Round and Round: Circle Calculations
Circle calculations involve finding circumference (the distance around), area (the space inside), and related measurements using the constant π (approximately 3.14159). From pizza slicing to satellite orbits, circles are everywhere, making these formulas some of the most practical in all of geometry.
Components of Circle Calculations
This section covers the key formulas and concepts:
- Circumference: The distance around a circle, calculated as C = 2πr or C = πd, where r is the radius and d is the diameter.
- Area: The space inside a circle, calculated as A = πr².
- Arc Length & Sector Area: A portion of the circumference (arc length = θ/360 × 2πr) and a slice of the circle's area (sector area = θ/360 × πr²), where θ is the central angle in degrees.
- Diameter, Radius & Relationships: The diameter is twice the radius (d = 2r), and all circle measurements flow from knowing just one of these values.
Examples of Circle Calculations
Circumference Examples
- A circle has radius 7 cm. C = 2π(7) = 14π ≈ 43.98 cm.
- A circle has diameter 20 inches. C = π(20) = 20π ≈ 62.83 inches.
- A circular track has radius 50 meters. One lap is C = 2π(50) = 100π ≈ 314.16 meters.
Area Examples
- A circle has radius 5 cm. A = π(5²) = 25π ≈ 78.54 cm².
- A circular garden has diameter 12 feet, so radius is 6 feet. A = π(6²) = 36π ≈ 113.10 square feet.
- A pizza has radius 9 inches. The total area is π(9²) = 81π ≈ 254.47 square inches.
Arc Length & Sector Area Examples
- A circle with radius 10 cm has a 90° arc. Arc length = 90/360 × 2π(10) = 1/4 × 20π = 5π ≈ 15.71 cm.
- A pizza slice is a 45° sector of a pizza with radius 9 inches. Sector area = 45/360 × π(81) = 1/8 × 81π ≈ 31.81 square inches.
- A windshield wiper sweeps a 120° arc with length 18 inches. The arc it covers is 120/360 × 2π(18) = 1/3 × 36π = 12π ≈ 37.70 inches.
Diameter & Radius Relationship Examples
- A wheel has circumference 94.25 cm. Solve C = πd: d = 94.25 ÷ π ≈ 30 cm, so the radius is 15 cm.
- A circular pond has area 201.06 m². Solve A = πr²: r² = 201.06 ÷ π ≈ 64, so r = 8 m and d = 16 m.
- A clock face has diameter 30 cm. The radius is 15 cm, and the circumference of the face is π(30) ≈ 94.25 cm.