Circumference and Area of a Circle Calculator
This calculator works out every key measurement of a circle, its radius, diameter, circumference and area, from just one figure you already know. It matters because circles turn up constantly in real life, from fencing a round garden to ordering turf, sizing a wheel, or working through NCEA geometry homework, and knowing one measurement lets you solve for all the others without hunting down separate formulas. Choose what you are solving from, radius, diameter or circumference, type in that single value, then pick your unit of measurement from millimetres through to yards and set how many decimal places you want shown. The calculator instantly returns all four measurements: radius, diameter, circumference and area, plus a full calculation panel showing the input you used, the formulas applied (C = 2πr for circumference and A = πr² for area), and the exact value of pi used in the working. A worked example below the results walks through the maths step by step so you can see how each figure was derived. Because the formulas are exact and use the standard value of pi, results are accurate to the decimal places you choose, with no rounding shortcuts. Use it to plan fencing or paving quantities, check a homework answer, or convert between radius, diameter and circumference on the fly, and remember that doubling the radius doubles the circumference but quadruples the area.
1. Input
Please enter a positive number.
2. Units
Full Calculation
Formulas Applied
Worked Example (default: radius = 5 cm)
Step 1: Diameter = 2 × r = 2 × 5 = 10.000 cm
Step 2: Circumference = 2 × π × r = 2 × 3.14159 × 5 = 31.416 cm
Step 3: Area = π × r² = 3.14159 × 25 = 78.540 cm²
Circumference and Area of a Circle: Explained
A circle is defined by a single measurement: its radius (the distance from the centre to the edge). From the radius, every other property of a circle follows exactly, with no approximation beyond the use of π.
The Two Key Formulas
The circumference is the perimeter of a circle: the total distance around the outside. The formula is:
C = 2πr (or equivalently, C = πd, where d is the diameter)
The area is the amount of flat space enclosed by the circle. The formula is:
A = πr²
Both formulas use π (pi), the ratio of any circle's circumference to its diameter. Pi is an irrational number: it cannot be expressed as a simple fraction and its decimal expansion never repeats. To many decimal places, π = 3.14159265358979…
Solving from Different Inputs
| Known value | Radius | Diameter | Circumference | Area |
|---|---|---|---|---|
| Radius (r) | r | 2r | 2πr | πr² |
| Diameter (d) | d / 2 | d | πd | π(d/2)² |
| Circumference (C) | C / (2π) | C / π | C | C² / (4π) |
Practical Applications
These formulas appear in a wide range of everyday situations:
- Fencing a round garden: The circumference tells you how many metres of fencing you need.
- Turf or paving: The area tells you how many square metres of material to order.
- Engineering and manufacturing: Wheel circumferences determine how far a vehicle travels per revolution. Pipe cross-sections use circle areas to calculate flow rates.
- Cooking: A round cake tin's area determines how much batter you need, or how to scale a recipe when switching between tin sizes.
- School maths: Circle problems frequently appear in NCEA Level 1 and Level 2 geometry.
Relationship Between Circumference and Area
The circumference and area are related through the radius. If you double the radius of a circle, the circumference doubles (linear relationship) but the area quadruples (squared relationship). This is why small differences in radius can have a large effect on area but a proportionally smaller effect on circumference.
Related Calculators
- Maths and Stats Calculators: all geometry and statistics tools.
- Circumference Calculator: circumference from radius or diameter.
- Area of a Circle Calculator: area from radius, diameter, or circumference.
- Arc Length and Sector Area Calculator: partial circle calculations.
- Annulus Calculator: area and perimeter of a ring (annulus) between two circles.
Method: Standard Euclidean geometry. Formulas: C = 2πr, A = πr². Pi used: JavaScript Math.PI (IEEE 754 double precision, 15 to 17 significant digits). Reference: Euclid's Elements, Book III; any standard mathematics textbook.
This calculator uses standard geometric formulas. Results are exact within the precision of floating-point arithmetic. For very large or very small values, verify results with the formulas given.