1.4 Circular measure

Cambridge International AS & A Level Mathematics 9709 (2026-2027) · Paper 1: Pure Mathematics 1 · official syllabus

CIRCULAR MEASURE LAB

Learn it by playing it

Four short missions on one live sector. The radius stays 2; the angle is yours. When a mission glows green, read why it worked, then take the next one.

1234
Mission 1 of 4

One radian, by hand

Swing the radius anticlockwise until the arc it cuts matches the radius itself. The half-degree slider can only land near the exact angle, so close counts here. Watch the arc length readout, not the angle.

Before you slide: is one radian MORE or LESS than 60 degrees?

arc = 1.4 vs radius 2 (angle 40 deg; exact 1 rad = 180/pi ~ 57.296 deg - the 0.5-deg slider can only land near it)

arc s = 1.40 · area A = 1.40 · rim P = 5.40

Study notes

Played the lab above? These notes are the reference version of what you just felt - use them to check exact formulas and the exam wording.

Radians and degrees 1.4

  • One radian is the angle at the centre of a circle made by an arc whose length equals the radius.
  • pi radians = 180 degrees. Convert degrees to radians by multiplying by pi/180, and radians to degrees by multiplying by 180/pi.
  • Learn the common ones: 30 deg = pi/6, 45 deg = pi/4, 60 deg = pi/3, 90 deg = pi/2, 120 deg = 2pi/3, 270 deg = 3pi/2.
  • At A Level, angles in arc, sector and (later) calculus formulas are in RADIANS - check your calculator mode.

Arc length and sector area 1.4

  • With theta in radians: arc length s = r*theta and sector area A = (1/2) r² theta. Both are just the fraction theta/(2pi) of the full circle.
  • The perimeter of a sector is the arc PLUS the two radii: r*theta + 2r. Losing the two radii is the classic slip.
  • Rearrange freely: theta = s / r and r = s / theta; given a sector area, theta = 2A / r².
  • If the angle is given in degrees, convert to radians FIRST - s = r*theta fails silently in degrees.

Triangles and segments 1.4

  • The triangle formed by two radii and a chord has area (1/2) r² sin(theta).
  • Segment area = sector area - triangle area = (1/2) r² (theta - sin(theta)).
  • Split harder problems into a circular part and a straight-edged triangle part, then add or subtract.

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