed, to make a lawned area. Given that 50 g of grass seed are needed for each square metre of lawn,

find the amount of grass seed needed, giving your answer to the nearest 10 g.

The triangle XYZ in Figure 1 has XY = 6 cm, YZ = 9 cm, ZX = 4 cm and angle ZXY = . The point W lies on the line XY. The circular arc ZW, in Figure 1 is a major arc of the circle with centre X and radius 4 cm.

Show that, to 3 significant figures, = 2.22 radians.

The region enclosed by the major arc ZW of the circle and the lines WY and YZ is shown shaded in Figure 1. Calculate

the area of this shaded region,

the perimeter ZWYZ of this shaded region.

The circle C with centre T and radius r has equation

Find the coordinates of the centre of C.

Show that r = 5

The line L has equation x = 13 and crosses C at the points P and Q as shown in Figure 1.

Find the y coordinate of P and the y coordinate of Q.

Given that, to 3 decimal places, the angle PTQ is 1.855 radians,

find the perimeter of the sector PTQ.

The shape shown in Figure 1 is a pattern for a pendant. It consists of a sector OAB of a circle centre O, of radius 6 cm, and angle AOB . The circle C, inside the sector, touches the two straight edges, OA and OB, and the arc AB as shown. Find

the area of the sector OAB,

the radius of the circle C.

The region outside the circle C and inside the sector OAB is shown shaded in Figure 1.

Find the area of the shaded region.

Figure 2 shows a closed box used by a shop for packing pieces of cake. The box is a right prism of height h cm. The cross section is a sector of a circle. The sector has radius r cm and angle 1 radian. The volume of the box is 300 cm³.

Show that the surface area of the box, S cm², is given by

Use calculus to find the value of r for which S is stationary.

Prove that this value of r gives a minimum value of S.

Find, to the nearest cm², this minimum value of S.

the length of the arc BC,

the area of the sector ABC.

The point D is the mid-point of AC. The region R, shown shaded in Figure 1, is bounded by CD, DB and the arc BC. Find

the perimeter of R, giving your answer to 3 significant figures,

the area of R, giving your answer to 3 significant figures.