Topic 4: Geometry and measures

Topic 4 is the largest visual strand of AQA GCSE Mathematics. It rewards students who can reason carefully from a small set of facts rather than memorise hundreds of separate results. Almost every problem reduces to a handful of core ideas: angles on lines and around points, parallel-line angle rules, the angle sum of triangles and polygons, the circle theorems, the area and volume formulae, Pythagoras, and the trigonometric ratios. Higher-tier students extend these with the sine and cosine rules, vectors, and similarity in three dimensions. The key skill is to write down what you know, mark it on the diagram, and give a reason for every step. This article works through each part in turn, with worked relationships and the language examiners expect you to use.

Angle facts and polygons

The foundation of every geometry proof is a short list of angle facts that you must quote by name. Angles on a straight line add to $180^{\\circ}$. Angles around a point add to $360^{\\circ}$. Vertically opposite angles (the pair formed when two lines cross) are equal. When a straight line crosses two parallel lines, three rules appear: corresponding angles (in matching positions, an F-shape) are equal; alternate angles (a Z-shape) are equal; and co-interior or allied angles (a C or U-shape) add to $180^{\\circ}$. Always state the reason, for example 'alternate angles are equal'.\n\nFor polygons, the interior angles of a triangle add to $180^{\\circ}$ and the angles of a quadrilateral add to $360^{\\circ}$. In general the sum of the interior angles of a polygon with $n$ sides is $(n-2)\\times 180^{\\circ}$, because the polygon can be split into $n-2$ triangles. The sum of the exterior angles of ANY convex polygon is always $360^{\\circ}$. For a regular polygon each exterior angle is $\\frac{360^{\\circ}}{n}$ and each interior angle is $180^{\\circ}-\\frac{360^{\\circ}}{n}$. A common task is to work backwards: if each exterior angle is $24^{\\circ}$ then $n=\\frac{360}{24}=15$ sides. Remember that at any vertex the interior and exterior angles lie on a straight line, so they add to $180^{\\circ}$.

Triangles and quadrilaterals

Triangles are classified by their sides and angles. An equilateral triangle has three equal sides and three $60^{\\circ}$ angles. An isosceles triangle has two equal sides and the two base angles equal, which is often the unlocking fact in a problem. A scalene triangle has all sides and angles different, and a right-angled triangle contains one $90^{\\circ}$ angle. The exterior angle of a triangle equals the sum of the two interior opposite angles, a result worth quoting directly.\n\nSpecial quadrilaterals each carry a bundle of properties. A square has four equal sides, four right angles, and diagonals that are equal and bisect each other at right angles. A rectangle has equal diagonals that bisect each other and opposite sides equal. A parallelogram has opposite sides equal and parallel, opposite angles equal, and diagonals that bisect each other. A rhombus is a parallelogram with four equal sides and diagonals that cross at right angles. A kite has two pairs of adjacent equal sides and one diagonal that bisects the other at right angles. A trapezium has exactly one pair of parallel sides. Knowing which properties belong to which shape lets you find missing angles quickly and justify each one.

Circle theorems

The circle theorems are a Higher-tier favourite and must each be quoted by name. First, the angle in a semicircle is $90^{\\circ}$: any triangle whose longest side is a diameter has a right angle opposite that diameter. Second, the angle at the centre is twice the angle at the circumference when both stand on the same arc. Third, angles in the same segment (subtended by the same chord) are equal. Fourth, opposite angles of a cyclic quadrilateral add to $180^{\\circ}$. Fifth, a tangent meets a radius at $90^{\\circ}$. Sixth, the two tangents drawn from an external point are equal in length, creating an isosceles arrangement. Seventh, the perpendicular from the centre to a chord bisects the chord.\n\nThe final and trickiest is the alternate segment theorem: the angle between a tangent and a chord equals the angle in the alternate segment. In practice you mark the tangent point, identify the chord, and match the tangent-chord angle to the inscribed angle on the far side of the chord. As with all proofs, every numerical step needs a named reason. A typical chain uses the angle at the centre to find an inscribed angle, then uses isosceles radii (two radii are always equal) to finish.

Perimeter, area, and sectors

Perimeter is the total distance around a shape and area is the space inside. Core area formulae are: rectangle $=$ base $\\times$ height; triangle $=\\frac{1}{2}\\times$ base $\\times$ perpendicular height; parallelogram $=$ base $\\times$ height; and trapezium $=\\frac{1}{2}(a+b)h$, where $a$ and $b$ are the parallel sides. For composite shapes, split the figure into standard pieces, find each area, and add or subtract.\n\nFor circles, circumference $=\\pi d=2\\pi r$ and area $=\\pi r^{2}$. A sector is a pie-slice with angle $\\theta$ at the centre. Its arc length is $\\frac{\\theta}{360}\\times 2\\pi r$ and its area is $\\frac{\\theta}{360}\\times \\pi r^{2}$ — both are simply the fraction $\\frac{\\theta}{360}$ of the full circle. The perimeter of a sector includes the arc PLUS the two straight radii, a step students often forget. Keep answers in terms of $\\pi$ when asked for an exact value, and watch units: areas are in square units and lengths in single units. If a question mixes centimetres and metres, convert first.

Surface area and volume of solids

Volume measures the space a solid occupies and is given in cubic units. For any prism, volume $=$ area of cross-section $\\times$ length. This covers cuboids (volume $=$ length $\\times$ width $\\times$ height) and cylinders (volume $=\\pi r^{2}h$, since the cross-section is a circle). For the curved shapes you are given formulae in the exam: a cone has volume $\\frac{1}{3}\\pi r^{2}h$ and a sphere has volume $\\frac{4}{3}\\pi r^{3}$. A pyramid has volume $\\frac{1}{3}\\times$ base area $\\times$ height.\n\nSurface area is the total area of all faces, found by adding the area of each surface. A cuboid has six rectangular faces. A cylinder has two circular ends of area $\\pi r^{2}$ each and a curved surface that unrolls into a rectangle of area $2\\pi rh$, giving total $2\\pi r^{2}+2\\pi rh$. A cone has curved surface area $\\pi rl$, where $l$ is the slant height, plus the base $\\pi r^{2}$; if you are given the vertical height instead, find $l$ by Pythagoras using $l^{2}=r^{2}+h^{2}$. A sphere has surface area $4\\pi r^{2}$. Always read whether the solid is solid or hollow (open top) before deciding which faces to include.

Pythagoras and trigonometry

Pythagoras' theorem links the three sides of a right-angled triangle: $a^{2}+b^{2}=c^{2}$, where $c$ is the hypotenuse, the longest side opposite the right angle. To find the hypotenuse you add the squares; to find a shorter side you subtract. In 3D, apply Pythagoras twice: first across the base to find a diagonal, then up to the top vertex. The space diagonal of a cuboid is $\\sqrt{l^{2}+w^{2}+h^{2}}$.\n\nRight-angled trigonometry uses SOHCAHTOA: $\\sin\\theta=\\frac{O}{H}$, $\\cos\\theta=\\frac{A}{H}$, and $\\tan\\theta=\\frac{O}{A}$, where O is opposite, A is adjacent, and H is hypotenuse relative to the angle $\\theta$. Label the sides first, pick the ratio that uses your two known quantities, and use the inverse function (such as $\\sin^{-1}$) to find an angle. You must learn the exact values: $\\sin 30^{\\circ}=\\frac{1}{2}$, $\\cos 60^{\\circ}=\\frac{1}{2}$, $\\tan 45^{\\circ}=1$, $\\sin 45^{\\circ}=\\cos 45^{\\circ}=\\frac{1}{\\sqrt{2}}$, $\\sin 60^{\\circ}=\\cos 30^{\\circ}=\\frac{\\sqrt{3}}{2}$, and $\\tan 30^{\\circ}=\\frac{1}{\\sqrt{3}}$, with $\\tan 60^{\\circ}=\\sqrt{3}$.\n\nFor non-right-angled triangles (Higher), use the sine rule $\\frac{a}{\\sin A}=\\frac{b}{\\sin B}=\\frac{c}{\\sin C}$ when you have a matching side-angle pair, and the cosine rule $a^{2}=b^{2}+c^{2}-2bc\\cos A$ when you have two sides and the included angle, or all three sides. The area of any triangle is $\\frac{1}{2}ab\\sin C$, where $C$ is the angle between sides $a$ and $b$.

Bearings, congruence, and similarity

A bearing is a direction measured clockwise from north, always written with three digits, so due east is $090^{\\circ}$ and south-west is $225^{\\circ}$. To solve bearings problems, draw a north line at each point, use parallel-line angle rules (north lines are parallel), and remember that the back bearing differs from the forward bearing by $180^{\\circ}$. Many bearings questions then need the sine or cosine rule to find a distance.\n\nTwo shapes are congruent if they are identical in size and shape. The four accepted conditions for congruent triangles are SSS (three sides), SAS (two sides and the included angle), ASA (two angles and a side), and RHS (right angle, hypotenuse, and one side). Naming the correct condition is essential in a proof.\n\nTwo shapes are similar if one is an enlargement of the other: corresponding angles are equal and corresponding sides are in the same ratio, the linear scale factor $k$. For areas the scale factor is $k^{2}$ and for volumes it is $k^{3}$. So if lengths double, area becomes four times larger and volume eight times larger. To find a missing length in similar shapes, set up a ratio of corresponding sides and solve.

Transformations, vectors, and constructions

There are four transformations. A translation slides a shape by a column vector $\\binom{x}{y}$, where $x$ is the horizontal and $y$ the vertical shift. A reflection flips a shape across a mirror line such as $y=x$ or $x=2$; you must state the line. A rotation turns a shape about a centre by an angle and direction, for example $90^{\\circ}$ clockwise about the origin. An enlargement scales a shape from a centre by a scale factor; a fractional factor shrinks it and a negative factor places the image on the opposite side of the centre, inverted. To describe a transformation fully, give its name and all its defining details.\n\nVectors are quantities with magnitude and direction, written as column vectors. They add and subtract component by component, and multiplying by a scalar stretches them: $2\\binom{3}{1}=\\binom{6}{2}$. If $\\overrightarrow{AB}=\\mathbf{b}-\\mathbf{a}$, then vector geometry problems are solved by travelling along known vectors. Parallel vectors are scalar multiples of each other, which is how you prove three points lie on a straight line (collinear).\n\nConstructions use only a pair of compasses and a straight edge, leaving all arcs visible. Key constructions are the perpendicular bisector of a line (equal arcs from both ends), the bisector of an angle, and the perpendicular from a point to a line. A locus is the set of all points satisfying a rule: the locus of points a fixed distance from a point is a circle, and the locus equidistant from two points is their perpendicular bisector. Combine loci with shading to answer region problems such as 'closer to A than B and within 3 cm of C'.