All AQA Level 2 Further Maths topic questions

AQA Level 2 Further Maths · Topic G

Geometry questions

Practice questions covering geometry from this specification. Every question below is the real AQA question as it is printed on the paper, with its mark scheme.

Questions to attempt

6 questions on Geometry, each with its mark scheme and worked method. Reveal the answer when you have had a go.

G1 · Knowledge of perimeter, area, surface area and volume of standard shapes; angle properties of parallel/intersecting lines, triangles, quadrilaterals and polygons; and understand and use circle theorems
Q1
ORIGINAL
1
A closed cylinder has radius $3$ cm and height $8$ cm. Work out its total surface area in terms of $\pi$ .
[3 marks]
G2 · Understand and construct geometrical proofs using formal arguments
Q2
ORIGINAL
2
In a circle with centre $O$ , the minor angle $AOB$ is $2t$ , where $0^\circ<t<90^\circ$ . A tangent is drawn at $A$ . Prove that the acute angle between the tangent and $AB$ is $t$ .
[3 marks]
G3 · Sine and cosine rules in scalene triangles; area of a triangle = 1/2 ab sinC
Q3
ORIGINAL
3
Two sides of a triangle are $7$ cm and $9$ cm, and their included angle is $60^\circ$ . Work out the exact length of the third side.
[3 marks]
G4 · Use of Pythagoras' theorem in 2D and 3D; recognise Pythagorean triples
Q4
ORIGINAL
4
A cuboid has side lengths $6$ cm, $6$ cm and $7$ cm. Work out the length of its body diagonal.
[3 marks]
G5 · Apply trigonometry and Pythagoras' theorem to 2 and 3 dimensional problems, including the angle between a line and a plane and between two planes
Q5
ORIGINAL
5
A square-based pyramid has base side $10$ cm. Its apex is vertically above the centre of the base, and a sloping edge from the apex to a base vertex is $13$ cm. Work out the angle that this edge makes with the base, to $1$ decimal place.
[3 marks]
G6 · Sketch and use graphs of y = sin x, y = cos x and y = tan x for angles of any size
Q6
ORIGINAL
6
Sketch $y=\tan x$ for $0^\circ\leq x\leq360^\circ$ . Mark its zeros and vertical asymptotes.
[3 marks]

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Printable question pack — G GeometryAll 100 questions across 10 specification points — 94 more than the 6 above — with the full mark scheme. Guest accessible · browser print.

G Geometry revision notes

Real past-paper questions

45 questions shown as printed, 12 reference-only · newest series first

Question and mark-scheme extracts © AQA. Reproduced for educational use; the original paper is linked on every question. Contact hello@oscarsongtutoring.co.uk for removal.

AQA 8365 · June 2025 · Paper 1 · Q16 · 3 marks
3 marks
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Examiner report
Many students were awarded at least two marks on this question by stating the sine rule correctly and then substituting in the correct trigonometrical values. The most common errors involved when attempting to manipulate the surds. © 2025 AQA
Text version
Triangle ABC has angle A = 45 degrees, angle B = 30 degrees and side BC = 10 cm.
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AQA 8365 · June 2025 · Paper 1 · Q21 · 2 marks
2 marks
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A small majority of students were awarded both marks. Some students attempted to manipulate the numerator, but this rarely led to any marks being awarded. Students need to take extra care with their working when asked to ‘show that’ something is true.
Text version
Consider 20sin^2(x) / (3 + cos^2(x) - 3sin^2(x)).
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AQA 8365 · June 2025 · Paper 1 · Q23 · 5 marks
5 marks
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A significant number of students were awarded four marks. A common error was to ignore the lower bound. Students who used the quadratic formula or completing the square on the quadratic inequality rarely managed to get to the answer and on this paper most of the quadratics will factorise into double brackets.
Text version
A trapezium has parallel sides (3x - 8) cm and (x + 5) cm, and perpendicular height 2x cm. Its area is less than 22 cm^2.
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AQA 8365 · June 2025 · Paper 2 · Q2 · 4 marks
4 marks
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This question was very well answered. The most common error was when attempting to work out the area of triangle COD with students either using an incorrect formula or having an incorrect value for OD.
Text version
C and E lie on the circle x^2 + y^2 = 144, centred at O. C is the point on the positive y-axis and E is the point on the positive x-axis. D lies on OE, CD is a straight line, and OD : DE = 5 : 1. The shaded region is the quarter-circle sector COE with triangle COD removed.
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AQA 8365 · June 2025 · Paper 2 · Q14 · 5 marks
5 marks
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Examiner report
Most students who started by writing Pythagoras’ theorem as (3w + 2)2 = (2w + 5)2 + w2 subsequently showed enough correct working to be awarded both marks. Mistakes were more likely when students attempted to expand straight away or wrote Pythagoras’ theorem in a less standard way. Question 14(b) Some errors were made when factorising or using the quadratic formula. The other common error was to link an expression to an incorrect trigonometric ratio. Overall, the question was well answered.
Text version
A right-angled triangle has perpendicular sides w cm and (2w + 5) cm, hypotenuse (3w + 2) cm, and angle x between the side (2w + 5) and the hypotenuse.
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AQA 8365 · June 2025 · Paper 2 · Q16 · 4 marks
4 marks
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Examiner report
There were lots of different ways to tackle this problem. Some expressions were ‘correct’ but they were not linked to the appropriate angle. Brackets were often omitted when angle BCD was worked out using the sum of the opposite angles of a cyclic quadrilateral is 180°. Most correct equations came from equating the angle sum of triangle BCD to 180°. The question was a good discriminator.
Text version
A, B, C and D lie on a circle. EBF is a tangent at B. Angle BAD is (4x - 15) degrees, angle DBC is (x + 20) degrees, and the angle between BF and BC is (x/2 + 60) degrees.
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AQA 8365 · June 2025 · Paper 2 · Q18 · 5 marks
5 marks
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The method or value for at least one of the appropriate lengths was correct for most students. A majority of students worked out all three lengths correctly but applying the cosine rule using these values proved more challenging. There were however many students who were awarded full marks. © 2025 AQA
Text version
ABCDEF is a triangular prism. ABCD is a rectangle, angle BCF = 90 degrees, AB = 14 cm, BC = 8 cm and CF = 6 cm. EF is parallel and equal to AB, and M is the midpoint of EF.
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AQA 8365 · June 2025 · Paper 2 · Q21 · 5 marks
5 marks
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Examiner report
A challenging question for many students, partly due to the unfamiliar nature of the question. Some who worked out than sin θ was – 5 3 gave –36.9 or 216.9 as the answer. Many students did a lot of work but were not awarded any marks. Question 21(b) Although a significant number of students worked out all four correct angles many only gave two of the correct ones. Others only worked out one correct angle or were not able to work out any and were not awarded any marks. The most common incorrect angles seen were 0° and 262.8°. Very few angles outside the given range were seen. This question had the most non- attempts on the paper.
Text version
This question concerns trigonometric values and equations for angles from 0 degrees to 360 degrees.
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AQA 8365 · June 2024 · Paper 1 · Q9 · 3 marks
3 marks
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Most students are getting 3D Pythagoras correct, some of them very quickly. There were quite a few misreads of the diagram with SC1 being awarded in these cases. Some students left their answer as 169 so lost the final A mark.
Text version
ABCDEFGH is a cuboid with perpendicular edge lengths 3 cm, 4 cm and 12 cm.
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AQA 8365 · June 2024 · Paper 1 · Q12 · 3 marks
3 marks
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Examiner report
Done well by most students though some made the expected error of trying to use cos 3 4 and didn’t score. The rest of the errors were down to carelessness. Some can’t follow BODMAS and do 244 – 240 and then try to work out 3 4 of it. A surprisingly large number couldn’t do 244 – 180. There are also a small number who couldn’t copy the cosine rule accurately from the formula sheet.
Text version
In triangle ABC, AB = 12 cm, AC = 10 cm and cos(A) = 3/4.
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AQA 8365 · June 2024 · Paper 1 · Q22 · 4 marks
4 marks
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Examiner report
Some excellent solutions here though not that many of them. Most of these follow our Alt 1 but a small number manipulate in different ways and still get to the correct answer using a valid method. Quite a few get to x 4 tan with incorrect working via a range of errors. cosx + sinx = 1 appeared more than once. There were quite a few non-attempts. Not enough students know how to use a CAST diagram or trigonometrical graphs and usually lost the final mark.
Text version
The function f is defined by f(x) = [3sin(x)cos(x) + sin^2(x)]/[12cos^2(x) + 4sin(x)cos(x)].
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AQA 8365 · June 2024 · Paper 2 · Q11 · 4 marks
4 marks
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Very well answered with many fully correct responses. Some rounded the value for AC to 17 and this resulted in the final answer being outside the accepted range.
Text version
Triangles ABC and ACD share side AC. In triangle ABC, BC = 18 cm, angle ABC = 59 degrees and angle BAC = 66 degrees. In triangle ACD, CD = 15 cm and angle ACD = 70 degrees.
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AQA 8365 · June 2023 · Paper 1 · Q7 · 4 marks
4 marks
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Many correct solutions. Some candidates don’t know how to find the area of a triangle sadly which limited them to a maximum of 1 mark. Too many candidates were expanding 4( 1 2 y2x) as 4 x 1 2 y2 + or x 4x. A small number of candidates found the correct answer but then rationalised the root pi in the denominator. As long as this was done correctly we didn’t penalise this.
Text version
A cone has radius r and perpendicular height x. A triangular prism has cross-section base y and perpendicular height y, and length x. The cone volume is four times the prism volume. Use volume of cone = (1/3) x base area x perpendicular height.
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AQA 8365 · June 2023 · Paper 1 · Q12 · 4 marks
4 marks
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[12a] Quite a few candidates used a trigonometrical graph to work it out the answer. Very few CAST diagrams seen though. This question was answered well.
Text version
Angles are measured in degrees in the interval from 0 to 360 inclusive.
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AQA 8365 · June 2023 · Paper 1 · Q20 · 6 marks
6 marks
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The first 4 marks were straight forward to mark. However there are numerous ways to gain the last 2 marks. The most popular methods included dividing the Isosceles triangle in half or using the sine rule however students could also use the cosine rule to get: 21 = r2 + r 2 – 2 x r x r x cos120 leading to 21 = 3 r 2 An unusual method that was seen but which is equally correct is to combine the area of a triangle formula with the sine rule to get abc r 4 where r is the radius of the circumcircle. This then leads to: Area of QPR = 1 2 x 4 x 5 x sin60 = 4 x 5 x r 4 21 which can be solved for r. A small number of students used the fact that it was a 30, 60, 90 triangle and used ratios. A small number of students wrote the answer without working and this had to be given the benefit of the doubt for all 6 marks.
Text version
P, Q and R lie on a circle with centre O. The minor central angle POR is 120 degrees, PQ = 4 cm and QR = 5 cm. Q lies on the major arc PR.
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AQA 8365 · June 2023 · Paper 2 · Q8 · 4 marks
4 marks
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Many fully correct responses were seen. Students sometimes muddled up the lengths of the sides and it was not uncommon to see BD or DC stated as 12.5 cm. However, many were able to use trigonometry correctly with their value for DC. Occasionally students used an incorrect formula for the area of a triangle, omitting to multiply by a half. Some students started by setting up the equation 0.5 × BC × AC × sin w = 25 but usually made no progress.
Text version
ABC is a triangle with perpendicular AD to BC. AD=4 cm, area ABC=25 cm^2, and BD:DC=2:3. Angle w is angle ACB.
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AQA 8365 · June 2023 · Paper 2 · Q12 · 3 marks
3 marks
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There were lots of fully correct solutions seen. However, some students worked out AC and then used triangle VAC as a right-angled triangle. It was also common to see students work out AC but think they had found AX and continue with the incorrect value. There were some responses that assumed AX to be 7.5 cm and made no progress. A few misconceptions were seen with students working out angle VAB or angle AVX.
Text version
VABCD is a pyramid with square horizontal base ABCD of side 15 cm. V is directly above the centre X of the base, and VA=28 cm.
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AQA 8365 · June 2023 · Paper 2 · Q18 · 3 marks
3 marks
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Part (a) differentiated well amongst students. The majority of students followed the instruction to select two answers but some only chose one. All of the distractors seemed equally popular. A small majority chose the correct option in part (b). Options 3 and 4 were both chosen quite often.
Text version
For 0 to 360 degrees, a cosine graph is supplied. Alpha is obtuse and cos(alpha)=-k, where k is positive.
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AQA 8365 · June 2023 · Paper 2 · Q20 · 3 marks
3 marks
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Many students were able to use one or both of the necessary trigonometric identities. Occasionally cos x was changed to 1 – sin x or 1 – sin2 x. Quite a few students omitted brackets and not all recovered them. Some students divided through by cos x or sin x at the start. A few students substituted values for the angle and made no progress.
Text version
Consider 2sin^3(x) + 2sin(x)cos^2(x) + 5tan(x)cos(x).
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AQA 8365 · June 2023 · Paper 2 · Q21 · 4 marks
4 marks
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This question was not well answered. Many students were able to correctly work out an expression for angle AOB using the angles of an isosceles triangle. Quite often obtuse angle AOD was stated as 180 – 7x or 14x or 7x. Those who correctly worked out that reflex angle AOD = 14x and reflex angle BOD = 12x were usually able to form a correct equation, although occasionally some omitted the brackets.
Text version
A, B, C, D and E lie on a circle with centre O. Segments OA and OB are radii. Angle BAO=2x, angle AED=7x and angle BCD=6x.
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AQA 8365 · June 2022 · Paper 1 · Q4 · 3 marks
3 marks
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[4b] Marks were awarded for intention in this sketch. Sketches with a clear vertex at (180 , –1) or lines that were straight lost the A mark. Some level of curvature needed to be seen at key points.
Text version
Angles are measured in degrees.
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AQA 8365 · June 2022 · Paper 1 · Q15 · 5 marks
5 marks
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The best way to get the first 3 B marks was to put them on the diagram. Flexibility was allowed in how reasons were stated. Unless it was clear which working applied to which angle the marks couldn’t be awarded (they could have been on the diagram). Some students ended with the correct answer but from incorrect methods which included double errors with negative signs. Three out of four reasons were needed for the final mark (the first 3 B marks and then the 4th mark when forming an equation). A number of students stated that ABPC was a cyclic quadrilateral (which it is). If they showed that ABP and ACP were right angles it was fine just to write that BAC + BPC = 180. Without stating ABP and ACP were right angles they have no evidence to support ABPC being a cyclic quadrilateral so this wasn’t accepted. A few wrote that it was a kite and opposite angles sum to 180 which is an incorrect statement and wasn’t accepted.
Text version
B, C and D lie on a circle with centre P. AB and AC are tangents. The angle BAC is x degrees and angle BDC is y degrees.
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AQA 8365 · June 2022 · Paper 1 · Q17 · 6 marks
6 marks
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[17a] Some students gained 1 mark in more than one way but didn’t get 2 marks on any method. It was common to use trigonometry to get one coordinate and then use Pythagoras to get the other one.
Text version
P lies in the first quadrant on x^2+y^2=16. OP makes an angle of 60 degrees with the positive x-axis.
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AQA 8365 · June 2022 · Paper 1 · Q18 · 3 marks
3 marks
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Most candidates spotted it was the sine rule (a few tried the cosine rule). Only a small number knew that sin135 = sin45 = 2 2 . Common errors were sin135 = sin90 + sin45 and sin135 = 3sin45.
Text version
In triangle RST, RS:ST=1:4. Angle SRT is 135 degrees and angle RTS is theta.
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AQA 8365 · June 2022 · Paper 1 · Q21 · 3 marks
3 marks
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Proofs were, for the most part, poorly constructed. Some students treated it as an equation and cross-multiplied ending with 1 = 1 (as soon as this happened we stopped marking so it was usually either 1 or 0 marks). Others try to work from both sides which makes it confusing to mark (examiners needed to look at which side gained the most marks). Substitutions are sometimes used without saying which or how. This can mean students didn’t gain the first M mark and hence usually ended up with 0. We condoned splitting the numerator from the denominator for the M marks as long as it was recovered for the A mark.
Text version
Consider (4cos^2(x)+3sin^2(x)-4)/cos^2(x).
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AQA 8365 · June 2022 · Paper 2 · Q11 · 2 marks
2 marks
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This novel question was not well answered. Many worked out 60° but never showed an inequality. Others substituted a = 6 and worked out 41.4°. The most common wrong answer was x > 60 with some of these scoring one mark by having cos x > 2 1 in the working lines.
Text version
A right-angled triangle has hypotenuse 4 cm, side adjacent to acute angle x equal to a-3 cm, and a>5.
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AQA 8365 · June 2022 · Paper 2 · Q14 · 3 marks
3 marks
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Different approaches were seen when forming an equation but most were correct. Some errors were made when eliminating brackets. A few thought the two angles were equal.
Text version
PQRS is cyclic. Angle PSR=4(x+15) degrees, and angle PQR is 40 degrees smaller than angle PSR.
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AQA 8365 · June 2022 · Paper 2 · Q16 · 4 marks
4 marks
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The angle included between the two given sides was often worked out correctly but a significant number of students thought that this was angle y. Those who worked out the perpendicular height usually worked this out correctly as 15 cm but often made no further progress. Most fully correct solutions started by using 2 1 absinC and then used base angles of an isosceles triangle.
Text version
An acute isosceles triangle has two equal sides of 16 cm, area 120 cm^2, and one of its equal base angles is y.
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AQA 8365 · June 2022 · Paper 2 · Q18 · 5 marks
5 marks
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Many students were able to correctly work out PG, PC and AC. Most knew to use the cosine rule in triangle APC but some substituted incorrectly, often omitting brackets, and many rearranged the formula incorrectly or inadvertently used it to find one of the other angles.
Text version
ABCDEFGH is a cuboid with AB=40 cm, BC=9 cm and CG=20 cm. P lies on HG with HP:PG=3:7, and AP=25 cm.
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AQA 8365 · June 2022 · Paper 2 · Q21 · 4 marks
4 marks
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This question was not well answered. A significant number rearranged correctly but often forget to consider the negative square root. Most students who scored well on this question worked using tan x with those converting to sin x or cos x rarely scoring more than one mark.
Text version
Angles are measured in degrees from 0 to 360 inclusive. Consider 2tan^2(x)=3.
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AQA 8365 · November 2021 · Paper 1 · Q6 · 2 marks
2 marks
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Text version
For 0 <= x <= 360 degrees, y=sin(x). You are given sin(220 degrees)=-k.
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AQA 8365 · November 2021 · Paper 1 · Q15 · 2 marks
2 marks
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Text version
3tan^2(x)=1 for 0 <= x <= 90 degrees.
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AQA 8365 · November 2021 · Paper 1 · Q18 · 4 marks
4 marks
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Text version
In triangle PQR, PQ=x cm, PR=7 cm, QR=3 cm and angle PQR=60 degrees.
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AQA 8365 · November 2021 · Paper 1 · Q23 · 6 marks
6 marks
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F, H, K and J lie on a circle. Chords HJ and KF intersect at L. EFG is tangent at F, and FH is parallel to JK. Angle FHJ=2x, angle EFH=3y, angle KFJ=98 degrees and angle KLJ=7y.
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AQA 8365 · November 2021 · Paper 2 · Q3 · 3 marks
3 marks
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A rectangle with side lengths 2d and 12de has the same area as a right triangle with perpendicular sides 9d and 8e^2.
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AQA 8365 · November 2021 · Paper 2 · Q4 · 3 marks
3 marks
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Two circles centred at the origin have equations x^2+y^2=100 and x^2+y^2=36. The shaded region is the first-quadrant quarter of the annulus between them.
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AQA 8365 · November 2021 · Paper 2 · Q5 · 3 marks
3 marks
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SQR is right-angled at S. P=(1,6) lies on SQ, Q=(1,1), R=(k,15), angle SPR=45 degrees, and M is the midpoint of QR.
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AQA 8365 · November 2021 · Paper 2 · Q10 · 3 marks
3 marks
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A sphere has radius 3k/2 cm and volume 972pi cm^3. Use V=(4/3)pi r^3.
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AQA 8365 · November 2021 · Paper 2 · Q17 · 5 marks
5 marks
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ABCDEFGH is a cube of side 32 cm. M lies on DH and N lies on CG, with MN horizontal across the back face and angle GMN=28 degrees. Line BM is drawn.
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AQA 8365 · November 2021 · Paper 2 · Q20 · 4 marks
4 marks
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Text version
Consider (2sin(x)+cos(x))/tan(x) - 1/sin(x).
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AQA 8365 · sample1 · Paper 2 · Q1 · 3 marks
3 marks
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Text version
A sketch shows the lines y=2x and y=6. They meet at Q, and P is (0,6). O is the origin.
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AQA 8365 · sample1 · Paper 2 · Q9 · 5 marks
5 marks
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A, B and C lie on a circle with centre O. Angle BAC is (2x-50) degrees, angle BOC is (x+62) degrees, and angle OCB is y degrees.
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AQA 8365 · sample1 · Paper 2 · Q12 · 7 marks
7 marks
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An L-shaped region is made from two rectangles. Its labelled boundary lengths are 30x, 15x, 10x, 20x and y centimetres, and its perimeter is 252 cm. Its area is A cm^2.
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AQA 8365 · sample1 · Paper 2 · Q14 · 3 marks
3 marks
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In a triangle, side y is opposite 18 degrees and side 2y is opposite the obtuse angle x.
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AQA 8365 · sample1 · Paper 2 · Q20 · 5 marks
5 marks
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VABCD is a pyramid with horizontal rectangular base ABCD. V is directly above the centre. VA=VB=VC=VD=10 cm, AB=8 cm, BC=6 cm, and M is the midpoint of BC.
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References open the specific AQA question paper when it is available. Missing, login-only or unverified question-paper sources are labelled on the affected row. Question content remains on the exam board's website.

June 20252 questions

Paper 1 · Q72 marks · Non-calculator · calculate · standardView question in paper (p.6)Mark scheme (p.13)
Paper 1 · Q124 marks · Non-calculator · prove · standardView question in paper (p.10)Mark scheme (p.18)

June 20243 questions

Paper 1: Q16 (p.14, 4 marks)

Paper 2: Q17 (p.16, 6 marks), Q22 (p.22, 5 marks)

June 20231 question

Paper 2: Q9 (p.8, 7 marks)

November 20212 questions

Paper 1: Q21 (p.20, 5 marks)

Paper 2: Q15 (p.13, 5 marks)

sample14 questions

Paper 1: Q7 (3 marks), Q9 (4 marks), Q12 (3 marks), Q20 (5 marks)

Question extracts are shown for study alongside a link to the original AQA paper. Reference-only rows carry series, paper, question number and marks.

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