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Circle Formulas

Circles are everywhere in mathematics and the physical world. Their formulas all stem from one fundamental constant: π (pi). Understanding why the formulas work — not just memorizing them — makes them much easier to apply.

In this lesson

  1. What pi is and where it comes from
  2. Area and circumference
  3. Arc length
  4. Sector area
  5. Real-world applications

1 What Pi Is and Where It Comes From

Pi (π ≈ 3.14159) is the ratio of a circle's circumference to its diameter. Take any circle, measure its circumference, divide by its diameter — you always get π, regardless of the circle's size. This constant relationship is why π appears in every circle formula.

π is irrational — its decimal expansion never terminates or repeats: 3.14159265358979... For most calculations, 3.14159 is more than sufficient. For rough estimates, 22/7 ≈ 3.143 is a useful fraction approximation.

The Fundamental Ratio

Circumference / Diameter = π. Always. This is the definition of π. Rearranging gives: Circumference = π × diameter = 2πr. This single relationship generates every circle formula.

2 Area and Circumference

Circumference = 2πr = πd (where r = radius, d = diameter)

Area = πr²

The area formula can be understood by "unrolling" a circle into thin concentric rings. Each ring at radius t has circumference 2πt and tiny thickness dt. Summing all rings from 0 to r gives: ∫₀ʳ 2πt dt = πr². This is the geometric intuition even if calculus notation isn't familiar yet.

Area and Circumference
A circle has radius 6 cm.
1Circumference = 2πr = 2 × π × 6 = 12π ≈ 37.70 cm
2Area = πr² = π × 6² = 36π ≈ 113.10 cm²
Answer: C ≈ 37.70 cm, A ≈ 113.10 cm²
Radius vs diameter

The most common circle error. If you're given the diameter (the full width), halve it to get the radius before using area or circumference formulas. A circle with diameter 10 has radius 5, area = π(5²) = 25π ≈ 78.5 — not π(10²) = 314.

3 Arc Length

An arc is a portion of a circle's circumference. If the central angle is θ degrees, the arc is (θ/360) of the full circumference.

Arc length = (θ/360) × 2πr where θ is the central angle in degrees.

Arc Length
A circle has radius 8 cm. Find the arc length for a central angle of 90°.
1Arc length = (θ/360) × 2πr
2= (90/360) × 2π × 8
3= (1/4) × 16π
4= 4π ≈ 12.57 cm
Answer: Arc length ≈ 12.57 cm — makes sense: 90° is ¼ of the circle, so the arc is ¼ of the circumference

4 Sector Area

A sector is a "pie slice" of a circle — the region bounded by two radii and an arc. Its area is the same fraction of the total circle area as the central angle is of 360°.

Sector area = (θ/360) × πr²

Sector Area
A pizza with radius 15 cm is cut into 8 equal slices. What is the area of one slice?
1Each slice has central angle = 360/8 = 45°
2Sector area = (45/360) × π × 15²
3= (1/8) × π × 225
4= 225π/8 ≈ 88.36 cm²
Answer: One slice ≈ 88.36 cm²

5 Real-World Applications

Engineering and manufacturing: circular cross-sections are used in pipes, cables, and columns. The area formula determines how much material flows through a pipe (proportional to r²) or how strong a cable is. Doubling the radius quadruples the area — which is why thick cables are so much stronger than thin ones.

Navigation: the Earth is approximately spherical, and great circle routes (the shortest path between two points on a sphere) are used in aviation. Arc length calculations on a sphere are the basis of GPS distance calculations.

Statistics: the normal distribution curve is related to a circle through Gaussian integral — ∫e^(−x²)dx = √π. This is why π appears in the formula for the normal distribution even though it has nothing obvious to do with circles.

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Practice Problems

A circle has diameter 14 cm. Find its area and circumference.
Radius = 14/2 = 7 cm. Area = π(7²) = 49π ≈ 153.94 cm². Circumference = 2π(7) = 14π ≈ 43.98 cm.
A sector has radius 10 and central angle 120°. Find its area.
Sector area = (120/360) × π × 10² = (1/3) × 100π = 100π/3 ≈ 104.72 sq units.
A circle has circumference 50 cm. What is its radius?
Circumference = 2πr. 50 = 2πr. r = 50/(2π) = 25/π ≈ 7.96 cm.
If you double a circle's radius, what happens to its area?
Area = πr². If r doubles to 2r: area = π(2r)² = 4πr². The area quadruples.