Circle Calculator
Calculate circle area, circumference, radius, and diameter.
Radius, diameter, and the squared term
Circle formulas all hang off the radius. Circumference is 2πr, area isπr², and the diameter is simply 2r. The most frequent error is substituting a diameter where a radius belongs — which inflates a circumference by a factor of two and an area by a factor of four, because the radius is squared.
That squaring is worth internalising. Doubling the radius doubles the circumference butquadruples the area. It is why a 16-inch pizza is not a third bigger than a 12-inch one but nearly twice the food, and why pricing by diameter consistently flatters the smaller size.
For parts of a circle, work in fractions of the whole. A sector's area is(θ/360) × πr² in degrees, and an arc's length is (θ/360) × 2πr. Radians simplify both — ½r²θ and rθ respectively — which is precisely why higher mathematics prefers them. π is irrational, so every answer is an approximation; 3.14159 is ample for everyday work and NASA uses about fifteen digits for interplanetary navigation.
Frequently Asked Questions
What is the formula for circle area and circumference?
Area = π × r²; Circumference = 2 × π × r, where r is the radius.
What are the circle formulas?
Area = pi x r^2; circumference = 2 x pi x r, or pi x d. Given the circumference, the radius is C / (2 x pi); given the area, the radius is sqrt(A / pi). Note that doubling the radius quadruples the area but only doubles the circumference.
What is the difference between a chord, an arc and a sector?
A chord is a straight line between two points on the circle. An arc is the curved portion between them. A sector is the pie-slice region bounded by two radii and an arc, with area (theta/360) x pi x r^2 in degrees. A segment is the smaller region cut off by a chord alone.
Why does pi appear in both formulas?
Because pi is defined as the ratio of circumference to diameter, so it is built into any measurement of a circle. The area formula follows from integrating the circumference over the radius, which is also why the derivative of pi x r^2 with respect to r is exactly 2 x pi x r.