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Triangles and polygons

Missing sides by Pythagoras, the 30-60-90 and 45-45-90 triangles, polygon angles and areas — geometry worked in the head, at growing sizes.

  • Geometry and trigonometry
  • Timed
  • Endless levels

How it plays

A figure described in a line — two sides of a right triangle, one side of a special triangle, a polygon of n sides, a shape to find the area of — and one thing to give. It starts with the missing side of a right triangle; special triangles, polygon angles, areas and "is this a right angle?" unlock as you climb, and the numbers grow without end.

The wrong options are the real mistakes: the legs added instead of squared, √2 used where it is √3, the long leg taken for the short, n × 180° for the angle sum, the ½ forgotten in a triangle, the slanted side used as the height.

Every triangle is built first — triples from Euclid's formula, special triangles from half an equilateral and half a square — so the working shows the squares, the ratio or the turn that gives the answer.

Keys

Space starts and skips ahead · 1–9 pick · Enter submits · Esc stops. On a phone, tap.

What it trains

This makes these geometry facts — Pythagoras, the special triangles, polygon angles, areas — quicker to apply. It claims nothing wider.

Learn

How to get better at triangles and polygons: the method, the techniques, what each new level asks of you, and the traps the wrong answers are made of.

Your approach to each question

  1. Name the figure first: a right triangle with two sides, a special triangle, a polygon, an area, or a “right or not?”.
  2. For a right triangle, find the hypotenuse — the longest side, opposite the right angle — before anything else.
  3. Look for a common factor and a known triple before you square anything. Most sides here are a triple scaled up.
  4. Sanity-check the size: a hypotenuse is longer than either leg but shorter than the two added; an interior angle of a regular polygon is under 180°.

Techniques and tricks

  • Know the triples

    Learn the small ones by heart: 3-4-5, 5-12-13, 8-15-17, 7-24-25, 20-21-29, 9-40-41, 12-35-37, 11-60-61. Spot one and the third side is free.

  • Divide out the scale first

    If both sides share a factor, divide it out, find the triple, then multiply back.

    e.g. Legs 15 and 36: divide by 3 to get 5 and 12, so 13, and the hypotenuse is 39.

  • Build any triple from two numbers

    Euclid’s formula: pick m larger than n, and the sides are m² − n², 2mn and m² + n². It is how this game makes every triple.

    e.g. m = 5, n = 2: 21, 20, 29.

  • The two special triangles

    30-60-90 is half an equilateral triangle: short leg, long leg, hypotenuse are 1 : √3 : 2. 45-45-90 is half a square: 1 : 1 : √2. Scale to the side you are given.

    e.g. Hypotenuse 10 in a 30-60-90: short leg 5, long leg 5√3. Long leg 12: short leg 12 ÷ √3 = 4√3, hypotenuse 8√3.

  • Polygons by their turns

    Walking round any polygon you turn 360°, so each exterior angle of a regular one is 360° ÷ n, and each interior angle 180° minus that. The interior angles sum to (n − 2) × 180°.

    e.g. 12 sides: exterior 30°, interior 150°, sum 1,800°. Interior 156°: exterior 24°, so 360 ÷ 24 = 15 sides.

  • One rule per area

    Triangle: ½ × base × height. Right triangle: ½ × the two legs. Trapezium: the average of the parallel sides × the gap. Parallelogram: base × the upright height, never the slanted side.

  • Right or not, by the last digit

    Compare a² + b² with the longest side squared. A quick first test: squares end only in 0, 1, 4, 5, 6 or 9, and the last digits must add up.

    e.g. 20, 21, 29: 400 + 441 = 841 = 29², so yes.

As the levels climb

  • It starts with the missing side of a right triangle. Special right triangles arrive at level 1, polygon angles at level 4, areas at level 7, and “right or not?” at level 10. The newest kind comes up more often for a while.
  • The numbers grow without end: larger triples and bigger scales, polygons with more sides, and from level 2 special triangles whose given side carries a √2 or √3 when you choose from options.
  • The clock tightens to 80% at level 3, 65% at level 6 and 55% at level 9, though it grows by 1.5 s with each step up in size. By then, spot triples and ratios rather than squaring.

Traps to avoid

  • The legs added instead of squared: legs 5 and 12 answered 17. Legs add as squares, never as lengths.
  • The other leg as a plain difference: hypotenuse 13 and leg 5 answered 8. Subtract the squares — 169 − 25 = 144 — and take the root, 12.
  • √2 and √3 swapped, or the long leg taken for the short. In a 30-60-90 the shortest side is opposite 30°, and √3 belongs to the long leg.
  • n × 180° for the angle sum. A hexagon’s angles add to 720°, not 1,080°: from one corner it cuts into four triangles, not six.
  • The ½ forgotten in a triangle’s area, the hypotenuse used as a height, or the slanted side of a parallelogram used in place of its height.

How to practise

  • In “Questions”, leave one or two kinds on to drill them — “Special right triangles” and “Angles of polygons” reward a few minutes on their own.
  • Set “Answer by” to “Typing the number” to produce the answer rather than recognise it; typed special triangles always come out whole.
  • Raise “Options shown” towards 8 so more of the real mistakes sit beside the truth and each must be ruled out.

Levels

Each question has a clock that comes from the work it asks for. There is no last level: at least one dial turns without end. Your level is judged on your last ten answers, every ten: 90% moves you up, below 80% moves you down, which holds most people near four in five right.

All of it is in the settings beside Start — how many answers a judgement looks at and how often it recalibrates, the thresholds, the time, which questions, how many options, and whether you see each verdict at once or review them together at the judgement. You can also jump to any level.

More geometry and trigonometry

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