UK GCSE · GCSE Mathematics (Higher Tier)

Geometry, Trigonometry and Circle Theorems

15 practice questions, each with the answer and an explanation.

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$A, B, C$ are points on the circumference of a circle with centre $O$ . Angle $∠ A O C = 13 0^{\circ}$ . What is the size of angle $∠ A B C$ ?
In a triangle $X Y Z$ , $X Y = 7 cm, Y Z = 9 cm$ , and angle $∠ X Y Z = 4 8^{\circ}$ . Calculate the area of triangle $X Y Z$ to 1 decimal place.
Points $P, Q, R, S$ lie consecutively on a circle, forming a cyclic quadrilateral. Angle $∠ P QR = 2 x + 1 5^{\circ}$ and angle $∠ P S R = 3 x - 1 0^{\circ}$ . Find the value of $x$ .
In triangle $A B C$ , side $a = 6 cm, b = 10 cm$ , and angle $C = 12 0^{\circ}$ . Calculate the length of side $c$ .
$A$ and $B$ are mathematically similar cylinders. Cylinder $A$ has surface area $36 cm^{2}$ and Cylinder $B$ has surface area $81 cm^{2}$ . If Cylinder $A$ has volume $48 cm^{3}$ , find t…
$T A$ is a tangent to a circle at point $A$ . $A B C$ is a triangle inscribed in the circle. If angle $∠ T A B = 6 8^{\circ}$ and angle $∠ A B C = 5 4^{\circ}$ , calculate the size of angle $∠ B A C$ .
A right-angled triangle has a hypotenuse of 13 cm and one shorter side of 5 cm. What is the length of the other short side?
In a right-angled triangle, the hypotenuse is 12 cm and one angle is $3 5^{\circ}$ . What is the length of the side opposite the $3 5^{\circ}$ angle, to 3 significant figures?
In triangle ABC, $A B = 8$ cm, $B C = 11$ cm and angle $A B C = 4 0^{\circ}$ . What is the area of the triangle, to 3 significant figures?
In triangle PQR, $P Q = 7$ cm, $P R = 9$ cm and angle $QP R = 6 0^{\circ}$ . Find the length of QR to 3 significant figures.
An angle at the centre of a circle is $12 8^{\circ}$ . What is the angle at the circumference standing on the same arc?
ABCD is a cyclic quadrilateral. Angle $A B C = 11 2^{\circ}$ . What is angle $A D C$ ?
Two similar solids have volumes of 54 cm cubed and 250 cm cubed. The surface area of the smaller solid is 90 cm squared. What is the surface area of the larger solid?
$a = (23)$ and $b = (- 1 4)$ . Work out $2 a - b$ .
A tangent from point P touches a circle at T. The centre is O, the radius is 5 cm and OP = 13 cm. How long is the tangent PT?

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Geometry, Trigonometry and Circle Theorems

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