Radians, arcs and sectors Worked Solutions — A-Level Maths

Fully worked, step-by-step solutions to A-Level Radians, arcs and sectors questions. See exactly how to solve problems on radians, degree-radian conversion, arc length, sector area.

radiansdegree-radian conversionarc lengthsector areasector perimeterexact trig values
A-Level70 questionsStep-by-step solutions
Question 1
2 markseasy
Convert 90∘90^\circ to radians, giving your answer as an exact multiple of π\pi.

Worked solution

  1. Write the conversion rule

    θrad=θdeg×π180\theta_{\text{rad}}=\theta_{\text{deg}}\times\dfrac{\pi}{180}

    Multiply the number of degrees by pi over 180.

  2. Substitute and simplify

    90×π180=π290\times\dfrac{\pi}{180}=\frac{\pi}{2}

    Cancel the common factor.

  3. State the angle in radians

    θ=π2\theta=\frac{\pi}{2}

    Leave the answer as an exact multiple of pi.

Answer
π2\frac{\pi}{2}
Question 2
2 markseasy
Convert 60∘60^\circ to radians, giving your answer as an exact multiple of π\pi.

Worked solution

  1. Write the conversion rule

    θrad=θdeg×π180\theta_{\text{rad}}=\theta_{\text{deg}}\times\dfrac{\pi}{180}

    Multiply the number of degrees by pi over 180.

  2. Substitute and simplify

    60×π180=π360\times\dfrac{\pi}{180}=\frac{\pi}{3}

    Cancel the common factor.

  3. State the angle in radians

    θ=π3\theta=\frac{\pi}{3}

    Leave the answer as an exact multiple of pi.

Answer
π3\frac{\pi}{3}
Question 3
2 markseasy
Convert 45∘45^\circ to radians, giving your answer as an exact multiple of π\pi.

Worked solution

  1. Write the conversion rule

    θrad=θdeg×π180\theta_{\text{rad}}=\theta_{\text{deg}}\times\dfrac{\pi}{180}

    Multiply the number of degrees by pi over 180.

  2. Substitute and simplify

    45×π180=π445\times\dfrac{\pi}{180}=\frac{\pi}{4}

    Cancel the common factor.

  3. State the angle in radians

    θ=π4\theta=\frac{\pi}{4}

    Leave the answer as an exact multiple of pi.

Answer
π4\frac{\pi}{4}
Question 4
2 markseasy
Convert π\pi radians to degrees.

Worked solution

  1. Write the conversion rule

    θdeg=θrad×180π\theta_{\text{deg}}=\theta_{\text{rad}}\times\dfrac{180}{\pi}

    Multiply the radian measure by 180 over pi.

  2. Substitute and simplify

    π×180π=180∘\pi\times\dfrac{180}{\pi}=180^\circ

    The factor of pi cancels.

  3. State the angle in degrees

    θ=180∘\theta=180^\circ

    This is the angle measured in degrees.

Answer
180∘180^\circ
Question 5
2 markseasy
Convert π2\frac{\pi}{2} radians to degrees.

Worked solution

  1. Write the conversion rule

    θdeg=θrad×180π\theta_{\text{deg}}=\theta_{\text{rad}}\times\dfrac{180}{\pi}

    Multiply the radian measure by 180 over pi.

  2. Substitute and simplify

    π2×180π=90∘\frac{\pi}{2}\times\dfrac{180}{\pi}=90^\circ

    The factor of pi cancels.

  3. State the angle in degrees

    θ=90∘\theta=90^\circ

    This is the angle measured in degrees.

Answer
90∘90^\circ

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