Grade 11 · Cambridge A-Level 9709 · Pure Mathematics 1
Welcome to Circular Measure!
Circular measure is the gateway to elegant mathematics. By measuring angles in radians rather than degrees, every formula involving circles becomes cleaner — arc length is simply rθ, sector area is ½r²θ, and calculus later relies on these definitions. This topic appears in every Cambridge A-Level 9709 Paper 1.
Understand and use radian measure, converting between radians and degrees
Know exact values of trig ratios at key angles in radians
Apply the arc length formula s = rθ to solve problems
Calculate the area of a sector using A = ½r²θ
Find the area and perimeter of a circular segment
Solve multi-step problems involving shaded regions between sectors/circles
Set up and solve equations where the angle θ is unknown
Radians
Definition, conversion, exact values
Arc Length
s = rθ, perimeter of sector
Sector Area
A = ½r²θ, derived from proportion
Segment Area
Sector minus triangle = ½r²(θ−sinθ)
Mixed Problems
Shaded regions, unknown θ, real-world
Visualiser
Interactive sector with live calculations
Start of content
Learn 1 — Radians
What is a Radian?
One radian is the angle subtended at the centre of a circle by an arc whose length equals the radius of the circle. If a circle has radius r and you lay a piece of string of length r along the circumference, the angle it subtends at the centre is exactly 1 radian.
Formal definition: If arc length s = radius r, then the angle θ = 1 radian.
More generally: θ (in radians) = arc length / radius = s/r
Why use radians? In calculus, the derivative of sin(x) is cos(x) ONLY when x is in radians. The arc length formula s = rθ and sector area A = ½r²θ are only this clean in radians. Degrees would require extra π/180 factors everywhere.
Converting Between Radians and Degrees
The full circle has arc length = circumference = 2πr. Dividing by r gives 2π radians. So:
2π radians = 360° ⟹ π radians = 180°
To convert degrees → radians: multiply by π/180 To convert radians → degrees: multiply by 180/π
Example 1: Convert 120° to radians.
120 × π/180 = 120π/180 = 2π/3 radians
Example 2: Convert 5π/6 to degrees.
(5π/6) × (180/π) = 5 × 30 = 150°
Exact Values in Radians
You must know these by heart for A-Level:
Degrees
Radians (exact)
sin
cos
tan
0°
0
0
1
0
30°
π/6
1/2
√3/2
1/√3
45°
π/4
1/√2
1/√2
1
60°
π/3
√3/2
1/2
√3
90°
π/2
1
0
undefined
180°
π
0
−1
0
270°
3π/2
−1
0
undefined
360°
2π
0
1
0
Memory trick: For 30/45/60 sin values — the sequence 1/2, 1/√2, √3/2 has numerators √1, √2, √3 over 2. Cos is the reverse. Tan = sin/cos.
Expressing Angles as Multiples of π
A-Level questions often give angles as fractions of π. Treat π as a symbol — do NOT substitute 3.14159... unless asked for a decimal approximation.
Example: A sector has angle 0.8 radians. Express this as a fraction of a full turn.
0.8 / (2π) = 0.4/π of a full turn. Alternatively, 0.8 rad ≈ 45.8°
Key rule: In all circular measure calculations, θ MUST be in radians. Never substitute a degree value directly into s = rθ or A = ½r²θ.
Learn 2 — Arc Length
The Arc Length Formula
An arc is a portion of a circle's circumference. For a sector with radius r and central angle θ (in radians):
s = rθ
Deriving s = rθ from First Principles
The full circumference of a circle is 2πr. A sector of angle θ is a fraction θ/(2π) of the full circle. Therefore:
Arc length = θ2π × 2πr = rθ
Rearranging s = rθ
The formula can be rearranged to find any one of the three quantities:
Finding s: s = rθ (given r and θ) Finding r: r = s/θ (given s and θ) Finding θ: θ = s/r (given s and r)
Example 1: r = 5 cm, θ = 1.4 rad. Find arc length.
s = 5 × 1.4 = 7 cm
Example 2: Arc length = 9 cm, θ = 0.6 rad. Find r.
r = 9 / 0.6 = 15 cm
Example 3: r = 12 cm, arc length = 8 cm. Find θ in radians.
θ = 8 / 12 = 2/3 rad ≈ 0.667 rad
Perimeter of a Sector
A sector has two straight edges (both of length r, the radius) and one curved edge (the arc). So:
Perimeter of sector = 2r + rθ = r(2 + θ)
Example: Sector with r = 7 cm and θ = 1.2 rad. Find perimeter.
P = 7(2 + 1.2) = 7 × 3.2 = 22.4 cm
When a question asks for the perimeter of a sector, don't forget BOTH radii. Students commonly add only one radius to the arc length.
Problems Involving Arc Length
Example (given degrees): A sector has r = 10 cm and angle 72°. Find the arc length.
Step 1 — Convert: 72° × π/180 = 2π/5 rad
Step 2 — s = rθ = 10 × 2π/5 = 4π ≈ 12.6 cm
Always check: is the angle in radians? If degrees are given, convert FIRST before applying s = rθ.
Learn 3 — Sector Area
The Sector Area Formula
A sector is the "pie slice" shape enclosed by two radii and an arc. Its area for angle θ in radians:
A = ½r²θ
Deriving A = ½r²θ from First Principles
Full circle area = πr².
A sector of angle θ is a fraction θ/(2π) of the full circle.
Area of sector = θ2π × πr² = θr²2 = ½r²θ
Using the Formula
Example 1: r = 8 cm, θ = 0.9 rad. Find sector area.
A = ½ × 64 × 0.9 = ½ × 57.6 = 28.8 cm²
Example 2: r = 6 cm, θ = π/3 rad. Find sector area.
A = ½ × 36 × π/3 = 18π/3 = 6π cm² ≈ 18.85 cm²
Finding r given area and θ: A = 50 cm², θ = 2 rad. Find r.
50 = ½ × r² × 2 = r² → r = √50 = 5√2 ≈ 7.07 cm
Relationship: A = ½rs
Since s = rθ, we can write A = ½r²θ = ½r(rθ) = ½rs. This useful form links sector area directly to arc length:
A = ½rs where s is the arc length
If you know the arc length s and radius r, you can find sector area immediately: A = ½rs. No need to find θ separately.
Problems Combining Arc Length and Area
Example: A sector has arc length 14 cm and radius 10 cm. Find the area.
Method 1 (using A = ½rs): A = ½ × 10 × 14 = 70 cm²
Method 2 (find θ first): θ = 14/10 = 1.4 rad, then A = ½ × 100 × 1.4 = 70 cm²
The formula A = ½r²θ only works when θ is in radians. If you must use degrees, either convert to radians first, or use A = (θ/360) × πr² — but the radian form is always preferred.
Learn 4 — Segment Area
What is a Segment?
A circular segment is the region between a chord and its arc. It is NOT the same as a sector. To find a segment, subtract the triangle from the sector:
Area of segment = Area of sector − Area of triangle
Area of the Triangle in a Sector
For a sector with two radii of length r and central angle θ, the triangle formed by the two radii and the chord has area:
Area of triangle = ½r² sinθ
This comes from the general formula: Area = ½ × a × b × sin(C), where a = b = r and C = θ.
Area = ½ × r × r × sinθ = ½r² sinθ
Full Segment Area Formula
Asegment = ½r²θ − ½r² sinθ = ½r²(θ − sinθ)
Example: Circle radius 9 cm, angle 1.4 rad. Find segment area.
A = ½ × 81 × (1.4 − sin 1.4)
sin 1.4 ≈ 0.9854
A = 40.5 × (1.4 − 0.9854) = 40.5 × 0.4146 ≈ 16.79 cm²
When computing ½r²(θ − sinθ), calculate sin θ on your calculator with the angle in radians. A common error is computing sin of the degree equivalent.
Perimeter of a Segment
The perimeter of a segment is the arc plus the chord (NOT the two radii — those are interior to the sector).
In the isosceles triangle OAB (O = centre, A and B on circle), the angle at O is θ. Using the sine rule or by splitting into two right triangles:
Each half-angle is θ/2, and the half-chord = r sin(θ/2)
Full chord = 2r sin(θ/2)
Example: r = 10 cm, θ = 1.2 rad. Find perimeter of segment.
Arc = 10 × 1.2 = 12 cm
Chord = 2 × 10 × sin(0.6) = 20 × 0.5646 ≈ 11.29 cm
Perimeter ≈ 12 + 11.29 = 23.29 cm
Complex Segment Problems
Finding angle given segment area: If A = ½r²(θ − sinθ) = 20 and r = 8, find θ.
½ × 64 × (θ − sinθ) = 20 → 32(θ − sinθ) = 20 → θ − sinθ = 0.625
This transcendental equation has no algebraic solution — use trial and improvement (iteration).
Try θ = 1.3: 1.3 − sin(1.3) = 1.3 − 0.9636 = 0.3364 (too small)
Try θ = 1.8: 1.8 − sin(1.8) = 1.8 − 0.9738 = 0.8262 (too big)
Try θ = 1.6: 1.6 − sin(1.6) = 1.6 − 0.9996 = 0.6004 (close)
Try θ = 1.61: 1.61 − sin(1.61) ≈ 0.614 → θ ≈ 1.60 rad
Segment area problems where θ is unknown often require iteration. Show your working clearly — Cambridge mark schemes award method marks for the iterative approach.
Learn 5 — Mixed Problems & Applications
Combining All Formulas
Exam questions rarely test a single formula in isolation. The most common pattern is a diagram with a shaded region that requires subtracting or adding areas of sectors, triangles, and segments.
Strategy: (1) Label all given information. (2) Identify which regions you need. (3) Write area of each piece. (4) Add or subtract. (5) Always state the answer with units.
Problems with Two Circles
Many Cambridge questions involve two concentric or overlapping circles. Typical structures:
Annular sector (ring sector): Sector of large circle minus sector of small circle, same angle θ.
Shaded area = ½R²θ − ½r²θ = ½θ(R² − r²)
Example: Two concentric circles, radii 4 and 9, sector angle 0.8 rad.
Shaded ring sector area = ½ × 0.8 × (81 − 16) = 0.4 × 65 = 26 cm²
Finding θ When Not Given Directly
Sometimes θ must be found from other conditions — typically the perimeter or area is given.
Example: Sector perimeter = 24 cm, area = 27 cm². Find r and θ.
Perimeter: r(2 + θ) = 24 → 2r + rθ = 24 … (1)
Area: ½r²θ = 27 → r²θ = 54 … (2)
From (2): rθ = 54/r. Substitute into (1): 2r + 54/r = 24
Multiply by r: 2r² − 24r + 54 = 0 → r² − 12r + 27 = 0
(r − 3)(r − 9) = 0 → r = 3 or r = 9
If r = 3: θ = 54/9 = 6 rad (valid, < 2π ≈ 6.28)
If r = 9: θ = 54/81 = 2/3 rad ✓ (Both technically valid — read question for context.)
Most likely intended answer: r = 9 cm, θ = 2/3 rad
Real-World Applications
Irrigation example: A sprinkler rotates through angle θ = 2.1 rad, reaching distance r = 12 m. Find the area irrigated.
A = ½ × 144 × 2.1 = ½ × 302.4 = 151.2 m²
Design problem: A logo is made from a circle of radius 5 cm. A circular segment of angle π/4 is cut off. Find the remaining area.
Full circle = 25π cm²
Segment removed = ½ × 25 × (π/4 − sin(π/4)) = 12.5 × (0.7854 − 0.7071) ≈ 12.5 × 0.0783 ≈ 0.979 cm²
Remaining ≈ 25π − 0.979 ≈ 77.57 cm²
Exam technique: Always write "θ is in radians" if there is any ambiguity. State the formula you are using before substituting. Leave answers in exact form (involving π) unless told to give a decimal.