1.05d - Radians, arc length and sector area | OCR A-Level Mathematics A

Define angles through arc length, convert between radians and degrees, and derive circle formulae that distinguish curved boundaries, sectors and segments.

An angle defined by arc length

A circular fan has radius $4\,\mathrm{cm}$ . Its radii enclose a small angle of $\pi/3$ , but the fabric covers the reflex sector. How much curved edging and fabric does it need? First we need an angle unit that talks directly to length and area.

On a circle of radius $4$ , an arc of length $4$ subtends one radian. Doubling the circle doubles both lengths, leaving the angle unchanged. Unlike degrees, radians are built from the circle itself.

You already know

A full circle has circumference $2\pi r$ , area $\pi r^2$ and angle $360^\circ$ . An angle and a length are different quantities; always match length units before forming a ratio.

The ratio of arc to radius defines the angle; it does not depend on the size of the circle.

Radian

One radian is the angle subtended at the centre of a circle by an arc equal in length to its radius. An arc of length $s$ on radius $r>0$ subtends angle $\theta=s/r$ radians, using matching length units.

A full circumference contains $2\pi$ radii, so a full turn is $2\pi$ radians. A surprisingly close near-fit: six arcs each one radius long almost go round a circle, but leave $(2\pi-6)r\approx0.283185r$ . Six radians is not a full turn.

Convert angle units

180^\circ=\pi\text{ radians},\qquad \theta_{\mathrm{rad}}=\theta_{\mathrm{deg}}\frac\pi{180},\qquad \theta_{\mathrm{deg}}=\theta_{\mathrm{rad}}\frac{180}\pi.

The conversion factors preserve the sign of a directed rotation.

Not in the formula sheet: learn it.

Keep exact multiples of $\pi$ when possible. For the reverse conversion, for example, $7\pi/6$ radians becomes $(7\pi/6)(180/\pi)=210^\circ$ .

Worked example 1: convert an anticlockwise turn

Question. Convert $135^\circ$ to an exact radian measure.

Plan. Multiply the degree measure by $\pi/180$ and simplify.

\begin{aligned} 135\frac\pi{180}&=\frac{135\pi}{180} &&\text{degrees to radians}\\ &=\frac{3\pi}{4} &&\text{cancel the common factor }45 \end{aligned}

Why this conversion factor? $180^\circ$ must become $\pi$ , so each degree is $\pi/180$ radians.

Answer. $3\pi/4$ radians.

Check. Multiplying by $180/\pi$ returns $135^\circ$ .

Change only the direction of rotation. A clockwise angle is negative in either unit.

Worked example 2: convert a clockwise turn

Question. Convert $-135^\circ$ to radians exactly.

Plan. Apply the same conversion and keep the negative sign.

\begin{aligned} -135\frac\pi{180}&=-\frac{135\pi}{180} &&\text{same unit conversion}\\ &=-\frac{3\pi}{4} &&\text{simplify} \end{aligned}

Why keep the sign? Conversion changes the unit, not the direction or number of turns.

Answer. $-3\pi/4$ radians.

Check. It is between $-\pi$ and $-\pi/2$ , matching a clockwise angle between $-180^\circ$ and $-90^\circ$ .

Rotations can be negative or exceed $2\pi$ . A physical sector instead needs the positive angle bounding the specified region; the reflex fan will need the larger angle.

Watch out

Wrong: $0.7$ radians means $0.7\pi$ radians. Right: a decimal angle already is its radian measure; multiply by $\pi$ only when the question actually contains it.

Make one more clockwise conversion without reducing away a full turn.

Arc length and sector area

The same fraction of a full turn selects the arc's fraction of circumference and the sector's fraction of circle area.

Sector

A region bounded by two radii and their joining arc. Its perimeter includes both radii; the arc is only the curved part of its boundary. A reflex sector uses an angle greater than $\pi$ and less than $2\pi$ .

For an angle $\theta$ radians, the fraction is $\theta/(2\pi)$ . Multiplying by the full circumference and area gives

s=\frac{\theta}{2\pi}(2\pi r)=r\theta,\qquad A=\frac{\theta}{2\pi}(\pi r^2)=\frac12r^2\theta.

These simple forms work because the full turn is $2\pi$ , not $360$ .

Arc, sector and boundary

s=r\theta,\qquad A=\frac12r^2\theta,\qquad P=r\theta+2r.

Use $r>0$ and the positive radian angle of the chosen sector, normally $0<\theta<2\pi$ . Arc length and perimeter have length units; area has squared units.

Not in the formula sheet: learn it.

The diagram separates the arc, full sector, triangle and segment. Those are different regions and boundaries.

A sector is bounded by two radii and an arc; a chord separates its triangle from its segment

Begin with the small sector between the fan's radii before finding the fabric's reflex region.

Worked example 3: the minor sector

Question. A circle has radius $4\,\mathrm{cm}$ . Find the arc length, area and perimeter of the sector with angle $\pi/3$ .

Plan. Use the positive angle directly in each formula.

\begin{aligned} s&=4\left(\frac\pi3\right)=\frac{4\pi}{3} &&\text{curved length}\\ A&=\frac12(4^2)\frac\pi3=\frac{8\pi}{3} &&\text{sector area}\\ P&=\frac{4\pi}{3}+8 &&\text{add both radii} \end{aligned}

Why add $8$ for the perimeter? The sector has two straight sides, each a radius.

Answer. Arc $4\pi/3\,\mathrm{cm}$ , area $8\pi/3\,\mathrm{cm}^2$ , perimeter $(8+4\pi/3)\,\mathrm{cm}$ .

Check. The angle is one sixth of a turn, so the arc and area are one sixth of their full-circle values.

Change only the chosen region: take the reflex sector between the same two radii. Its angle is the rest of a full turn.

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1.05d - Radians, arc length and sector area

An angle defined by arc length

Arc length and sector area

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