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Radius 5 → area 78.54, circumference 31.42. Know only the area of 100? The radius is 5.64. Sectors and arcs included.
Circle problems give you one measurement and ask for three others — this calculator converts freely between radius, diameter, circumference, and area, then extends to sector area and arc length for the geometry course and the workshop alike. Everything flows from two formulas: A = πr² and C = 2πr.
From any one measurement — radius, diameter, circumference, or area — get all four, plus sector and arc: r=5 → area 78.54, circumference 31.42.
Invert the formulas: from area, r = √(A/π) — an area of 100 gives r = 5.64; from circumference, r = C/(2π) — a circumference of 31.42 gives r = 5. These inversions are the real-world cases: you can wrap a tape around a tree or pipe (circumference) but cannot measure its radius directly; you know a circular patio covers 200 sq ft but need the diameter for the edging order. Pipe-sizing is the professional version: HVAC ducts and plumbing are sold by diameter while airflow capacity scales with area — a 6-inch duct carries 28.3 sq in, and upgrading to 8-inch (50.3 sq in) nearly doubles capacity for 33% more width.
Because area scales with the SQUARE of the radius — A = πr² means doubling r multiplies area by 4, tripling by 9. The pizza illustration is the classic: a 16-inch pizza (201 sq in) has twice the area of an 11.3-inch, and comfortably beats two 10-inch pizzas (157 sq in) — "two mediums" deals rarely beat one large on food-per-dollar. The same square law runs engineering: doubling pipe diameter quadruples flow capacity, doubling a circle of light coverage quadruples the area lit (which is why lumens spread thin so fast), and cell-tower coverage areas explode with range. Sectors scale linearly with angle instead: a 90° sector of an r=10 circle is exactly a quarter of its 314.16 area — 78.54, with a 15.71 arc.
Anywhere a partial circle does work. Food: a pizza slice cut 8 ways from a 14-inch pie has area 19.2 sq in and a 5.5-inch crust edge — the crust-to-middle debate, settled by math. Landscaping: a sprinkler head sweeping 120° covers one-third of its full circle — radius 20 ft gives 419 sq ft of coverage, not the 1,257 the box implies. Construction: curved walkways need arc lengths for edging material (a 90° curve with 10-ft radius needs 15.7 ft of border), and circular driveways are sector minus sector (outer arc minus inner arc). Sports: the shot-put sector is 34.92° — legal throws must land inside its arcs. The formulas are just fractions of the whole: sector area = πr² × θ/360, arc = 2πr × θ/360.