C5.3; E5.3 - Circle area and circumference

Circumference and diameter

If a circle's radius doubles, do both its boundary length and its area double? Start by imagining a thread wrapped once round its edge. Straightening the thread measures the circle's circumference.

For every circle, circumference divided by diameter is the same constant, $\pi$. Its decimal expansion begins $3.14159\ldots$ and never terminates or repeats. The circumference is therefore a little more than three times the diameter.

[DIAGRAM: asset_name: Radius diameter and circumference; asset_slug: radius_diameter_circumference; caption: The diameter contains two radii; the circumference follows the curved boundary.; description: A circle has a horizontal diameter labelled 7 centimetres, centre O, a radius in a separate direction labelled 3.5 centimetres, and an outside arrow identifying the full circumference. The radius ends exactly on the circle and the diameter passes through its centre.]

An answer such as $7\pi\,\mathrm{cm}$ is exact: it retains the precise constant $\pi$. Replacing $\pi$ with a finite decimal approximation makes the answer approximate. On a calculator paper, use the calculator's $\pi$ key and round only at the end when a decimal is required.

Describe the same circle using its diameter instead. Only the type of supplied length changes.

The two calculations agree because they describe the same circle with different measurements.

Give the next circumference exactly, without replacing $\pi$.

Area and radius

Area measures the surface inside the circle. Its formula contains the radius squared, so changing a radius has a different effect on area than on circumference.

To picture why, cut a circle into many thin equal wedges and alternate them pointing up and down. Their curved edges form two increasingly straight rows. The arrangement approaches a rectangle of height $r$ and width half the circumference, $\pi r$. Its area is therefore $\pi r\times r=\pi r^2$. More, thinner wedges make the picture closer; cutting and rearranging do not change the total area.

If a diameter is given, find $r=d\div2$ first. The expression $\pi d^2$ would give four times the correct area.

[DIAGRAM: asset_name: Doubling a circle radius; asset_slug: doubled_circle_radius; caption: Doubling every length makes four times the area.; description: Two exact circles are drawn at the same scale with radii 4 and 8 centimetres. Their areas are labelled 16 pi and 64 pi square centimetres. A radius line reaches the boundary in each; captions distinguish circumference factor 2 from area factor 4.]