Circles: Circumference, Area, Arcs & Sectors
Two circle formulas, then the fraction trick for arcs and sectors. Every problem in this skill uses π = 3.14 and rounds to 2 decimal places.
1 Key ideas
Circumference is the distance around; area is the space inside. Arcs and sectors are fractions of those.
Circumference
C = 2πr or C = πd
r = 5: C = 2 × 3.14 × 5 = 31.40.
Area
A = πr²
r = 4: A = 3.14 × 16 = 50.24.
Arc & sector — the fraction trick
arc = (θ/360) × C
sector = (θ/360) × A
A 90° sector is ¼ of the circle.
The diameter trap: the formulas need the radius. If a problem gives the
diameter (across the whole circle), halve it first: r = d ÷ 2.
House rules for this skill: π = 3.14 in every problem, and every answer is rounded to
2 decimal places. Write both rules at the top of your work.
Quick check 1 — A circle has radius 5 cm. Its circumference is… (π = 3.14)
Quick check 2 — A circle has diameter 10 cm. Its circumference is… (π = 3.14)
Quick check 3 — A circle has radius 4 cm. Its area is… (π = 3.14)
Quick check 4 — A circle has diameter 6 cm. Its area is… (π = 3.14)
2 Worked examples
Arcs, sectors, and working backwards from circumference or area.
Example 1 — arc length
- r = 10 cm, central angle 90°. First the full circumference: C = 2 × 3.14 × 10 = 62.80.
- Take the fraction: 90/360 = 1/4.
- Arc = ¼ × 62.80 = 15.70 cm.
Example 2 — sector area
- r = 6 cm, central angle 120°. Full area: A = 3.14 × 36 = 113.04.
- Fraction: 120/360 = 1/3.
- Sector = ⅓ × 113.04 = 37.68 cm².
Example 3 — working backwards
- A circle has circumference 62.80 cm. Find the radius.
- C = 2πr → 62.80 = 2 × 3.14 × r = 6.28r.
- r = 62.80 ÷ 6.28 = 10 cm.
Quick check 5 — r = 8 cm, central angle 180°. The arc length is… (π = 3.14)
Quick check 6 — r = 10 cm, central angle 90°. The sector area is… (π = 3.14)