Circumference and area of circles

Where \(\pi\) comes from

Wrap a string around a can, a plate, a bicycle wheel. Divide the length of the string by the diameter. Every time, you get the same number, a little more than \(3\).

\(\pi\) is irrational: its decimal expansion never ends or repeats. In answers you can leave it as \(\pi\) (an exact answer) or round with \(\pi \approx 3.14\) (an approximate answer).

Circumference and area

Where does \(\pi r^2\) come from? Slice a circle into many thin wedges and lay them side by side, alternating up and down. The result is close to a rectangle with height \(r\) and base half the circumference, \(\pi r\). Its area is \(\pi r \cdot r = \pi r^2\).

Arc length and sector area

An arc with central angle \(\theta\) is the fraction \(\frac{\theta}{360°}\) of the whole circle. The same fraction applies to the sector, the pie-slice region between two radii.

Shaded regions

Many problems ask for the area of a region left over when one shape is cut out of another. Find both areas and subtract.

Practice