A-Level Radians, arcs and sectors Practice Questions

Free A-Level Radians, arcs and sectors practice questions with full step-by-step worked solutions. Covers radians, degree-radian conversion, arc length, sector area. Practise exam-style problems and check your method.

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.
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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
Which of the following is π2\frac{\pi}{2} radians expressed in degrees?
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Worked solution

  1. Recall that pi radians equals 180 degrees

    π rad=180∘\pi\text{ rad}=180^\circ

    Use this to convert.

  2. Multiply by 180 over pi

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

    The factor of pi cancels.

  3. Select the correct option

    90∘90^\circ

    This matches the correct conversion.

Answer
90∘90^\circ
Question 3
3 marksintermediate
Which of the following is the exact value of tan⁡(π3)\tan\left(\frac{\pi}{3}\right)?
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Worked solution

  1. Identify the angle in radians

    θ=π3\theta=\frac{\pi}{3}

    Note the exact radian measure requested.

  2. Convert the angle to degrees

    π3=60∘\frac{\pi}{3}=60^\circ

    This helps recognise the special angle.

  3. Recognise it as a standard angle

    60∘ is a special angle60^\circ\text{ is a special angle}

    Its trigonometric ratios are known exactly.

  4. Write down the exact value

    tan⁡(π3)=3\tan\left(\frac{\pi}{3}\right)=\sqrt{3}

    Read the value from the special triangles or unit circle.

  5. Confirm with a decimal check

    tan⁡(π3)≈1.732\tan\left(\frac{\pi}{3}\right)\approx 1.732

    A quick numerical check agrees with the exact value.

  6. Select the correct option

    3\sqrt{3}

    This is the exact value.

Answer
3\sqrt{3}
Question 4
5 markshard
A sector has radius r=8r=8 and angle θ=π6\theta=\frac{\pi}{6} radians. Which of the following is the exact area of the triangle formed by the radii?
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Worked solution

  1. State the given radius and angle

    r=8, θ=π6 radr=8,\ \theta=\frac{\pi}{6}\text{ rad}

    Write down the information provided in the question.

  2. Express the angle in degrees

    θ=π6=30∘\theta=\frac{\pi}{6}=30^\circ

    Converting helps you picture the size of the sector.

  3. Recall the area of the triangle between the two radii

    T=12r2sin⁡θT=\tfrac{1}{2}r^2\sin\theta

    The two radii and the chord form a triangle.

  4. Calculate the triangle area

    T=12×82×sin⁡π6=16T=\tfrac{1}{2}\times 8^2\times\sin \frac{\pi}{6}=16

    Use the included-angle area formula.

  5. Recall the arc length formula

    s=rθs=r\theta

    The arc length equals the radius times the angle in radians.

  6. Calculate the arc length

    s=8×π6=4π3s=8\times \frac{\pi}{6}=\frac{4 \pi}{3}

    Substitute the radius and angle.

  7. Recall the sector area formula

    A=12r2θA=\tfrac{1}{2}r^2\theta

    The area of a sector uses the square of the radius.

  8. Calculate the sector area

    A=12×82×π6=16π3A=\tfrac{1}{2}\times 8^2\times \frac{\pi}{6}=\frac{16 \pi}{3}

    Substitute the values and simplify.

  9. Recall the perimeter of a sector

    P=2r+rθP=2r+r\theta

    The perimeter is the two radii plus the arc.

  10. Compare with the options and select

    T=16T=16

    The matching option is the answer.

Answer
1616
Question 5
8 markschallenging
A sector has radius r=12r=12 and angle θ=π2\theta=\frac{\pi}{2} radians. Which of the following is the exact area of the triangle formed by the radii?
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Worked solution

  1. State the given radius and angle

    r=12, θ=π2 radr=12,\ \theta=\frac{\pi}{2}\text{ rad}

    Write down the information provided in the question.

  2. Express the angle in degrees

    θ=π2=90∘\theta=\frac{\pi}{2}=90^\circ

    Converting helps you picture the size of the sector.

  3. Recall the area of the triangle between the two radii

    T=12r2sin⁡θT=\tfrac{1}{2}r^2\sin\theta

    The two radii and the chord form a triangle.

  4. Calculate the triangle area

    T=12×122×sin⁡π2=72T=\tfrac{1}{2}\times 12^2\times\sin \frac{\pi}{2}=72

    Use the included-angle area formula.

  5. Recall the arc length formula

    s=rθs=r\theta

    The arc length equals the radius times the angle in radians.

  6. Calculate the arc length

    s=12×π2=6πs=12\times \frac{\pi}{2}=6 \pi

    Substitute the radius and angle.

  7. Recall the sector area formula

    A=12r2θA=\tfrac{1}{2}r^2\theta

    The area of a sector uses the square of the radius.

  8. Calculate the sector area

    A=12×122×π2=36πA=\tfrac{1}{2}\times 12^2\times \frac{\pi}{2}=36 \pi

    Substitute the values and simplify.

  9. Recall the perimeter of a sector

    P=2r+rθP=2r+r\theta

    The perimeter is the two radii plus the arc.

  10. Calculate the perimeter

    P=2×12+6π=6π+24P=2\times 12+6 \pi=6 \pi + 24

    Add the two straight edges to the arc length.

  11. Recall the area of a segment

    S=12r2(θ−sin⁡θ)S=\tfrac{1}{2}r^2(\theta-\sin\theta)

    The segment is the sector minus the triangle.

  12. Calculate the segment area

    S=12×122(π2−sin⁡π2)=−72+36πS=\tfrac{1}{2}\times 12^2\left(\frac{\pi}{2}-\sin \frac{\pi}{2}\right)=-72 + 36 \pi

    Subtract the triangle from the sector.

  13. Recall the chord length

    c=2rsin⁡θ2c=2r\sin\tfrac{\theta}{2}

    The chord subtends the angle at the centre.

  14. Calculate the chord length

    c=2×12×sin⁡π4=122c=2\times 12\times\sin \frac{\pi}{4}=12 \sqrt{2}

    Halve the angle and use the chord formula.

  15. Compare with the options and select

    T=72T=72

    The matching option is the answer.

Answer
7272

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