Exam-Style Questions on Circle
Problems on Circle adapted from questions set in previous Mathematics exams.
GCSE Higher
The diagram shows a sector of a circle, centre O, with one angle shown as being 110°. If the radius of the circle is 8cm calculate:
(a) The area of the shaded area.
(a) The perimeter of the shaded area.
Worked Solution
This draft emoji design includes a circle and a kite.
AB and AC are both tangents to the circle centre O.
The radius of the circle is 7cm. The dashed line AO is 12cm.
Calculate the length of the arc BDC.
(a) Find \(x\), the size of angle ABC if B, C and D are all points on the circle to which AB is a tangent. The diagram is not drawn to scale.
(b) Give a reason for your answer
The diagram shows a circle with equation \(x^2+y^2=13\).
A tangent to the circle touches the circle at the point J. The x-coordinate of J is 2. The tangent intersects the x-axis at K. Find the coordinates of the point K.
(a) Find the coordinates of the point at which the curve \(y = k^x\) intersects the y-axis.
The equation of circle \(A\) is \(x^2+y^2=25\). This circle is translated by the vector \( \begin{pmatrix} 0 \\ 4 \\ \end{pmatrix} \) to give circle \(B\).
(b) Draw a sketch of circle \(B\) clearly labelling the points of intersection with the y-axis.
A circle with equation \(x^2 + y^2 = 6 \) meets a one of its tangents at point \(S\).
The coordinates of \(S\) are:
$$\left( \frac{\sqrt 5}{3} , \frac{7}{3} \right)$$
Find the equation of the tangent to the circle at the point \(S\).
The diagram, not drawn to scale, shows chords joining points on the circumference of a circle. Prove that triangle ADE is similar to triangle BCE.
IB Standard
The following diagram shows a circle with centre O and radius 9 cm.
The points A, B and C lie on the circumference of the circle, and AÔC = 1.2 radians.
(a) Find the length of the arc ABC.
(b) Find the perimeter of the minor sector OAC.
(c) Find the area of the minor sector OAC.
Two identical small yellow circles are drawn inside one large circle, as shown in the diagram. The centres of the small circles lie on the diameter of the large circle. The part of the large circle that is outside both small circles is painted red.
Find the fraction of the large circle that is painted red.
(a) The circumference of a circle is \(16 \pi \) cm and its centre is at the origin. Find the equation of the circle.
(b) The line \(12x+ ay = b\) is a tangent at the point (6, 5) to a different circle with centre at the origin. Find the values of a and b.
The diagram below shows an isosceles right-angled triangle and two arcs of circles. The larger arc is part of a circle with its centre at D and the smaller arc is part of a circle with its centre at A.
The right-angled triangle has two sides of length 10cm.
The point D is the midpoint of BC.
(a) Calculate the area of the red segment.
(b) Calculate the area of the yellow region.
Kausar Begum Khan, Kenya
"This is a fabulous website! I cannot thank you enough."
Aruna Upadhyayula, Bahrain
"Transum is so resourceful. The resources are so child-friendly that my students love its usage.It has a wide range of topics and caters to most of my needs. 'Thankyou" will be too small a word to express my gratitude to the content developers of this site."
Ahmed Jaja, Ghana
"It was very useful for me.I recommend it to every math student.Thank you."
Madiwa Otieno, Kenya
"A very great site.Nice questions."
Yuli, Woodmansterne
"This website is amazing it helps me with my maths to get good marks in my upcoming test. Thanks."