Circle Area and Circumference

Both quantities depend only on the radius: the area is $A = \pi{}r^{2}$, the circumference $U = 2\pi{}r$. Example with $r = 5 cm$: $A = \pi{}\cdot{}25 \approx{} 78.5 cm^{2}$ and $U = 10\pi{} \approx{} 31.4 cm$. Watch the units: the area carries square centimetres, the circumference ordinary centimetres; this doubles as a good self-check for which formula you need. If the diameter is given instead of the radius, halve first $(r =\frac{ d}{2})$ or use $U = \pi{}d$ and $A = (\frac{\pi{}}{4})d^{2}$ directly. For partial circles you scale proportionally: a sector with central angle α has area (α/360°)·πr² and arc length (α/360°)·2πr. Almost no circle task needs more than these building blocks.

Rearrange the respective formula. From the circumference: $r =\frac{ U}{(2\pi{})}$, a simple division. Example: $U = 62.8 cm$ gives $r =\frac{ 62.8}{6.283} \approx{} 10 cm$. From the area: $r = \sqrt{(\frac{A}{\pi{}})}$, where after dividing by π you must still take the square root, because the radius appears squared in the area formula. Example: $A = 78.5 cm^{2}$ gives $r = \sqrt{(\frac{78.5}{3.1416})} = \sqrt{25} \approx{} 5 cm$. The most common mistake is forgetting the root in the area rearrangement; a result like $r = 25 cm$ for $A = 78.5 cm^{2}$ should immediately raise suspicion, since such a circle would be huge. Plausibility check: substitute back and verify that the original quantity reappears.

π is defined as the ratio of circumference to diameter of a circle: $\pi{} =\frac{ U}{d} \approx{} 3.14159$. This ratio is the same for every circle, whatever its size, because all circles are similar to each other. From the definition, $U = \pi{}d = 2\pi{}r$ follows immediately. That the same π also appears in the area formula is shown by the classic decomposition idea: cutting the circle into many thin sectors and laying them alternately up and down produces approximately a rectangle with sides $\frac{U}{2}$ and r, so $A = (\frac{U}{2})\cdot{}r = \pi{}r^{2}$. Archimedes sandwiched the circle between polygons and thus proved $\frac{3 10}{71} < \pi{} <\frac{ 3 1}{7}$. π is irrational, its decimal expansion never terminates; for exams the calculator's π key suffices.

By different amounts, and that is precisely the key insight: the circumference grows linearly with r, so it doubles: $U = 2\pi{}r$ becomes $2\pi{}(2r) = 2\cdot{}U$. The area grows quadratically, it quadruples: $A = \pi{}r^{2}$ becomes $\pi{}(2r)^{2} = 4\pi{}r^{2} = 4\cdot{}A$. Example: $r = 5 cm$ has $U \approx{} 31.4 cm$ and $A \approx{} 78.5 cm^{2}$; $r = 10 cm$ has $U \approx{} 62.8 cm$ but $A \approx{} 314.2 cm^{2}$. In general the circumference scales with the factor k, the area with $k^{2}$. This scaling logic explains everyday phenomena: a pizza with twice the diameter holds four times the topping, and a pipe with twice the radius transports (at equal flow speed) four times as much, because the cross-section grows quadratically.

Rarely as a task of their own, but constantly as a building block. In calculus: solids of revolution and volume integrals use circular cross-sections $A(x) = \pi{}\cdot{}f(x)^{2}$, and optimization problems tune cans or enclosures with circular and semicircular shapes (material via the circumference, content via the area). In geometry: composite areas from rectangles, half and quarter circles, annuli as the difference of two circle areas $(A = \pi{}(R^{2} - r^{2}))$, plus sectors and arc lengths. In stochastics, circles appear in geometric probabilities (target area over total area). Radian measure itself is circle logic: an angle in rad is the arc length on the unit circle, the full angle 360° corresponds to the circumference 2π. So the two small formulas carry a surprising amount of upper-school material.