Geometry questions fail differently from arithmetic questions. An arithmetic slip usually produces an answer that is not among the options, so you notice. A geometry question answered with the circumference formula instead of the area formula, or with metres instead of square metres, produces a perfectly plausible number that happens to be wrong — and there is no internal signal to catch it.
The defence is a small set of formulas you know exactly, plus a unit check on every answer. Those two habits cover nearly every geometry question on the paper.
Organise geometry into three questions, asked in this order:
The unit question is the cheapest error check available. If a question about painting a wall produces an answer in metres, you used a perimeter formula, and you can see that in one second.
Angles. Angles on a straight line add to 180 degrees; around a point they add to 360 degrees; the three angles of any triangle add to 180 degrees. When a question gives angles as multiples of x, write the sum as an equation: if the angles are 3x, 2x and 4x, then 9x = 180, so x = 20 and the angles are 60, 40 and 80 degrees.
Exterior angles. Walk around any convex polygon turning at each corner; you always make one full turn, so the exterior angles — one per vertex — sum to 360 degrees no matter how many sides the polygon has. A triangle is the special case where each exterior equals the sum of the two opposite interior angles. For a regular n-gon each exterior is 360/n: a regular hexagon turns 60 degrees at every vertex.
Clock angles. Both hands move continuously, so the hour hand is never exactly on a number once minutes have passed. Put both hands in degrees from 12 o'clock (0 degrees, clockwise): minute hand = 6 × minutes; hour hand = 30 × hours + 0.5 × minutes. The angle between them is the absolute difference; if that exceeds 180, take 360 minus it (the smaller angle). At 2:25 the minute hand is at 150 and the hour hand at 60 + 12.5 = 72.5, so the gap is 77.5 degrees — the hour hand has already crawled past the 2.
Triangles. Area is one half times base times height: for a base of 12 and a height of 5, the area is 30 square units. Note that the height must be perpendicular to the base, not one of the slanted sides.
Pythagoras. In a right triangle, the square on the hypotenuse equals the sum of the squares on the other two sides. With legs 9 and 12: 81 + 144 = 225, so the hypotenuse is 15. The standard triples — 3, 4, 5 and its multiples 6, 8, 10 and 9, 12, 15 — appear constantly; recognising them saves the calculation.
Rectangles and squares. For a rectangle 14 by 9, the area is 126 square units and the perimeter is 2 x (14 + 9) = 46 units. For a square, area is the side squared and perimeter is four times the side.
Circles. Circumference is two pi r and area is pi r squared. With the common instruction to take pi as 22/7 and a radius of 7: circumference is 2 x 22/7 x 7 = 44, and area is 22/7 x 49 = 154.
Trapezium. Area is the average of the parallel sides times the height. With parallel sides 8 and 12 and a height of 5, the average is 10 and the area is 50.
Volume. A cube of side 4 has volume 64 and surface area 6 x 16 = 96. A cylinder of radius 7 and height 10 has volume 22/7 x 49 x 10 = 1540 cubic units.
A rectangular field is 40 metres by 30 metres. A path of uniform width 2 metres runs around the inside edge. A farmer wants to fence the entire outer boundary. What length of fencing is needed, and what area does the path occupy?
Fencing is a perimeter, so one dimension: 2 x (40 + 30) = 140 metres.
The path is an area, so two dimensions. The inner rectangle, excluding the path, is 40 - 2 - 2 = 36 metres by 30 - 2 - 2 = 26 metres. Its area is 36 x 26 = 936 square metres, and the whole field is 40 x 30 = 1200 square metres. The path occupies 1200 - 936 = 264 square metres.
Two checks worth making. First, the units differ correctly between the two answers — metres for fencing, square metres for the path — which confirms both formulas were the right kind. Second, an estimate: the path is roughly a band of area (140 x 2) minus the four corner squares that get counted twice, which is 280 - 16 = 264. The estimate matches exactly, because it is the same computation arranged the other way.
Clock and similarity, worked. At 25 minutes past 2: minute hand 6 × 25 = 150 degrees from 12; hour hand 30 × 2 + 0.5 × 25 = 72.5 degrees; difference 77.5 degrees. Two similar triangles with areas 9 and 16: length ratio is the square root of the area ratio, so altitudes are in 3 : 4. Both are the same habit — name which quantity scales linearly and which scales by the square before touching the numbers.
A right triangle has legs of 9 and 12. How long is the hypotenuse?
Taking pi as 22/7, what is the circumference of a circle of radius 7?
A triangle's angles are 3x, 2x and 4x. What is the largest angle?
The clock shows 25 minutes past 2. What is the angle between the hands?
Two similar triangles have areas in the ratio 9 : 16. What is the ratio of their corresponding altitudes?
What is the sum of the exterior angles of any convex polygon (one at each vertex)?
Without looking back: how do you tell an area question from a perimeter question, and what does a right angle let you use?
The answer is: ask whether the answer should be in single units (a length, perimeter or circumference) or square units (an area). A right angle allows Pythagoras — or recognition of a 3, 4, 5 style triple.
Fill in the blanks from memory:
A triangle with base 12 and perpendicular height 5 has area square units. A rectangle 14 by 9 has perimeter . A cuboid measuring 2 by 3 by 5 has volume cubic units. For similar shapes, areas scale by the of the length ratio. At 2:25 the angle between the clock hands is degrees. The exterior angles of any convex polygon sum to degrees.
A cylindrical water tank has a diameter of 14 metres and a height of 10 metres. You need to paint the curved side only. What area must be painted, using pi as 22/7?
The curved side of a cylinder unrolls into a rectangle whose width is the circumference and whose height is the cylinder's height. The diameter is 14, so the radius is 7, and the circumference is 2 x 22/7 x 7 = 44 metres. The height is 10 metres, so the area is 44 x 10 = 440 square metres.
Two traps are deliberately built into this question. The first is the diameter: using 14 as the radius would give 880 square metres, double the right answer, and 880 is a plausible-looking option. The second is "curved side only" — including the two circular ends would add 2 x 154 = 308 square metres, which is the answer to a question that was not asked. Both errors are avoided by writing down what the question wants before choosing a formula.
Geometry measures isolated shapes. The next lesson turns that into formula discipline: perimeter, area, surface area and volume, plus the unit conversions that decide whether an otherwise correct answer is accepted.