Circles break the rules that polygons follow — no straight sides, no corners, just one continuous curve governed by a single irrational number, π. But they also reward you: every circle measurement (circumference, area, arc length, sector area) flows from just the radius. Learn the handful of formulas below and you’ll own every circle problem on the test.
The Two Parts You Must Never Confuse
- Radius (r): center to edge — half the diameter.
- Diameter (d): straight across through the center — d = 2r.
Nearly every circle formula wants the radius. If a problem hands you a diameter, your first move is always to halve it.
Circumference: Distance Around the Circle
Circumference is the circle’s “perimeter” — the length of one trip around the edge:
C = 2π r or C = π d
Both give the same answer since d = 2r.
Worked Example 1: Circumference from radius
Find the circumference of a circle with radius 7 inches (use π ≈ 22/7).
- Write the formula: C = 2 × π × r
- Substitute: C = 2 × (22/7) × 7
- Cancel the 7s: C = 2 × 22 = 44 inches
Choosing π ≈ 22/7 was deliberate — when the radius is a multiple of 7, the fraction cancels beautifully. Match your π approximation to the numbers you’re given.
Area: Space Inside the Circle
A = π r^2
The exponent belongs to the radius alone — square r first, then multiply by π. Computing (π × r)² squares π too, and your answer will be wrong by a factor of π.
Worked Example 2: Area from diameter
Find the area of a circular tabletop with diameter 10 meters (use π ≈ 3.14).
- Find the radius: r = d ÷ 2 = 10 ÷ 2 = 5 meters
- Square the radius: 5² = 25
- Multiply by π: A = 3.14 × 25 = 78.5 m²
Worked Example 3: Working backwards — area from circumference
A running track’s circular lane measures 62.8 meters around. What’s the area inside it? (π ≈ 3.14)
- Find the radius from C = 2πr: r = 62.8 ÷ (2 × 3.14) = 62.8 ÷ 6.28 = 10 meters
- Apply the area formula: A = 3.14 × (10)² = 3.14 × 100 = 314 m²
Partial Circles: Arc Length and Sector Area
Real problems often ask about a slice of a circle, defined by a central angle θ (in degrees).
- Arc length (a piece of the circumference): Arc = (θ/360) × 2πr
- Sector area (a pie slice): Sector = (θ/360) × πr²
The logic is simple: θ/360 is the fraction of the whole circle you’re taking.
Worked Example 4: Sector area
Find the area of a sector with radius 6 cm and central angle 90°.
- Angle fraction: 90 ÷ 360 = 1/4
- Square the radius: 6² = 36
- Multiply: Area = (1/4) × π × 36 = 9π cm² (≈ 28.27 cm²)
Worked Example 5: Arc length
Find the arc length for a circle of radius 12 cm with a central angle of 60°.
- Angle fraction: 60 ÷ 360 = 1/6
- Full circumference: 2πr = 2 × π × 12 = 24π
- Multiply: Arc = (1/6) × 24π = 4π cm (≈ 12.57 cm)
3 Mistakes to Avoid
- Squaring the diameter: plugging d = 8 straight into πr² gives 64π — four times the correct 16π. Halve the diameter first.
- Squaring π with the radius: (πr)² ≠ πr². Only r gets the exponent.
- Rounding π too early: in multi-step arc and sector problems, keep π symbolic (as 9π, 4π) until the final step. Early rounding to 3.14 compounds the error.
Exam shortcut: whenever the central angle divides 360 evenly (90°, 60°, 45°, 30°), work with the fraction (1/4, 1/6, 1/8, 1/12) instead of decimals — the arithmetic stays clean and exact.