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Circles

Circumference, area, arc length and sector area, with π kept exact or worked out — the circle formulas, fast and without mixing them up.

  • Geometry and trigonometry
  • Timed
  • Endless levels

How it plays

A circle given by its radius or diameter, or an arc or sector given by its angle, and one measure to give. It starts with the circumference; area, arcs, sectors, angles in radians and working back to the radius unlock as you climb, and the radius grows without end.

Choose how π is shown: kept exact (12π, 3π/2), or multiplied out to one decimal place using π = 3.14159…. The answer is computed exactly either way; only its writing changes.

The wrong options are the formulas mixed up: 2πr for πr², the diameter squared instead of the radius, the 2 dropped from an arc, 180 used where it is 360. The working substitutes into the formula.

Keys

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

What it trains

This makes the circle formulas quicker and surer to apply. It trains that, not mathematical ability in general.

Learn

How to get better at circles: 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. Read whether you are given the radius or the diameter. If it is the diameter, halve it before anything else.
  2. Name the formula before you compute: C = 2πr, A = πr², arc = θ/360 × 2πr, sector = θ/360 × πr². In radians: arc = rθ, sector = ½r²θ.
  3. Work out only the number in front of π, and keep π itself to the end.
  4. With π “Worked out to 1 decimal place”, multiply that number by 3.1416 last, and round to one decimal place.

Techniques and tricks

  • Angles as fractions of a turn

    Every angle here is a simple fraction of 360°: 30° is 1/12, 45° is 1/8, 60° is 1/6, 90° is 1/4, 120° is 1/3, 135° is 3/8, 150° is 5/12, 270° is 3/4. Take that fraction of the whole circumference or area.

    e.g. Arc of 120°, radius 9: a third of 18π is 6π.

  • Arc length in one step

    The arc in degrees is θ × r ÷ 180, times π. Do the division first when it comes out whole.

    e.g. Arc of 45°, radius 8: 45 × 8 ÷ 180 = 2, so 2π.

  • Sector from the arc

    A sector’s area is half the radius times its arc. If you have the arc already, one multiplication gives the sector.

    e.g. Radius 9, 120°: arc 6π, so the sector is ½ × 9 × 6π = 27π.

  • Radians need no 360

    In radians the angle already is the turn’s fraction times 2π, so arc = rθ and sector = ½r²θ, nothing more.

    e.g. Radius 6, angle 5π/6: arc = 6 × 5π/6 = 5π; sector = ½ × 36 × 5π/6 = 15π.

  • Work back to the radius

    From a circumference, the radius is the number in front of π halved. From an area, it is the square root of that number.

    e.g. C = 14π gives r = 7; A = 49π gives r = 7.

  • π to one decimal place

    Multiply by 3, then add 0.1416 times the number. For small numbers, 0.14 times is close enough.

    e.g. 12π ≈ 36 + 1.7 = 37.7; 81π ≈ 243 + 11.47 = 254.5

As the levels climb

  • Only circumference at level 0. Area arrives at level 1, arc length at level 4, sector area at level 7, arcs in radians at level 10 and working back to the radius at level 12. The newest kind comes up more often for a while.
  • The largest radius grows by 4 every few levels without end: 6, then 10 at level 2, 14 at level 5, 18 at level 8. Bigger radii mean bigger squares for area, so know your squares to 30.
  • The clock tightens to 80% at level 3, 65% at level 6 and 55% at level 9, though it also grows a second with each step up in radius. Under the tighter clock, lean on the fraction-of-a-turn trick rather than the full formula.

Traps to avoid

  • 2πr and πr² swapped: the circumference of radius 5 offered as 25π, or its area as 10π. Length goes with r, area with r².
  • The diameter squared: a circle of diameter 10 has area 25π, not 100π. Squaring the diameter gives four times too much.
  • Arc length with the 2 dropped: an arc of 90° on radius 4 is 2π; using θ/360 × πr gives π. The sector answer offered for the arc is the same kind of mix-up.
  • Working back from C = 14π and answering 14. That is the diameter; the radius is 7.
  • The ½ dropped from a sector in radians: radius 6 and angle π/3 make a sector of 6π, not 12π.

How to practise

  • In “Questions”, tick one or two kinds to drill the ones you mix up — Arc length and Sector area together are the classic pair.
  • Set “π” to “Worked out to 1 decimal place” once the exact answers are quick; the question then gets extra time for the multiplying.
  • Set “Answer by” to “Typing the number”. With π kept exact you type the number in front of π, so 3π/2 is 1.5.
  • Raise “Options shown” towards 8: more of the classic mistakes appear beside the truth, and you must rule each one 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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