Five hands-on experiments built around the circle's most famous number. Drag, slide, and watch the formulas from the chapter play out in real time.
The circumference-to-diameter ratio
C/D ratio
Drag the gold handle to resize the circle. Below it, the diameter (gold bar) and circumference (cyan bar) are drawn to the same scale — however big or small the circle gets, the cyan bar stays the same multiple of the gold one: that multiple is π.
drag the gold handle to resize the circle
Readings
Radius r—
Diameter D = 2r—
Circumference C = 2πr—
Ratio C ÷ D—
However far you drag the handle, C ÷ D never moves from 3.14159… — that fixed ratio is what we call π.
Arc length and sector area
§6.4 & §6.10.1
Drag point B around the circle. The shaded sector's arc length and area both scale directly with the angle θ it subtends — each is simply that fraction of the whole circle's circumference or area.
drag B around the circle
Readings
Radius r—
Angle θ—
Arc length = 2πr × θ/360°—
Sector area = πr² × θ/360°—
Try θ = 90° or 180° — the arc length and area should land exactly on the quarter- and semicircle values from the chapter.
Trapping π between polygons
Archimedes, 250 BCE
The blue polygon sits inside the circle; the red one sits snugly around it. Both are regular polygons with the same number of sides. Slide to add more sides — watch both perimeters converge on the true circumference, squeezing π tighter from both directions.
slide to change the number of sides
Readings
Inscribed perimeter ÷ diameter—
Circumscribed perimeter ÷ diameter—
True π3.14159…
At 6 sides this matches the book exactly: 3 < π < 2√3 ≈ 3.46.
Why relay lanes need a stagger
400 m track
Each lane is a wider stadium shape than the one inside it. Slide the lane width and the innermost radius to match any track — the stagger between neighbouring lanes turns out to depend only on the lane width, never on the radius.
sliders change every lane at once
Readings
Stagger between any two adjacent lanes—
One lap in lane 1—
Stagger = 2π × (lane width) — it never depends on the radius, so a smaller school track needs the same stagger as an Olympic one.
Two roads to the same area
Heron, c. 60 CE
Drag any vertex of the triangle. One area comes from the familiar ½ × base × height. The other comes from Heron's strange-looking formula using only the three side lengths. Watch them agree, however oddly you reshape the triangle.
drag any vertex A, B or C
Readings
Sides a, b, c—
Semi-perimeter s—
½ × base × height—
√(s(s−a)(s−b)(s−c))—
The two areas should match to within rounding, for any triangle you drag.