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LearnAQA8300First teaching 2015Papers 1–3 · Higher tierGeometry and measures

Papers 1–3 · Module

Geometry and measures

17 subtopics≈ 530 min in total

Start the module: Angles and polygons Open graphing calculator

Angles and polygons
A handful of angle facts (straight lines $180^\circ$ , points $360^\circ$ , triangles $180^\circ$ , and the three parallel-line pairs) solve almost every angle problem. For polygons, start with the exterior angles: they always add up to $360^\circ$ .
≈ 30 min
Circle theorems
Eight short theorems describe how angles behave in a circle. Learn each one with its exact wording, spot which one a diagram is using, and give it as your reason.
≈ 35 min
Similarity and congruence
Similar shapes are enlargements of each other: same angles, all lengths multiplied by the same scale factor $k$ . Areas then scale by $k^2$ and volumes by $k^3$ . Congruent shapes are identical: $k=1$ .
≈ 30 min
Bearings, scale drawings and symmetry
A bearing is an angle measured clockwise from north, written with three figures. To reverse a bearing, add or subtract $180^\circ$ . Map scales turn real distances into drawn ones, and symmetry describes how a shape fits onto itself.
≈ 25 min
Constructions and loci
A locus is the set of all points that obey a rule. Points a fixed distance from a point make a circle; equidistant from two points, the perpendicular bisector; equidistant from two lines, the angle bisector. Ruler-and-compass constructions draw these exactly, and you leave the arcs to show your method.
≈ 30 min
Perimeter, area and units
Perimeter is the distance around a shape; area is the space inside it. Five area formulas cover almost every straight-sided shape, and compound shapes are just these pieces added or subtracted. When converting area units, square the length conversion.
≈ 30 min
Circles, arcs and sectors
Circumference $2\pi r$ and area $\pi r^2$ . A sector is a fraction $\dfrac{\theta}{360}$ of the circle, so its arc and area are that fraction of the circumference and area. Its perimeter is the arc plus two radii.
≈ 30 min
Prisms and cylinders
A prism has the same cross-section all the way through, so its volume is cross-section area $\times$ length. A cylinder is a prism with a circular cross-section. Surface area is the total area of every face: unfold the solid into a net and add.
≈ 30 min
Pyramids, cones and spheres
A pyramid or cone holds exactly one third of the prism with the same base and height. A sphere has volume $\tfrac43\pi r^3$ and surface area $4\pi r^2$ . Composite solids are built from these pieces: add the volumes, and count only the outside surfaces.
≈ 35 min
Pythagoras' theorem
In a right-angled triangle, the square on the hypotenuse equals the sum of the squares on the other two sides: $a^2+b^2=c^2$ . Add the squares to find the hypotenuse; subtract to find a shorter side.
≈ 25 min
Right-angled trigonometry
Label the sides opposite, adjacent and hypotenuse relative to the angle. Then SOHCAHTOA tells you which ratio links the two sides you know (or want). Use the ratio to find a side, or the inverse function to find an angle.
≈ 35 min
Sine rule, cosine rule and area
For triangles without a right angle: use the sine rule when you know a side and its opposite angle, and the cosine rule when you know two sides and the angle between them, or all three sides. The area is $\tfrac12ab\sin C$ .
≈ 40 min
Trigonometry in three dimensions
Every 3D problem is a chain of 2D right-angled triangles. Find a triangle on the base first (often a diagonal), then stand a vertical triangle on it. The angle between a line and a plane is the angle between the line and its shadow on the plane.
≈ 35 min
Reflections, rotations and translations
Translations slide, reflections flip and rotations turn; all three keep a shape exactly the same size. To "describe fully" you must give every detail: the vector, the mirror line, or the centre, angle and direction.
≈ 30 min
Enlargements and combined transformations
An enlargement multiplies every distance from a fixed centre by the scale factor $k$ . If $0<k<1$ the shape shrinks; if $k$ is negative the image is on the opposite side of the centre and upside down. Combine transformations by doing them in order.
≈ 30 min
Column vectors and magnitude
A vector has a size and a direction. Write it as a column $\dbinom xy$ : $x$ across, $y$ up. Add, subtract and scale vectors component by component, and find the length (magnitude) with Pythagoras.
≈ 25 min
Vector geometry
To find a vector between two points, follow any route along known vectors, adding as you go and subtracting when you travel backwards. Two vectors are parallel if one is a multiple of the other; points are collinear if they are joined by parallel vectors sharing a point.
≈ 35 min