GCSE Mathematics (9-1)
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    19.3 Arc length and sector area — Practise with help

    Warm-up: remember

    1. Find 14\frac14 of 360∘360^\circ. 2. State the circumference of a circle with radius rr.
    Check your answers
    1. 90∘90^\circ. 2. 2πr2\pi r.

    Practise with help

    Attempt each question first. Open the hint, then the next step, when needed.

    1. A sector has radius 10 cm and angle 72°. Work out its arc length in terms of π.

    Hint

    An arc is the same fraction of the circumference as its angle is of 360∘360^\circ.

    Next step

    Find the fraction 72360=15\frac{72}{360}=\frac15, then take it of 2π×102\pi\times10.

    Full working

    15×2π×10=4π cm\frac15\times2\pi\times10=4\pi\text{ cm}.

    2. A sector has radius 6 cm and angle 120°. Work out its area in terms of π.

    Hint

    Sector area is the same fraction of the circle area as its angle is of 360∘360^\circ.

    Next step

    Find the fraction 120360=13\frac{120}{360}=\frac13 of π×62\pi\times6^2.

    Full working

    13×π×36=12π cm2\frac13\times\pi\times36=12\pi\text{ cm}^2.

    3. A sector has radius 5 cm and angle 180°. Work out its perimeter in terms of π.

    Hint

    The perimeter of a sector includes the arc and both radii.

    Next step

    Find the arc as half the circumference, then add the two radii: 5π+2×55\pi+2\times5.

    Full working

    The arc is 12×2π×5=5π\frac12\times2\pi\times5=5\pi. The perimeter is 10+5π cm10+5\pi\text{ cm}.

    Try the remaining questions without opening the full working.

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