Radian Measure (Edexcel A Level Maths: Pure): Exam Questions

Figure 1 shows a sector of a circle with centre and radius cm.

The angle is radians.

The area of the sector is 11 cm².

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

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

In the design

• is a sector of a circle centre and radius

• sector is congruent to sector

• is a sector of a circle centre and radius

• is a straight line

Given that the size of angle is radians, write down, in terms of , the size of angle

Show that the area of the logo is

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

Given that is small and is measured in radians, use the small angle approximations to find an approximate value of

Given that is small and measured in radians, use the small angle approximations to show that

where , and are integers to be found.

The diagram below shows the sector .

Find the area of the shaded segment, giving your answer to 3 significant figures.

Find the perimeter of the sector , to 2 decimal places.

A parachutist, modelled by the point below, is heading towards the ground using a parachute, modelled by the shaded segment.

The radii and are cords that connect the parachutist to the parachute.

The area of sector is  m².

Find the length of one of the cords.

Given that is small and measured in radians, use small angle approximations to show that 

where , and are constants to be found.

Hence find two approximate solutions to the equation

Find all the solutions to the equation

in the interval  , giving your answers in exact form.

Find all the solutions to the equation

in the interval  , giving your answers to 3 significant figures.

The sector  is shown in the diagram below, where

• the central angle of the sector is radians

• the line segments  and  have lengths of cm and cm respectively

•  is parallel to

Find an expression for the exact area of sector in terms of .

Find the exact area of triangle .

The shaded shape , where is the arc of the sector, has an area of

cm²

Find the value of .

The diagram shows a prism where

• the uniform cross section of the prism is a sector

• the radius of the sector is cm

• the central angle of the sector is radians

• the height of the prism is cm

Given that the volume of the prism is cm³, find the value of .

Given that  is small and measured in radians, use the small angle approximations to find an approximate value of

A sector of a circle centre has a radius of cm and a central angle of  radians, where is a non-zero constant.

The area of sector is three times the length of the arc .

Find the value of .

Given that is small and measured in radians, use the small angle approximations to find and simplify an approximate value of

A company makes toys for children.

Figure 5 shows the design for a solid toy that looks like a piece of cheese.

The toy is modelled so that

• face is a sector of a circle with radius cm and centre

• angle radians

• faces and are congruent

• edges and are perpendicular to faces and

• edges and have length cm

Given that the volume of the toy is show that the surface area of the toy, , is given by

making your method clear.

Figure 1 shows the plan view of a stage.

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

• angle radians

• arc length m

• is a straight line of length m

Show that m.

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

Given that the total length of the front of the stage, , is m, find the total area of the stage, giving your answer to the nearest square metre.

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

The curve has equation where

Given that

• 

• the curve has a stationary point with coordinate

• is small

use the small angle approximation for to estimate the value of to 3 decimal places.

Circle has equation .

Circle has equation .

The circles meet at points and as shown in Figure 3.

Show that angle =0.635 radians to 3 significant figures, where is the origin.

The region shown shaded in Figure 3 is bounded by and

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

Figure 1 shows a plot of part of the curve with equation where is measured in radians.

Diagram 1, shown below, is a copy of Figure 1.

Use Diagram 1 to show why the equation

has only one real root, giving a reason for your answer.

Given that the root of the equation is , and that is small, use the small angle approximation for to estimate the value of to 3 decimal places.

Given that is small and measured in radians, use small angle approximations to show that 

can be approximated by

where and are constants to be found.

Hence find three approximate solutions to the equation

Comment on the validity of your solutions in part (b).

The sector  is shown in the diagram below, where

• the central angle of the sector is radians

• the radii  and  are both equal to

• cm

Find an expression for the shaded area in terms of and .

Show that, for values of near to zero, an approximation of the shaded area is given by the formula

where and are constants to be found.

The diagram below shows the sector .

Show that the area of the shaded segment is

cm²

Find, in terms of , an expression for the percentage of the sector that the segment occupies.

Given that is small and measured in radians, use the small angle approximations to find approximate solutions to the equation

Determine which out of the solutions in part (a) are more accurate, giving a reason for your answer.

A sector of a circle centre has a radius of cm and a central angle of  radians.

The chord has length  cm.

Show that 

Given that

• 

• the area of the sector is cm²

find the exact value of .

The sector  is shown in the diagram below, where

• the central angle of the sector is radians

• the radii  and  are both equal to

• The line  is perpendicular to the line

• 

Show that the area of the shaded shape  is

cm² 

where  and are constants to be found.