Ch 4 · 1.4 Circular Measure · A Level Mathematics 9709
Topic 4 of 8 in Pure Mathematics 1 — A Level Mathematics 9709 — 47 lessons.
Introduction
Degrees are a human choice (360 is easy to divide). Mathematicians prefer the radian, which comes from the circle itself, and makes the formulae for arcs, sectors and, later, calculus of sin x and cos x beautifully simple. Pizza slices, windscreen wipers and running tracks are all sector problems. (Syllabus section 1.4, Paper 1.)
Key ideas — What a radian is
One radian is the angle at the centre of a circle subtended by an arc equal in length to the radius. The whole circumference is 2π r, so a full turn is 2π radians:
🧑🏫 Teacher's explanation
Imagine you have a piece of string exactly the same length as the radius of a circle. If you take that string and wrap it along the curved edge (the circumference) of the circle, the angle formed at the center by that string is exactly 1 radian. It is a 'natural' way to measure angles because it links the angle directly to the linear distance traveled along the edge.
📖 Definitions
🔍 Worked example — Easy
Convert 150^° into radians, giving your answer in terms of π.
🔍 Worked example — Standard
A sector of a circle has a radius of 8 cm and an angle of 3π/4 radians. Calculate the arc length of the sector.
🔍 Worked example — Exam-style
A sector of a circle has an area of 20 cm^2 and an arc length of 10 cm. Find the radius of the circle.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Arc length and sector area
For a sector of radius r and angle θ in radians:
🧑🏫 Teacher's explanation
Imagine you are standing at the center of a circular park. If you walk along the edge, the distance you travel is the arc length. If you want to know how much grass is in the slice of the park you just walked through, you are looking for the sector area.
📖 Definitions
🔍 Worked example — Easy
A sector of a circle has a radius of 8 cm and an angle of 2 radians. Calculate the arc length s.
🔍 Worked example — Standard
A sector of a circle has a radius of 6 cm and an arc length of 3π cm. Find the area of the sector.
🔍 Worked example — Exam-style
A sector of a circle has a radius of 8 cm and an angle of 1.5 radians. Calculate the area of the segment formed by the chord connecting the endpoints of the radii. (Give your answer to 3 significant figures).
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Triangles inside circle problems
You also need the triangle facts from earlier work:
🧑🏫 Teacher's explanation
Imagine you are eating a slice of pizza. The crust is the 'arc', the whole slice is the 'sector', and the straight line cutting across the crust is the 'chord'. When we work with circular measure, we are often looking at the relationship between these three parts: the curved edge, the wedge-shaped area, and the straight-line distance between the ends of the crust.
📖 Definitions
🔍 Worked example — Standard
A chord of a circle with radius 10 cm subtends an angle of 120^° at the centre. Find the length of the chord.
🔍 Worked example — Exam-style
A sector of a circle has an arc length of 12 cm and an area of 48 cm^2. Find the radius and the angle in radians.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Worked examples — Example 1: arc, area and segment
A sector has radius 8 cm and angle 1.2 rad. Find the arc length, the sector area and the area and perimeter of the segment cut off by the chord.
🔍 Worked example — Easy
A sector of a circle has a radius of 10 cm and an angle of 2.5 rad. Calculate the arc length and the area of the sector.
🔍 Worked example — Standard
A sector has radius 8 cm and angle 1.2 rad. Find the arc length, the sector area, and the area of the segment cut off by the chord.
🔍 Worked example — Exam-style
A sector has radius r and angle θ radians. Show that the perimeter of the segment is rθ + 2rsin(θ/2).
⚠️ Common errors
Worked examples — Example 2: finding the angle
A sector of radius 6 cm has area 30 cm². Find its angle and its perimeter.
🔍 Worked example — Easy
A sector of a circle has a radius of 5 cm and an angle of 2 rad. Calculate the arc length.
🔍 Worked example — Standard
A sector of radius 6 cm has area 30 cm^2. Find its angle θ in radians.
🔍 Worked example — Exam-style
A sector of radius 6 cm has area 30 cm^2. Find its perimeter.
⚠️ Common errors
Worked examples — Example 3: sector and tangent
OAB is a sector with centre O, radius 10 cm and angle AOB = π/3. The tangent at A meets OB extended at C. Find the exact area and the perimeter of the region bounded by AC, CB and the arc AB.
🔍 Worked example — Easy
A sector of a circle has radius r = 8 cm and an angle $ heta = 2$ radians. Calculate the arc length s.
🔍 Worked example — Standard
A sector has a radius of 6 cm and an angle of π/3 radians. Find the area of the sector.
🔍 Worked example — Exam-style
OAB is a sector with centre O, radius 10 cm and angle AOB = π/3. The tangent at A meets OB extended at C. Find the exact area of the region bounded by AC, CB and the arc AB.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) (a) 5π/6; (b) 143.2^°. (2) s = 4 cm, A = 10 cm². (3) 2r + 2r = 20 ⇒ r = 5; A = tfrac12 × 25 × 2 = 25 cm². (4) tfrac12 × 36 (π/2 - 1 ) = 9π - 18 cm². (5) A = tfrac12 r s ⇒ 50 = 5r ⇒ r = 10 cm; θ = s/r = 1 rad.