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 .

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.

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.

Complete the table.

A sector of a circle,, is such that it has radius cm and the angle at its centre, , is  radians.

(i) Find the length of the arc .

(ii) Find the area of the sector .

Given that  is small, write an approximation in terms of  for

                                             sin cos tan² 

The sector of a circle is shown below.

Find the area and the perimeter of the sector.

A rainbow-shaped logo is formed from two semicircles as shown below.
The radius of the smaller semicircle is 1 cm less than the radius of the larger semicircle.

Find the area of the logo.

The diagram below shows the sector of a circle .

(i) Find the area of the sector , giving your answer to 3 significant figures.

(ii) Find the area of the triangle , giving your answer to 3 significant figures.

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

(i) Find the length of the arc .

(ii) Find the perimeter of the sector .

The canopy of a parachute and the outermost connecting cords form a sector of a circle as shown in the diagram below, with the parachutist modelled as a particle at point .

The area of the sector is  m².

Find the length of one of the connecting cords on the parachute.

Given that is small, show that 

sin  tan  cos 

Hence, or otherwise, find approximate solution(s) to the equation

sin tan cos 

Find all solutions to the equation cos   in the interval  , giving your answers in radians as multiples of .

Find all solutions to the equation  sin   in the interval  , giving your answer in radians to three significant figures.

A plastic puzzle piece is in the form of a prism with a cross-section that is the sector of a circle, as shown in the diagram below.  The radius of the sector is cm, and the angle at the centre is radians.

The height of the puzzle piece is cm.

(i) Work out the area of the cross-section.

(ii) Hence, or otherwise, work out the volume of the puzzle piece.

The circle sector  is shown in the diagram below.

The angle at the centre is radians , and the line
segments  and  have lengths of cm and cm respectively.

Additionally,  is parallel to , so that  and .

Show that the area of the sector is   cm².

Show that the area of the triangle is cm².

Given that the area of the shaded shape is cm².  find the 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.

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.

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.

Complete the table.

Given that  is small, write an approximation in terms of  for

The canopy of a parachute and the outermost suspension lines (cords) form a sector of a circle as shown in the diagram below, with the parachutist modelled as a particle at point .

The area of the sector is  m².

The parachute is made with 40 equal length suspension lines.

Disregarding any additional lengths of cord that might be required for knots, fastenings, etc., find the total length of the suspension lines required to make the parachute.

A sector of a circle, , is such that it has radius cm and the angle at its centre, , is  radians.

Given that the area of the sector in cm² is three times the length of the arc  in cm, find the value of .

The diagram below shows the sector of a circle .

Given that the area of triangle cm², find the area of the shaded segment. Give your answer correct to 3 significant figures.

Find the perimeter of the sector , giving your answer correct to 3 significant figures.

When is small, show that the equation

cos²  tan⁴ 

can be written as

Hence find approximate solutions to the equation

cos² tan⁴ 

Comment on the validity of your solutions.

The circle sector  is shown in the diagram below.

The angle at the centre is  radians , and the radii  and  are each equal to cm

Additionally, is parallel to , so that   and .

In the case when cm, show that the area of the shaded shape is given by sin .

.

Show that for small values of , the area of is approximately .

The diagram shows a prism with cross-section in the shape of a sector of a circle. 

The radius of the sector is cm, and the angle at the centre is radians.

The height of the prism is cm.

Given that the volume of the prism is cm³, find the possible value(s) of .

A rainbow-shaped logo is formed from three semicircles as shown below.
Each of the inner semicircles has a radius that is cm less than that of the next semicircle further out.

Find, in simplest terms, the ratio of the outer area of the logo, to the inner area . 

Find all the solutions to the equation tan in the interval  , giving your answers in radians as multiples of .

Find all the solutions to the equation sin² sin  in the interval     ,  giving your answers in radians to three significant figures.

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.

Complete the table.

Given that  is small, find an approximate expression in terms of  for

The canopy of a parachute and the outermost suspension lines (cords) form a sector of a circle as shown in the diagram below, with the parachutist modelled as a particle at point

The area of the sector  is   m²  .

The length of the arc  is   m .

Find the length of one suspension line and the angle  that the parachutist makes with the two outermost suspension lines.

Given that is small, find approximate solution(s) to the equation

Comment on the relative reliability of the solution(s) you found in part (a).

The diagram below shows the sector of a circle .

Show that the area of the shaded segment is given by  cm².

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

An evil wizard has captured a unicorn, and is threatening to kill it unless you can answer the following question:

“I wish to create a rainbow-shaped mosaic for the floor of the throne room in my castle. 

 The mosaic is to be formed from four semicircles as shown below.

The innermost semicircle is to have a radius of metres, and each of the outer semicircles must have a radius that is a constant  metres greater than the radius of the next semicircle further in.

 The rainbow mosaic must be exactly 22 metres across.

Moreover, I wish the inner part of the area (labelled  in the diagram) to take up exactly one quarter of the total area of the rainbow.

Find me the required values of and , or the unicorn dies.  Bwah-ha-ha-ha-ha!”

Solve the wizard’s problem and save the unicorn.

A sector of a circle, , is such that it has radius cm  and the angle at its centre, , is  radians.  The chord has length  cm.

Show that  .

Given that   and that the area of the sector is   cm² , find the value of .

Find all the solutions to the equation cos sin cos  in the interval      , giving your answers in radians as multiples of .

Find all the solutions to the equation  cos² cos  in the interval , giving your answers in radians to three significant figures.

The diagram below shows the sector of a circle with centre .  The radii  and  are each equal to cm, and the angle at the centre, , is equal to  radians.  The line  is perpendicular to the line .

Given that  , show that the area of the shaded shape  is given by

cm².