MATHS

Question 1:

A circular lake in the nature reserve has a radius of 45 m.

(i) Calculate the area of the lake.
Answer (b)(i): .......................................... m² [2]

(ii)


A fence is placed along part of the circumference of the lake.
This arc subtends an angle of 210° at the centre of the circle.
Calculate the length of the fence.
Answer (b)(ii): .......................................... m [2]

Question 2:

The total area of each of the following shapes is X.
The area of the shaded part of each shape is kX.
For each shape, find the value of k and write your answer below each diagram.

Question 3:



The diagram shows a sector of a circle of radius 3.8 cm.
The arc length is 7.7 cm.

(i) Calculate the value of y.

y = .............................................. [2]

(ii) Calculate the area of the sector.

........................................ cm² [2]

Question 4:



In the diagram OKL is a sector of a circle, centre O and radius 8 cm.
OKM is a straight line and ML is a tangent to the circle at L.
Angle LOK = 44°.
Calculate the area shaded in the diagram.

Answer(c) .......................................... cm² [5]

Question 5:



The diagram shows a triangle and a sector of a circle.
In triangle ABC, AB = AC = 8 cm and angle BAC = 56°.
Sector OPQ has centre O, sector angle x and radius 6.5 cm.

(a) Show that the area of triangle ABC is 26.5 cm² correct to 1 decimal place.
Answer(a):

[2]

(b) The area of sector OPQ is equal to the area of triangle ABC.

(i) Calculate the sector angle x.
Answer(b)(i):

[3]

(ii) Calculate the perimeter of the sector OPQ.
Answer(b)(ii):

[3]

(c) The diagram shows a sector of a circle, radius r cm.



(i) Show that the area of the shaded segment is:
\[\frac{1}{4}r^2\left(\frac{\pi}{3} - 1\right)\] cm²

Answer(c)(i):
[4]

(ii) The area of the segment is 5 cm².
Find the value of r.

Answer(c)(ii) r = .............................................. [3]

Question 6:

(a)


The area of this sector is r² square centimetres.
Find the value of w.

Answer(a):
w = .............................................. [3]

(b)


The perimeter of this sector is \(\frac{7r}{10} + 2r\) centimetres.
Find the value of x.

Answer(b):
x = .............................................. [3]

Question 7:

The perimeter of each of the three shapes is 60 cm.
Find x in each part.

(a) Rectangle


x = .......................................... cm [2]

(b) Triangle


x = .......................................... cm [3]

(c) Sector


x = .......................................... cm [3]

Question 8:



The diagram shows a sector of a circle, a triangle and a rectangle.
The sector has centre O, radius x cm and angle 270°.
The rectangle has length 2x cm.
The total area of the shape is k x² cm².

(a) Find the value of k.

k = ................................................ [5]

(b) Find the value of x when the total area is 110 cm².

x = ................................................ [2]

Question 9:

(a)


(i) Calculate the area of the triangle.

............................................ cm² [2]

(ii) Calculate the perimeter of the triangle and show that it is 23.5 cm, correct to 1 decimal place.
Show all your working.

............................................ cm [5]

(b)


The perimeter of this sector of a circle is 28.2 cm.
Calculate the value of c.

c = .................................................. [3]

Question 10:



The diagram shows a circle, centre O. The straight line ABC is a tangent to the circle at B.
OB = 8 cm, AB = 15 cm and BC = 22.4 cm.
AO crosses the circle at X and OC crosses the circle at Y.

(a) Calculate angle XOY.

Angle XOY = ............................................... [5]

(b) Calculate the length of the arc XBY.

.......................................... cm [2]

(c) Calculate the total area of the two shaded regions.

........................................ cm² [4]