A circle's area, circumference and diameter from its radius.
The area of a circle is S = πr², where r is the radius. With a radius of 5 cm the area is 3.14159 × 25 = 78.54 cm². If you know the diameter, halve it to get the radius — that is the most common slip here, because a diameter is easier to measure than a radius.
The circumference is C = 2πr, or C = πd where d is the diameter. The two are the same formula, since a diameter is two radii. With a diameter of 10 cm the circumference is 3.14159 × 10 = 31.42 cm. Unlike area, circumference grows linearly — double the radius, double the circumference.
Because the radius is squared in the formula. If r becomes 2r, the area becomes π(2r)² = 4πr² — four times larger. The same relation explains why a pizza of twice the diameter gives four times the food, and why a pipe's capacity grows much faster than its diameter.
For practical work 3.14159 is plenty. In construction and craft even 3.14 gives an error below 0.05%, smaller than the accuracy of a tape measure. This calculator uses full machine precision, so rounding the result is the only place any accuracy is lost.