Geometry · GCSE Maths
Arc length and sector area
GCSE Maths arc length and sector area: use θ/360 of the full circumference or circle area, with θ in degrees — or θ/2π and θ/2 for radian-ready Higher work.
Arc length is a fraction of the circumference. Sector area is a fraction of the circle area. θ/360 of 2πr or πr² when θ is in degrees.
The important bits
What you need to know
- 1
Circumference of a full circle is C = 2πr. Arc length for angle θ° at the centre is (θ/360) × 2πr.
- 2
Area of a full circle is A = πr². Sector area for angle θ° is (θ/360) × πr².
- 3
θ must be the angle at the centre, between the two radii that bound the sector. It is not an angle on the circumference unless the question clearly gives the centre angle.
- 4
A semicircle is θ = 180°: arc = πr, area = ½πr². A quarter circle is θ = 90°: arc = ½πr, area = ¼πr².
- 5
Leave answers in terms of π when asked for exact values. 120° of a circle radius 6 cm: arc = (120/360) × 2π × 6 = 4π cm.
- 6
The minor sector uses the smaller angle θ. The major sector uses 360° − θ. Read which sector the question names.
- 7
Perimeter of a sector includes two radii plus the arc: P = 2r + (θ/360) × 2πr. Do not give only the arc when “perimeter” is asked.
- 8
Higher: θ in radians — arc = rθ, sector area = ½r²θ. At GCSE most papers use degrees; check the formula sheet.
Quotations worth analysing
Short evidence. Real method.
“Arc length = (θ/360) × 2πr”
θ is the centre angle in degrees. Using the circumference angle or the diameter instead of the centre angle is the usual wrong fraction.
“Sector area = (θ/360) × πr²”
Same fraction θ/360 as the arc, but applied to πr² not 2πr. Mixing the two denominators produces arc-length answers on area questions.
“Perimeter = 2r + arc”
“Length of the sector” in some papers means perimeter. Read whether the boundary or only the arc is wanted.
Go deeper
θ/360 is the fraction of the circle
A 60° sector is 60/360 = 1/6 of the full turn. Arc = 1/6 of circumference = (1/6) × 2πr = πr/3. Area = 1/6 of πr². For r = 9 cm and θ = 120°: fraction = 120/360 = 1/3. Arc = (1/3) × 2π × 9 = 6π cm. Area = (1/3) × π × 81 = 27π cm². Write the fraction as a separate line; it is a method mark. Simplify the fraction before multiplying: 72/360 = 2/10 = 1/5. Calculator papers may want decimals; give π form first if the question says “exact”, then convert only if instructed. A chord length is not an arc length — a chord is a straight line across the circle.
Go deeper
Minor sector, major sector, and perimeter
If θ = 80°, the minor sector is 80° and the major is 280°. A question asking for “the sector” usually means the minor one unless it says major. Major sector area = (280/360) × πr². Perimeter of the minor sector: walk along one radius, along the arc, along the other radius. That is 2r + arc. For r = 5 cm, θ = 72°: arc = (72/360) × 2π × 5 = 2π cm, perimeter = 10 + 2π cm. Shaded-region questions often subtract a sector from a triangle: find sector area, find triangle area with ½r² sin θ (Higher) or split a right-angled triangle, then subtract. Draw the pieces before you combine formulas.
Go deeper
Radians on Higher papers
When θ is in radians, arc length = rθ and sector area = ½r²θ. π radians = 180°, so θ radians = θ × 180/π degrees if you need to convert. π/3 radians = 60°. For r = 8 cm and θ = π/4 radians: arc = 8 × π/4 = 2π cm. Area = ½ × 64 × π/4 = 8π cm². The formula sheet on your paper tells you which form to use. If only degrees appear, stay with θ/360. Mixing a radian formula with a degree angle without converting is a silent way to lose every mark on the question.
See the idea in action
A sector has radius 12 cm and angle 150°. Find the arc length and area, giving exact answers. Step 1: Fraction = 150/360 = 5/12. Step 2: Arc = (5/12) × 2π × 12 = 5π cm. Step 3: Area = (5/12) × π × 12² = (5/12) × 144π = 60π cm². Step 4: If perimeter is required: 2 × 12 + 5π = 24 + 5π cm. Check: 150° is less than 180°, so arc is less than half the circumference π × 12 = 12π, and 5π < 12π.
Exam technique
Turn knowledge into marks
Write θ/360 as a simplified fraction before you multiply. Label whether the question wants arc, area, or full sector perimeter.
Common mistakes
Do not give these marks away
- 01
Using θ/360 × πr² for arc length, or θ/360 × 2πr for sector area.
- 02
Using an angle at the circumference instead of the centre angle θ.
- 03
Giving only the arc when the question asked for the perimeter of the sector.
A sector has radius 6 cm and angle 90°. The arc length is
A3π cm
B6π cm
C9π cm
D36π cm
Show the answer
3π cm. (90/360) × 2π × 6 = ¼ × 12π = 3π cm. 6π cm is half the circumference (semicircle arc). 36π cm used πr².
Quick questions
If this is the bit you searched
What angle do I use in the formula?
The angle at the centre between the two radii of the sector, in degrees unless the question uses radians.
How do I find the perimeter of a sector?
Add the arc length to twice the radius: 2r + (θ/360) × 2πr.
When do I leave π in the answer?
When the question asks for an exact value or gives answers “in terms of π”. Otherwise use π on the calculator and round as instructed.
What is the difference between arc and chord?
The arc is the curved part of the circumference. The chord is the straight line joining the two endpoints on the circle.