Circumference and Area of a Circle Calculator
Enter any one measurement of a circle - radius, diameter, circumference, or area - and this calculator instantly finds the other three. Choose your unit, pick your known value, and see the full working with the formulas applied to your exact numbers.
Circle formulas: the four key relationships
Every circle can be described by just four numbers: radius (r), diameter (d), circumference (C), and area (A). They are connected by two fundamental formulas. The circumference formula is C = 2 pi r (equivalently, pi times the diameter). The area formula is A = pi r squared. Because all four properties are linked through the radius, knowing any one of them is enough to calculate the other three. This calculator does exactly that: enter your known value, select which property it represents, and get the rest instantly.
How to solve for each circle property
If you know the radius, multiply by 2 to get the diameter, multiply by 2 pi to get the circumference, and multiply pi by the radius squared to get the area. If you know the diameter, divide by 2 for the radius, then apply the same formulas. If you know the circumference, divide by 2 pi to recover the radius, then proceed as normal. If you know the area, divide by pi and take the square root to find the radius. In symbols: r = square root of (A / pi), d = 2r, C = 2 pi r, A = pi r squared.
Why pi appears in both formulas
Pi (approximately 3.14159) is the ratio of a circle's circumference to its diameter, the same constant for every circle regardless of size. That constant connection is why both formulas share pi. The circumference formula C = pi d just restates the definition of pi. The area formula A = pi r squared can be understood by imagining cutting the circle into many thin wedge-shaped slices, rearranging them into a shape that approaches a rectangle with height r and width pi r, giving an area of pi r squared.
Practical uses of circle calculations
Circle calculations come up in everyday tasks far more often than most people expect. Finding how much fencing to go around a circular pond needs the circumference. Estimating how much paint to cover a circular ceiling or how much turf to lay in a round garden bed needs the area. Pipe and wheel sizing in engineering, lens and sensor dimensions in optics, pizza size comparisons (area grows as the square of the radius, so a 14-inch pizza has nearly four times the area of a 7-inch one), road curve geometry, and rotary actuator design all reduce to these two formulas. The unit switch in this calculator lets you work in the units your project already uses, so no manual conversion is needed.
Common circle sizes at a glance
| Radius (cm) | Diameter (cm) | Circumference (cm) | Area (cm²) |
|---|---|---|---|
| 1 | 2 | 6.28 | 3.14 |
| 2 | 4 | 12.57 | 12.57 |
| 5 | 10 | 31.42 | 78.54 |
| 7 | 14 | 43.98 | 153.94 |
| 10 | 20 | 62.83 | 314.16 |
| 15 | 30 | 94.25 | 706.86 |
| 20 | 40 | 125.66 | 1,256.64 |
| 25 | 50 | 157.08 | 1,963.50 |
| 50 | 100 | 314.16 | 7,853.98 |
| 100 | 200 | 628.32 | 31,415.93 |
Radius, diameter, circumference, and area for frequently used circle sizes in centimetres.
Frequently asked questions
What is the formula for the circumference of a circle?
The circumference equals 2 times pi times the radius: C = 2 pi r. Because the diameter d equals 2r, you can also write it as C = pi d. Pi is approximately 3.14159. For example, a circle with radius 7 cm has circumference 2 times 3.14159 times 7, which is about 43.98 cm.
What is the formula for the area of a circle?
The area equals pi times the radius squared: A = pi r squared. For a radius of 7 cm, that is 3.14159 times 49, which is approximately 153.94 cm squared. Because area scales with the square of the radius, doubling the radius gives four times the area, not twice.
How do I find the radius from the circumference?
Divide the circumference by 2 pi: r = C / (2 pi). If the circumference is 62.83 m, the radius is 62.83 / 6.28318, which is approximately 10 m. Once you have the radius you can find the diameter, area, and the rest.
How do I find the radius from the area?
Divide the area by pi, then take the square root: r = square root of (A / pi). For an area of 314.16 cm squared, r = square root of (314.16 / 3.14159) = square root of 100 = 10 cm. This reversal is built into the calculator: select Area as your known value, enter it, and all other properties are shown automatically.
What is the difference between circumference and perimeter?
For a circle, circumference and perimeter mean the same thing: the total distance around the outside edge. The word circumference is used specifically for circles and ellipses, while perimeter is the general term used for any closed flat shape.
Does the area of a circle include the boundary?
In practical applications such as painting, tiling, or planting, the area formula A = pi r squared gives the surface enclosed by the circle, and whether or not you count the edge line itself makes no difference to the numerical result. Mathematically, the circle (the curve) and the disk (the filled region) are distinct, but for all real-world measurement purposes they are treated as identical.
How does a change in radius affect the area and circumference?
Circumference grows linearly with radius: doubling the radius doubles the circumference. Area grows as the square of the radius: doubling the radius quadruples the area, and tripling it gives nine times the area. This is why large circles are disproportionately bigger in terms of surface than they appear from the boundary alone.