The arc equals the radius: One radian is the angle at the centre of a circle for which the arc length equals the radius. Since the whole circumference is 2πr, a full turn is 2π radians.
One radian is the angle whose arc (the curved edge) is exactly as long as the radius — about 57.3°. So 2π ≈ 6.28 of them fit around a full circle.
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Radians are 'pure' numbers: A radian has no degree symbol — an angle written as a multiple of π (or a plain number) is in radians.
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Multiply by π/180 or 180/π: Degrees → radians: multiply by π/180. Radians → degrees: multiply by 180/π. (Both come from 180° = π.)
IB-style question — convert both ways
Convert (a) 60° to radians and (b) 3π/4 radians to degrees.
Step by step
- (a) × π/180.
- (b) × 180/π.
Final answer
(a) π/3; (b) 135°.
60° and π/3 are two names for the SAME angle — 60 × π/180 = π/3. Converting never changes the angle, just how it's written.
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π cancels: When converting radians to degrees, the π in the angle cancels the π in 180/π — leaving a clean number.
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Learn the key fractions of π: The angles you'll use constantly: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, 360° = 2π.
Small
- π/6, π/4
Medium
- π/3, π/2
Large
- π, 2π
The common exact angles around the circle, shown in both degrees and radians (π/6, π/4, π/3, π/2, …). Picture where each one lands.
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Build the rest by adding: Other angles combine these: 120° = 2π/3, 270° = 3π/2, etc. Knowing the basic six lets you place any common angle.
Set the mode to match: Trig of an angle in radians must be computed with the GDC in radian mode. A full circle is 2π ≈ 6.28, so radian angles are small numbers — don't confuse them with degrees.
Most calculus uses radians: When angles appear with calculus (derivatives/integrals of sin and cos), they're in radians — keep the GDC in radian mode there.
A classic lost mark: sin(30) in radian mode is NOT sin(30°). Check whether the question is in degrees or radians and set the mode accordingly.
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