Practice Maths

Radian Measure, Arcs and Sectors

Key Terms

A radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. It is the natural unit for measuring angles in higher mathematics.
There are 2π radians in a full rotation, so 2π rad = 360°.
Conversion: degrees → radians: multiply by π/180.   radians → degrees: multiply by 180/π.
Arc length
of a sector: l = rθ   (where θ is in radians, r is the radius).
Area of a sector
: A = ½r²θ   (where θ is in radians).
Area of a segment
(sector minus triangle): Aseg = ½r²(θ − sinθ).
Always check: if the angle is given in degrees, convert to radians before applying the formulas.
Key Formulas — Radians, Arcs and Sectors
Conversion to radiansθrad = θdeg × π/180
Conversion to degreesθdeg = θrad × 180/π
Arc lengthl = rθ
Area of sectorA = ½r²θ
Area of segmentAseg = ½r²(θ − sinθ)
θ must be in radians for all formulas above.
Degrees Radians (exact) Radians (approx.)
30°π/60.524
45°π/40.785
60°π/31.047
90°π/21.571
120°2π/32.094
180°π3.142
270°3π/24.712
360°2π6.283

A sector with radius r and central angle θ (in radians): arc length l = rθ, sector area A = ½r²θ

x y r r θ l = rθ A = ½r²θ O
Hot Tip Always check whether an angle is given in degrees or radians before applying l = rθ or A = ½r²θ. If in degrees, convert first: multiply by π/180. A quick sanity check: a full circle (2π rad) gives arc length 2πr ✓ and area πr² ✓.

Worked Example 1 — Arc length and sector area

Question: A sector has radius 8 cm and central angle π/3 radians. Find the arc length and area of the sector.

Arc length: l = rθ = 8 × π/3 = 8π/3 cm ≈ 8.38 cm

Area of sector: A = ½r²θ = ½ × 64 × π/3 = 32π/3 cm² ≈ 33.5 cm²

Worked Example 2 — Converting and finding unknowns

Question: A sector has area 54 cm² and radius 6 cm. Find the central angle in degrees.

Step 1: Use A = ½r²θ: 54 = ½ × 36 × θ = 18θ

Step 2: θ = 54/18 = 3 radians

Step 3: Convert to degrees: 3 × 180/π = 540/π ≈ 171.9°

What IS a Radian? A Proper Definition

One radian is the angle subtended at the centre of a circle by an arc whose length equals the radius. If you take a circle of radius r and mark off an arc of length r along the circumference, the angle at the centre that “opens out” to that arc is exactly 1 radian. This is a genuine geometric definition — the radian is built from the circle itself.

Since the full circumference is 2πr and each radian corresponds to an arc of length r, there are 2πr / r = 2π radians in a full revolution. This gives the fundamental conversion: 2π rad = 360°, or equivalently π rad = 180°. Every conversion formula follows from this single relationship.

Why Radians Are More Natural Than Degrees

Degrees are a human invention: the Babylonians divided a circle into 360 parts because 360 has many factors. Radians, by contrast, are inherent to the mathematics of circles and periodic functions.

The key evidence for naturalness: the derivative of sin(x) is cos(x) ONLY when x is measured in radians. If x were in degrees, the derivative would be (π/180)cos(x) — an ugly constant appears. All of calculus becomes cleaner in radians. Similarly, the Taylor series sin(x) = x − x³/6 + x&sup5;/120 − … and the formula e^(iπ) + 1 = 0 all require radian measure. From Year 12 onwards, angles in mathematics are always in radians unless explicitly stated otherwise.

Conversion: Degrees to Radians and Back

Since π rad = 180°: to convert degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π.

A useful memory technique: degrees → radians = multiply by π/180 (“divide by 180 and multiply by π”); radians → degrees = multiply by 180/π (“multiply by 180 and divide by π”).

Common conversions to know without calculation: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 120° = 2π/3, 180° = π, 270° = 3π/2, 360° = 2π. Deriving these is faster than using the formula each time.

Arc Length: s = rθ

The formula s = rθ (arc length = radius × angle in radians) comes from proportionality. In a full circle (angle 2π), the arc is the full circumference 2πr. For angle θ, the arc is the fraction θ/(2π) of the full circumference: s = (θ/(2π)) × 2πr = rθ. This derivation shows why θ must be in radians — the proportionality only works when the angle is measured in radians, the natural unit of angle.

The formula is beautifully simple: arc length equals radius times angle. A radian is exactly the angle that makes arc length equal radius (s = r × 1 = r). This is the definition of radian restated algebraically.

Sector Area: A = ½r²θ

Similarly, a sector of angle θ is the fraction θ/(2π) of the full circle: A = (θ/(2π)) × πr² = ½r²θ. This derivation requires θ in radians. Alternatively: A = ½ × r × s = ½ × r × rθ = ½r²θ (area of a sector as base-times-height divided by 2, where the “base” is the arc and the “height” is the radius — an analogy with triangle area ½bh).

An important extension: the area of a circular segment (the region between a chord and its arc) equals the area of the sector minus the area of the triangle. Aseg = ½r²θ − ½r²sin(θ) = ½r²(θ − sinθ).

Exam Tip: θ MUST be in radians for the formulas s = rθ and A = ½r²θ. If an angle is given in degrees, convert to radians first: multiply by π/180. This is a one-step conversion that students frequently forget, leading to answers that are wrong by a factor of π/180 or 180/π. Write the conversion clearly in your working so the examiner can follow your method.
Exam Tip: Arc length (s = rθ) gives the curved distance along the circumference. Chord length (the straight-line distance between the two endpoints of the arc) is a different quantity: chord = 2r sin(θ/2). Do not confuse these two measurements. A question asking for “the length of arc AB” wants s = rθ; a question asking for “the length AB” where A and B are points on a circle wants the chord or straight-line distance.

Mastery Practice

See Answers ➔
  1. Fluency

    Convert each angle from degrees to radians. Give exact answers.

    1. (a) 60°
    2. (b) 150°
    3. (c) 225°
    4. (d) 315°
    5. (e) 270°
  2. Fluency

    Convert each angle from radians to degrees. Give exact answers.

    1. (a) π/4
    2. (b) 5π/6
    3. (c) 4π/3
    4. (d) 7π/4
    5. (e) 5π/2
  3. Fluency

    Find the arc length of each sector.

    1. (a) r = 5 cm, θ = π/3
    2. (b) r = 12 cm, θ = 2π/3
    3. (c) r = 9 cm, θ = 1.4 rad
    4. (d) r = 7 cm, θ = 60° (convert first)
  4. Fluency

    Find the area of each sector. Give exact answers where possible.

    1. (a) r = 6 cm, θ = π/2
    2. (b) r = 10 cm, θ = π/5
    3. (c) r = 4 cm, θ = 2.5 rad
    4. (d) r = 8 cm, θ = 45° (convert first)
  5. Understanding

    Find the unknown quantity in each sector.

    Rearrange: From l = rθ: r = l/θ or θ = l/r. From A = ½r²θ: θ = 2A/r² or r = √(2A/θ).
    1. (a) Arc length l = 15 cm, radius r = 5 cm. Find θ in radians and degrees.
    2. (b) Arc length l = 8π cm, θ = 2π/3. Find r.
    3. (c) Area A = 27 cm², radius r = 6 cm. Find θ.
    4. (d) Area A = 50 cm², θ = π/4. Find r.
  6. Understanding

    Perimeter of a sector.

    Note: The perimeter of a sector = arc length + 2 radii = rθ + 2r = r(θ + 2).
    1. (a) Find the perimeter of a sector with r = 7 cm and θ = π/3.
    2. (b) A sector has perimeter 30 cm and radius 9 cm. Find θ and the area.
    3. (c) A sector has perimeter 40 cm and θ = π/2. Find r and the area.
  7. Understanding

    Clock hands — understanding context.

    The minute hand of a clock is 14 cm long. Use radians to answer the following.
    1. (a) What angle (in radians) does the minute hand sweep in 20 minutes? Give an exact answer.
    2. (b) How far does the tip of the minute hand travel in 20 minutes? Give an exact answer.
    3. (c) What is the area swept by the minute hand in 45 minutes?
  8. Problem Solving

    Windscreen wiper.

    Challenge. A windscreen wiper sweeps through an angle of 120°. The wiper blade itself is 40 cm long and the nearest edge of the blade is 10 cm from the pivot point (so the blade extends from r = 10 cm to r = 50 cm from the pivot).
    1. (a) Convert 120° to radians.
    2. (b) Find the area swept by the large sector (r = 50 cm, θ = 2π/3).
    3. (c) Find the area swept by the small sector (r = 10 cm, θ = 2π/3).
    4. (d) Hence find the area of the windscreen cleaned by the wiper. Give your answer to the nearest cm².
  9. Problem Solving

    Irrigation system.

    Challenge. A rotating irrigation arm of length 80 m waters a sector of a field. The arm can rotate through an angle of 240°. Water costs $0.02 per m² to deliver.
    1. (a) Convert 240° to radians.
    2. (b) Find the area watered in exact form and to the nearest m².
    3. (c) Calculate the cost of watering this area.
    4. (d) If the angle were changed to 300°, by what percentage would the area (and cost) increase?
  10. Problem Solving

    Segment area and rearranging formulas.

    Challenge. A chord divides a circle of radius 10 cm into two parts. The central angle subtended by the chord is π/2 radians.
    1. (a) Find the area of the minor sector.
    2. (b) Find the area of the triangle formed by the two radii and the chord. (Area of triangle = ½r²sinθ.)
    3. (c) Hence find the area of the minor segment (the region between the chord and the arc).
    4. (d) Find the area of the major segment.