G Geometry — AQA 8365 spec coverage

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

Know the standard formulae for perimeters, areas, surface areas and volumes, and keep all measurements in compatible units before substituting.
Angle facts include angles on a line and at a point, vertically opposite and alternate angles, triangle and quadrilateral sums, and interior or exterior angles of polygons.
Circle theorems include: the angle at the centre is twice the angle at the circumference, angles in the same segment, the angle in a semicircle, cyclic quadrilaterals, tangent-radius perpendicularity and the alternate segment theorem.
State the theorem or angle fact when reasons are requested; an unlabelled numerical angle may not earn a reasoning mark.
For a composite shape, split or subtract familiar regions and avoid counting a shared face in a surface-area calculation.

Tier 1 · Easy

2 marks

ORIGINAL

Work out the interior angle of a regular $12$ -sided polygon.

Tier 2 · Standard

3 marks

ORIGINAL

A closed cylinder has radius $3$ cm and height $8$ cm. Work out its total surface area in terms of $\pi$ .

Tier 3 · Hard

4 marks

ORIGINAL

Two radii of a circle of radius $10$ cm enclose an angle of $120^\circ$ . Work out the exact area of the minor sector with the triangle formed by the two radii removed.

G2

Understand and construct geometrical proofs using formal arguments

A geometrical proof is a connected chain of statements, each supported by a stated theorem, definition or established result.
Use precise labels such as $\angle ABC$ so it is clear which angle each statement concerns.
Congruence can establish equal corresponding sides or angles; name a valid test such as SSS, SAS, ASA or RHS.
Do not assume a diagram is drawn to scale, and do not use the result to be proved as one of your reasons.
For circle proofs, identify the relevant chord, arc, radius or tangent when naming a theorem.

Tier 1 · Easy

2 marks

ORIGINAL

In isosceles triangle $ABC$ , $AB=AC$ . Point $D$ is the midpoint of $BC$ . Prove that $AD$ is perpendicular to $BC$ .

Tier 2 · Standard

3 marks

ORIGINAL

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 $t$ .

Tier 3 · Hard

4 marks

ORIGINAL

From a point $P$ outside a circle with centre $O$ , tangents touch the circle at $A$ and $B$ . Prove that $OP$ is the perpendicular bisector of $AB$ .

G3

Sine and cosine rules in scalene triangles; area of a triangle = 1/2 ab sinC

Use the sine rule when you know an opposite side-angle pair; keep each side matched with its opposite angle.
Use the cosine rule for three known sides, or for two sides and their included angle.
The area formula $A=\frac12 ab\sin C$ uses the angle $C$ included between sides $a$ and $b$ .
The sine rule can produce two angles because $\sin\theta=\sin(180^\circ-\theta)$ ; check the diagram, interval and triangle angle sum.
Keep full calculator values during working and round only the final answer to the accuracy requested.

Tier 1 · Easy

2 marks

ORIGINAL

Two sides of a triangle are $8$ cm and $5$ cm, and their included angle is $30^\circ$ . Work out the area.

Tier 2 · Standard

3 marks

ORIGINAL

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.

Tier 3 · Hard

4 marks

ORIGINAL

In triangle $ABC$ , $A=30^\circ$ , $a=6$ cm and $b=10$ cm. Work out both possible values of angle $C$ . Give each answer to $1$ decimal place.

G4

Use of Pythagoras' theorem in 2D and 3D; recognise Pythagorean triples

In a right-angled triangle, $a^2+b^2=c^2$ , where $c$ is the hypotenuse opposite the right angle.
Common triples such as $3,4,5$ , $5,12,13$ and their multiples can give exact lengths without a calculator.
For a cuboid, find a face diagonal first and then use a second right-angled triangle, or use $d^2=l^2+w^2+h^2$ directly.
Do not use Pythagoras unless a right angle is known, and check that the longest side has been used as the hypotenuse.

Tier 1 · Easy

1 mark

ORIGINAL

A right-angled triangle has perpendicular sides $5$ cm and $12$ cm. Work out the hypotenuse.

Tier 2 · Standard

3 marks

ORIGINAL

A cuboid has side lengths $6$ cm, $6$ cm and $7$ cm. Work out the length of its body diagonal.

Tier 3 · Hard

4 marks

ORIGINAL

A cuboid has base dimensions $8$ cm by $9$ cm and body diagonal $17$ cm. Work out its volume.

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

Draw or identify a right-angled cross-section containing the required line and its perpendicular projection onto the plane.
The angle between a line and a plane is the angle between the line and its projection on that plane, not usually an angle with an arbitrary edge.
For the angle between two planes, use a cross-section perpendicular to their line of intersection; the angle between the two cross-section lines is the required dihedral angle.
Use Pythagoras to find missing face or body diagonals before applying $\sin$ , $\cos$ or $\tan$ .
Mark the right angle and label every calculated length so that the final trigonometric ratio uses the correct triangle.

Tier 1 · Easy

2 marks

ORIGINAL

A straight ramp is $10$ m long and rises vertically by $6$ m. Work out the angle the ramp makes with the horizontal, to $1$ decimal place.

Tier 2 · Standard

3 marks

ORIGINAL

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.

Tier 3 · Hard

4 marks

ORIGINAL

A cuboid $ABCDEFGH$ has $AB=12$ cm, $BC=5$ cm and vertical edge $BF=8$ cm. The base is $ABCD$ and $EFGH$ is the top face. Work out the acute angle between plane $ABGH$ and the base plane $ABCD$ , to $1$ decimal place.

G6

Sketch and use graphs of y = sin x, y = cos x and y = tan x for angles of any size

$y=\sin x$ and $y=\cos x$ have period $360^\circ$ and range $-1\leq y\leq1$ .
$y=\tan x$ has period $180^\circ$ , zeros at multiples of $180^\circ$ , and vertical asymptotes at $90^\circ+180^\circ n$ .
A trigonometric sketch should be smooth and pass through the exact key points; do not join plotted points with straight line segments.
Extend graphs to angles of any size by repeating the correct period in both directions.
Use the graph to read signs, repeated values, intersections and approximate solutions, while excluding points where $\tan x$ is undefined.

Tier 1 · Easy

1 mark

ORIGINAL

Write down the coordinates of the minimum point of $y=\cos x$ for $0^\circ\leq x\leq360^\circ$ .

Tier 2 · Standard

3 marks

ORIGINAL

Sketch $y=\tan x$ for $0^\circ\leq x\leq360^\circ$ . Mark its zeros and vertical asymptotes.

Tier 3 · Hard

3 marks

ORIGINAL

Use the graphs of $y=\sin x$ and $y=\cos x$ to state the range of $x$ for which both values are positive and $\sin x>\cos x$ , where $0^\circ\leq x\leq360^\circ$ .

G7

Use the definitions of sin, cos and tan for any positive angle up to 360 degrees (measured in degrees only)

Use a reference angle and the quadrant to determine the sign: sine is positive in quadrants I and II, cosine in I and IV, and tangent in I and III.
For a point $(x,y)$ at distance $r$ from the origin, $\cos\theta=x/r$ , $\sin\theta=y/r$ and $\tan\theta=y/x$ .
Angles are measured anticlockwise from the positive $x$ -axis and are in degrees for this specification.
When using inverse trigonometric functions, use the quadrant information to convert the calculator's reference angle into the required angle from $0^\circ$ to $360^\circ$ .

Tier 1 · Easy

1 mark

ORIGINAL

Work out the exact value of $\cos120^\circ$ .

Tier 2 · Standard

3 marks

ORIGINAL

$180^\circ<\theta<270^\circ$ and $\sin\theta=-\frac{12}{13}$ . Work out the exact values of $\cos\theta$ and $\tan\theta$ .

Tier 3 · Hard

4 marks

ORIGINAL

$0^\circ<\theta<360^\circ$ , $\tan\theta=-\frac7{24}$ and $\cos\theta>0$ . Work out $\sin\theta$ exactly and $\theta$ to $1$ decimal place.

G8

Knowledge and use of 30, 60, 90 triangles and 45, 45, 90 triangles

A $30^\circ$ - $60^\circ$ - $90^\circ$ triangle has side ratio $1:\sqrt3:2$ , opposite those angles respectively.
A $45^\circ$ - $45^\circ$ - $90^\circ$ triangle has side ratio $1:1:\sqrt2$ .
These ratios give the exact values $\sin30^\circ=1/2$ , $\cos30^\circ=\sqrt3/2$ and $\sin45^\circ=\cos45^\circ=\sqrt2/2$ .
Keep exact surd values unless a decimal accuracy is requested, and match each side to the angle it is opposite.

Tier 1 · Easy

1 mark

ORIGINAL

Write down the exact value of $\sin30^\circ$ .

Tier 2 · Standard

2 marks

ORIGINAL

A right-angled isosceles triangle has hypotenuse $14$ cm. Work out the exact length of each equal side.

Tier 3 · Hard

4 marks

ORIGINAL

A regular hexagon has side length $8$ cm. Work out the exact distance between a pair of opposite sides and hence the exact area of the hexagon.

G9

Know and use tan = sin / cos and sin^2 + cos^2 = 1

Use $\tan x=\frac{\sin x}{\cos x}$ and $\sin^2x+\cos^2x=1$ to replace one trigonometric form with another.
Useful rearrangements include $1-\sin^2x=\cos^2x$ and $1-\cos^2x=\sin^2x$ .
In an identity proof, start from one side and transform it through valid equal expressions; do not assume the target statement.
Factor before substituting identities, and keep brackets around complete numerators and denominators.
Cancellation is valid only where the original expression is defined.

Tier 1 · Easy

2 marks

ORIGINAL

$x$ is acute and $\sin x=\frac35$ . Work out the exact value of $\tan x$ .

Tier 2 · Standard

2 marks

ORIGINAL

Simplify $\frac{1-\cos^2x}{\sin x}$ .

Tier 3 · Hard

4 marks

ORIGINAL

Prove that $\frac{\sin x}{1+\cos x}+\frac{1+\cos x}{\sin x}=\frac{2}{\sin x}$ for values of $x$ where both sides are defined.

G10

Solution of simple trigonometric equations in given intervals

Find a reference angle, use the signs of the trigonometric function to choose the correct quadrants, and list every solution in the stated interval.
For squared equations, rearrange or factor before taking values; do not lose the positive or negative possibilities introduced by a square.
Use the periods $360^\circ$ for sine and cosine and $180^\circ$ for tangent to generate repeated solutions.
Check whether interval endpoints are included and give angles in degrees to the requested accuracy.
Substitute solutions back into the equation when factorisation or cancellation could have introduced or removed values.

Tier 1 · Easy

1 mark

ORIGINAL

Solve $\sin x=1$ for $0^\circ\leq x\leq360^\circ$ .

Tier 2 · Standard

2 marks

ORIGINAL

Solve $\cos x=-\frac{\sqrt3}{2}$ for $0^\circ\leq x\leq360^\circ$ .

Tier 3 · Hard

4 marks

ORIGINAL

Solve $2\sin^2x+\sin x-1=0$ for $0^\circ\leq x<360^\circ$ .