Circular Measure (Radians) (CIE A Level Maths: Pure 1)

Complete the table.

Degrees	Radians	sin	cos	tan
45°
60°
270°				n/a

A sector of a circle, , is such that it has radius 3.4 cm and the angle at its centre, , is  radians.

Find the length of the arc  .

Find the area of the sector  .

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 .

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

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

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 .

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

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 8 cm, and the angle at the centre is 1.2 radians.

The height of the puzzle piece is 2 cm.

The circle sector  is shown in the diagram below.

The angle at the centre is radians, and the line segments and  have lengths of 8 cm and cm respectively.

 Additionally, is parallel to , so that and .

Show that the area of the sector is   .

Show that the area of the triangle is .

Given that the area of the shaded shape  is ,  find the value of .

Complete the table.

Degrees	Radians	sin	cos	tan
135°
180°				0

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 .

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 is three times the length of the arc in cm, find the value of a.

The diagram below shows the sector of a circle .

Given that the area of triangle , 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.

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 AB , so that and .

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

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 , 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 1 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 .

Complete the table.

Degrees	Radians	sin	cos	tan
45°
150°
				-1

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 .

The length of the arc is .    

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

.

The diagram below shows the sector of a circle .

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

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 , find the value of .

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.