Cambridge O Level Additional Mathematics · Syllabus 4037 · Circular Measure
Radian
What is Radian?
The unit of angle defined by the ratio of the arc a central angle cuts off to the radius of the circle, so that an angle of one radian is the angle for which the arc length is exactly equal to the radius, approximately 57.3 degrees. Because it is one length divided by another, an angle in radians is a pure number carrying no physical unit, which is why the arc-length and sector-area formulas contain no conversion constant and are valid only when the angle is measured this way.
This definition is part of the Circular Measure chapter in Cambridge O Level Additional Mathematics.
Radian in context
A radian is the angle for which the arc it cuts off equals the radius, defined as the ratio \(\theta=\dfrac{s}{r}\) of arc length to radius. Because it is one length divided by another, it carries no unit and is a pure number, so a full turn equals \(2\pi\) radians and \(180^\circ=\pi\) radians is the single conversion anchor. Two formulas follow directly from that ratio with no extra constants: arc length \(s=r\theta\) and sector area \(A=\tfrac12r^2\theta\). Both are valid only when \(\theta\) is in radians; neither formula is printed in the examination formula list, so both must be recalled.
A circular segment is the region cut off by a chord, found by subtracting from a sector the triangle on its two radii: \(A=\tfrac12r^2(\theta-\sin\theta)\), with \(\theta\) in radians throughout, including inside the sine. Minor and major regions on the same chord or pair of radii always add to the whole circle, so a major sector or segment is usually found by subtracting the minor one from \(\pi r^2\) rather than substituting the reflex angle directly. A perimeter is traced around the exposed boundary of a region, not quoted from a formula: an ordinary sector's perimeter is \(2r+r\theta\), but a shared internal edge inside a compound figure is never part of it.
Common mistakes with Radian
- Taking a sine in degree mode on a radian angle. Why it fails \(\sin 1.2\) means the sine of \(1.2\) radians, which is \(0.932\). A calculator in degree mode returns \(\sin 1.2^\circ=0.0209\) — the sine is out by a factor of about \(45\). The segment answer that follows is not out by \(45\); it is out by rather less, because the sine sits inside a subtraction. For \(r=8\), \(\theta=1.2\) it returns \(37.7\ \mathrm{cm^2}\) instead of \(8.57\ \mathrm{cm^2}\), about four times too large; at other angles the factor is different again. What is constant is not the size of the error but its shape: the triangle almost vanishes, so the segment comes out as nearly the whole sector. Fix Set radian mode at the start of the topic. Two sanity checks: for \(0<\theta<\pi\), \(\sin\theta\) should be a healthy fraction, not something starting \(0.0\); and a segment should be a modest slice of its sector, never \(98\%\) of it.
- Using \(s=r\theta\) with \(\theta\) in degrees. Why it fails \(s=r\theta\) is the definition \(\theta=s/r\) rearranged, and that definition measures the angle in radii of arc. A degree is a different and much smaller unit, so a degree value is about \(57.3\) times the radian value and the arc comes out about \(57.3\) times too long. Fix Convert on a separate line first. Then check the arc against \(2\pi r\); an arc longer than the circumference is impossible.
- Using \(\dfrac{\pi r^2\theta}{360}\) while \(\theta\) has already been converted to radians. Why it fails This is the degree formula fed a radian value. The \(360\) in the denominator is only correct if the numerator angle is measured in degrees; with radians the correct denominator is \(2\pi\). The answer comes out about \(57.3\) times too small. Fix Choose one system before writing anything. In radians: \(\tfrac12r^2\theta\). In degrees: \(\dfrac{\theta}{360}\times\pi r^2\). Never a hybrid.
- Forgetting that \(180^\circ=\pi\) radians, and reaching instead for “\(360=\pi\)” or “\(90=\pi\)”. Why it fails \(\pi\) radians is a half turn, because the full turn is \(2\pi\) — which is itself a consequence of the circumference containing \(2\pi\) radii. Fix Anchor on the right angle instead, which is harder to misremember: \(90^\circ=\pi/2\). Doubling it gives \(180^\circ=\pi\).
- Converting degrees to radians by multiplying by \(\dfrac{180}{\pi}\). Why it fails That multiplier is about \(57.3\), so it makes the number bigger — but radians are the larger unit, so a given angle has fewer of them and the number must get smaller. Fix Test the multiplier on \(90^\circ\). It must produce \(\pi/2\approx1.57\), not \(5157\).
- Converting radians to degrees by multiplying by \(\dfrac{\pi}{180}\). Why it fails The mirror image of the previous entry: that multiplier is about \(0.0175\) and shrinks the number, whereas degrees are the smaller unit and there must be more of them. Fix Same test. \(\pi/2\) must come out as \(90\).
- Taking a sine in degree mode on a radian angle. Why it fails \(\sin1.2\) asks for the sine of \(1.2\) radians, roughly \(69^\circ\), which is \(0.932\). In degree mode the calculator returns \(\sin1.2^\circ=0.0209\), the sine of a nearly flat angle. Every segment computed from it is wrong by a wide margin. Fix Set RAD at the start of the topic. Recognise the signature: a sine near \(0.02\) for an angle that ought to be substantial.
Examiner tips on Radian
- The one-line reminder to carry forward. Write it on the inside cover of your notes: radians, then radius, then trace the boundary. Those three steps in that order prevent the large majority of the marks lost in this topic, and none of them requires remembering a formula.
Questions students ask about Radian
What is a radian?
A radian is the angle for which the arc it cuts off equals the radius, defined as \(\theta=\dfrac{s}{r}\), the ratio of arc length to radius. Because it is one length divided by another it carries no unit and is a pure number, roughly \(57.3^\circ\). A full turn is \(2\pi\) radians, and the anchor identity is \(180^\circ=\pi\) radians; every other conversion is a multiple or fraction of that.
How do you convert between degrees and radians?
Use the single identity \(180^\circ=\pi\) radians. To convert degrees to radians, multiply by \(\dfrac{\pi}{180}\); to convert radians to degrees, multiply by \(\dfrac{180}{\pi}\). Check you have chosen the right multiplier by testing it on \(90^\circ\), which must give \(\pi/2\); radians are the larger unit, so a given angle has fewer of them, and converting to degrees must make the number bigger. Avoid rounding \(180/\pi\) to \(57\) or \(57.3\), which introduces an error large enough to change the final significant figure.
Why does using degrees in \(s=r\theta\) or \(A=\tfrac12r^2\theta\) give the wrong answer?
Both formulas come directly from the definition \(\theta=s/r\), which measures the angle in radii, so they are only valid when \(\theta\) is in radians. A degree value is about \(57.3\) times too large, so an arc length or sector area calculated with degrees is roughly \(57.3\) times too big, while still looking like a plausible number. Convert the angle to radians first, before substituting into either formula, and check a sector area against \(\pi r^2\) — it should never exceed it.
Why do you evaluate \(\sin\theta\) in radian mode when finding a segment area?
Because \(\theta\) in \(A=\tfrac12r^2(\theta-\sin\theta)\) is already in radians, so the sine must be taken of that same radian value, not converted to degrees first. \(\sin1.2\) means the sine of \(1.2\) radians, which is \(0.932\); a calculator left in degree mode instead returns \(\sin1.2^\circ=0.0209\), out by a factor of about \(45\). Check your calculator is in radian mode before evaluating, since this error is silent — nothing warns you the mode was wrong.