- Angles on a straight line add to 180 degrees; angles around a point add to 360.
- Vertically opposite angles are equal.
- Angles in a triangle add to 180 degrees; angles in a quadrilateral add to 360.
- The base angles of an isosceles triangle are equal.
Watch out: Using 360° instead of 180° for angles on a straight line, or the other way round for angles around a point.
- Corresponding angles, in an F shape, are equal.
- Alternate angles, in a Z shape, are equal.
- Co-interior angles, in a C or U shape, add to 180 degrees.
- Always name the angle fact you use as your reason.
Watch out: Confusing alternate angles (Z shape, equal) with co-interior angles (C shape, sum to 180°).
- The exterior angles of any polygon add to 360 degrees.
- One exterior angle of a regular polygon is 360 divided by the number of sides.
- The interior angles add to (n minus 2) times 180 degrees.
- At each vertex, the interior and exterior angle add to 180 degrees.
Watch out: Using n instead of (n - 2) in the interior angle sum formula, forgetting to subtract 2 for the two triangles that cannot be formed at the starting vertex.
- A square has four equal sides and four right angles; a rhombus has four equal sides but no right angles.
- A parallelogram has two pairs of parallel sides and opposite angles equal.
- A trapezium has exactly one pair of parallel sides.
- Lines of symmetry and order of rotational symmetry are common exam questions.
Watch out: Confusing a rhombus with a square, forgetting that a rhombus does not need right angles even though both shapes have four equal sides.
- Perimeter is the distance all the way round; area is the space inside, in square units.
- Rectangle area is length times width; triangle area is one half times base times perpendicular height.
- Parallelogram area is base times perpendicular height; trapezium is one half of (a plus b) times height.
- Keep units consistent and give area in square units.
Watch out: Using a sloping side instead of the perpendicular height when finding the area of a triangle or parallelogram.
- Split the shape into rectangles, triangles and other basic shapes.
- Find any missing lengths using the fact that opposite sides must match up.
- Add the parts, or subtract a cut-out from a larger shape.
- Work in one set of units throughout.
Watch out: Splitting an L-shape into rectangles that overlap or leave a gap, rather than two rectangles that together make up the whole shape exactly once.
- Circumference is pi times diameter, which is the same as 2 times pi times radius.
- Area is pi times radius squared; always square the radius, not the diameter.
- The diameter is twice the radius.
- For an arc or sector, take the fraction angle over 360 of the whole circle.
Watch out: Using the diameter instead of the radius in the area formula without halving it first, since area = π × radius², not π × diameter².
- A prism has the same cross-section all the way through.
- Volume of a prism is the area of the cross-section times the length.
- Surface area is the total of all the faces; sketch the net so you count them all.
- Volume is in cubic units and surface area in square units.
Watch out: Forgetting to double the sum of the three face areas when finding the surface area of a cuboid, since each face has an identical opposite face.
- Volume of a cylinder is pi times radius squared times height.
- The curved surface area is 2 times pi times radius times height, because the label unrolls into a rectangle.
- Total surface area adds the two circular ends, each pi times radius squared.
- Check whether the question gives the radius or the diameter.
Watch out: Confusing the volume formula (π × radius² × height) with the curved surface area formula (2 × π × radius × height), especially mixing up which one squares the radius.
- Sphere volume is four thirds pi r cubed; sphere surface area is 4 pi r squared.
- Cone volume is one third pi r squared h; curved surface area is pi r l, where l is the slant height.
- Pyramid volume is one third times base area times perpendicular height.
- These formulae are given in the exam, but you must know which letter is which.
Watch out: Forgetting the (1/3) in the pyramid and cone volume formulae, effectively calculating the volume of a prism or cylinder instead.
- Similar shapes are the same shape with every length in the same ratio, the linear scale factor.
- The area scale factor is the linear scale factor squared.
- The volume scale factor is the linear scale factor cubed.
- Matching angles in similar shapes are equal.
Watch out: Using an area or volume ratio directly as a linear scale factor, instead of taking a square root (for area) or cube root (for volume) first.
- Congruent shapes are identical: the same size and the same shape.
- Triangles are congruent if you can show SSS, SAS, ASA or RHS.
- AAA only proves similarity, not congruence.
- In a proof, state the condition and give a reason for each matching pair.
Watch out: Treating SSA (two sides and a non-included angle) as a valid congruence condition, when it does not guarantee the triangles are identical.
- a squared plus b squared equals c squared, where c is the hypotenuse, opposite the right angle.
- Add the squares to find the hypotenuse; subtract to find a shorter side.
- It works only in right-angled triangles.
- It also gives the distance between two points from their coordinates.
Watch out: Adding the squares of two sides when one of them is actually the hypotenuse, instead of subtracting to find a shorter side.
- The angle at the centre is twice the angle at the circumference on the same arc.
- The angle in a semicircle is 90 degrees.
- Angles in the same segment are equal.
- Opposite angles of a cyclic quadrilateral add to 180 degrees, and a tangent meets a radius at 90 degrees.
Watch out: Not stating the name of the circle theorem used as a reason, which usually costs a mark even when the numerical answer is correct.
- A translation uses a column vector: the top number is right or left, the bottom is up or down.
- A rotation needs a centre, an angle and a direction.
- A reflection needs the equation of the mirror line.
- An enlargement needs a centre and a scale factor; a negative scale factor puts the image on the other side.
Watch out: Mixing up the rules for 90° clockwise and 90° anticlockwise rotation about the origin, since one negates the x-value and the other negates the y-value.
- Constructions must be done with compasses and a straight edge, leaving your arcs showing.
- The perpendicular bisector is the locus of points the same distance from two points.
- The angle bisector is the locus of points the same distance from two lines.
- The locus of points a fixed distance from a point is a circle.
Watch out: Erasing or not showing the compass arcs used in a construction, which loses method marks even if the final line is drawn in the correct place.
- A bearing is measured clockwise from north and is always written with three figures, such as 072 degrees.
- Draw a north line at the point you are measuring from.
- A back bearing differs from the forward bearing by 180 degrees.
- Use alternate angles between the parallel north lines to work bearings out.
Watch out: Forgetting to write a bearing using three figures, for example writing 65° instead of 065°.
- A vector has magnitude and direction, written as a column vector or in bold.
- Adding vectors means following one and then the other, nose to tail.
- A scalar multiple like 3a is three times as long in the same direction; minus a is the same length in the opposite direction.
- Parallel vectors are scalar multiples of each other, which you can use to prove points lie on a straight line.
Watch out: Subtracting the coordinates the wrong way round when finding a column vector, rather than always taking the end point minus the start point.
