Geometry and trigonometry

Exam tip: Area uses r²; circumference uses the diameter. Write the formula before substituting so a slip in the calculator still earns the method mark.

Common error: A cuboid has six faces in three matching pairs. Adding only three face areas gives half the correct surface area.

Exam tip: Keep the unrounded volume in your calculator for part (b). Using 354 gives 371.7, which still rounds to 372, but early rounding can change the final figure in other questions.

Exam tip: Label the sides from the angle you want: BC is opposite angle A and AB is adjacent to it. Writing O, A and H on the sides first helps you choose the right ratio.

Exam tip: Carry the unrounded value of QR from your calculator into part (b). Using a value rounded too early can change the final digit.

Common error: Subtracting a negative coordinate is easy to get wrong: from −3 to 5 is a distance of 8, not 2. Sketch the points to check the horizontal and vertical distances.

Exam tip: Write reasons using the correct geometric words, such as 'alternate angles are equal' or 'co-interior angles add up to 180°'. Informal names like 'Z angles' or 'C angles' do not earn the reason mark.

Common error: The angle is measured from the eyepiece, not from the ground. Add the 1.5 m height of the instrument after using tan to find the height above the eyepiece.

Common error: The ladder is the hypotenuse, so you subtract the squares. Adding them gives 6.74 m, which is longer than the ladder and cannot be right.

Common error: The angles of a quadrilateral add up to 360°, not 180°. Using 180° gives \(x = 15\), and then the angle \((3x − 20)\)° would be 25°. This looks reasonable but is wrong, so always check that your four angles add up to 360°.

Common error: Using a radius of 9 m instead of 4.5 m, or forgetting to halve the circle, is the most frequent slip with semicircles.

Common error: Do not divide 360 by the interior angle. The exterior angles of any polygon add up to 360°, so 360 must be divided by the exterior angle.

Common error: The two equal angles are opposite the two equal sides. Because PQ = PR, the equal angles are at Q and R, not at P.

Common error: Multiplying 7.5 × 12.3 × 6, or forgetting the ½, gives the wrong area. Always add the parallel sides first. Then divide grams by 1000 to get kilograms, not by 100.

Exam tip: Angles on a straight line add up to 180°. Write the equation first, then collect like terms. That way the method mark is earned even if the algebra slips.

Exam tip: A triangular prism has five faces: two triangles and three rectangles. Use Pythagoras to find the width of the sloping rectangle first.

Exam tip: The height of an isosceles triangle cuts the base in half and makes two right-angled triangles. Use 5.5 cm, not 11 cm, in Pythagoras' theorem.

Exam tip: Bearings are measured clockwise from north and written with three figures. The back bearing differs from the original bearing by 180°.

Exam tip: The pentagon and the square have the same side length, so AE = AG. Spotting this hidden isosceles triangle is the key step, and stating it earns the reason mark.

Common error: 1 m² = 10 000 cm², not 100 cm². Square the length conversion factor for areas and cube it for volumes. Check the time unit too: a rate per second gives an answer in seconds, so divide by 60 to get minutes.

Common error: AB is the opposite side in both right-angled triangles and BC, BD are adjacent, so divide 8 by tan. Multiplying by tan gives lengths that are far too short. Keep full calculator values until the final subtraction.

Exam tip: Whenever a radius meets a tangent, mark the right angle straight away. This turns the problem into right-angled trigonometry.

Exam tip: When polygons fit exactly around a point, their interior angles add up to 360°. A regular polygon exists only if 360 divided by its exterior angle is a whole number.

Exam tip: Draw a north line at every point and measure bearings clockwise from north. A back bearing differs from the original bearing by exactly 180°.

Exam tip: Match corresponding vertices first. Going from the small triangle to the large one, multiply by the scale factor. Going back, divide by it.

Exam tip: A tangent meeting a radius gives a right angle. Once you see it, Pythagoras and SOHCAHTOA can both be used.

Common error: Multiplying the area by the length scale factor gives 144 cm², which is wrong. Areas scale by the square of the length scale factor.

Exam tip: Two radii always form an isosceles triangle. Look for this whenever the centre is joined to two points on the circle, and state 'radii are equal' as your reason.

Common error: Do not set the two opposite angles equal to each other. In a cyclic quadrilateral, opposite angles add up to 180°. Check your answer: here 100° + 80° = 180°.

Common error: Using 3.24 as the length scale factor is wrong. Take the square root of the area ratio to get the length ratio, then cube that for volume.

Exam tip: For the alternate segment theorem, the angle between the tangent and chord AB equals the angle subtended by AB on the opposite side of the chord.

Common error: The perimeter of a sector includes the two straight radii as well as the arc. Giving only the arc length loses the last two marks.

Common error: Using the perpendicular height 8 cm in πrl instead of the slant height gives 48π for the curved surface, which is wrong.

Exam tip: ½ab sin C needs the angle between the two given sides. Check that 52° is the included angle before substituting.

Exam tip: The angle between a line and a plane is measured to the line's projection on that plane. Here the projection is the base diagonal, not an edge of the base.

Exam tip: Find the angle between the two directions first by subtracting the bearings: 125° − 40° = 85°. The cosine rule uses the angle between the two known sides.

Exam tip: Write down k, k² and k³ at the start. Then use the right one for length, area or volume.

Common error: Pairing the wrong sides gives the wrong answer. Match sides that lie opposite equal angles. AX is opposite angle ABX (50°), and DX is opposite the equal angle DCX (50°).

Common error: For the area formula ½ab sin C, the angle must be the one between the two sides you use. Here BC and AC meet at C, so you need angle ACB. Keep the unrounded value of AC in your calculator: using 11.8 gives 50.8 rather than 50.7.

Common error: Do not double the angle at the centre. The angle at the circumference is half the angle at the centre, not twice it. Always give the full circle theorem as your reason, not just the numbers.

Exam tip: In the alternate segment theorem, the matching angle is in the segment on the other side of the chord. Trace the chord, then look across it.

Common error: Only opposite angles of a cyclic quadrilateral add up to 180°. Adding neighbouring angles, such as angle A and angle B, gives the wrong equation.

Exam tip: Leaving the volume as 281.25π lets the π cancel when you divide by 25π, giving an exact answer with no rounding error.

Common error: Using DB ÷ AD = 1.5 as the scale factor is wrong. The similar triangles are ADE and ABC, so compare AB with AD. For areas, square the length scale factor.

Common error: Multiplying the surface area by the volume ratio 3.375 is wrong. Find the length scale factor first, then square it for areas.

Exam tip: In a 'show that' part, write every step including the equation with π on both sides; just stating 10.7 earns no marks.

Common error: For the angle between a face and the base, use the line from M to the midpoint of AB (length 5 cm), not the line from M to a corner. The two angles in (b) and (c) are different.

Common error: \((x + 7)^2\) is not \(x^2 + 49\); you must include the middle term \(14x\). Also reject \(x = -3\), because a length cannot be negative.

Common error: Both statues are made of the same metal, so the ratio of their masses equals the ratio of their volumes. Take the cube root to get the length scale factor, not the square root. Then square the length scale factor to get the area scale factor.

Exam tip: OT bisects angle ATB because the two tangents from T are equal in length and the figure is symmetrical. Triangle OAT is right-angled at A, so you can use either the 24° angle at T or the 66° angle at O.

Exam tip: The largest angle is always opposite the longest side. Using the area from part (b) with the 12 m side as the base avoids a second long trigonometry calculation.

Exam tip: A frustum is a large cone minus a similar small cone. Find the missing radius with similar triangles before using the volume formula. Alternatively, cube the length scale factor to find the small cone's volume.

Exam tip: Angles ADB and ACB both stand on the chord AB, on the same side of it. Spotting this pair is the key step. When a question asks for reasons, name each circle theorem in full.