Math · Live
Circle Calculator —
area, circumference, diameter, radius.
Enter any one measurement — radius, diameter, circumference, or area and the calculator instantly computes the other three, with a step-by-step derivation and an SVG visualization.
Inputs
Enter any one value
Type a value in any field — the other three update instantly.
- π (pi)
- 3.14159265358979…
- Input type
- Radius
- Radius
- 5
- Ratio C/d
- 3.14159265
- Ratio A/r²
- 3.14159265
Circle properties
r = 5
Visualization
Radius, diameter, and area to scale within the view.
Math notepad
Step-by-step derivation from r = 5
Given: radius r = 5
Step 1: diameter d = 2r:
d = 2 × 5
d = 10
Step 2: circumference C = 2πr:
C = 2 × π × 5
C = 2 × 3.14159265358979… × 5
C = 31.415927
Step 3: area A = πr²:
A = π × r²
A = 3.14159265358979… × (5)²
A = 3.14159265358979… × 25
A = 78.539816
Formulas
Circle relationship chart
| Find | Given radius r |
|---|---|
| Radius r | r |
| Diameter d | 2r |
| Circumference C | 2πr |
| Area A | πr² |
Field guide
Circle formulas explained.
A circle is defined by a single number — its radius. Every other property (diameter, circumference, area) follows directly from that one value through a handful of formulas involving π (pi). This calculator lets you start from any of the four measurements and derives the other three instantly.
The four properties of a circle
- Radius (r): the distance from the center of the circle to any point on its edge. All other properties are derived from r.
- Diameter (d): the longest chord across the circle, passing through the center. Exactly twice the radius.
- Circumference (C): the perimeter of the circle; the total distance around the edge.
- Area (A): the amount of 2D space enclosed within the circle.
The formulas
Where π (pi) is the mathematical constant approximately equal to 3.14159265358979… It is the ratio of a circle's circumference to its diameter, and appears in every circle formula.
Solving for the radius from any input
Because all four properties depend on r, we can solve for the radius first and then derive everything else:
Once r is known, d = 2r, C = 2πr, and A = πr² follow immediately.
A worked example: radius = 5
Start with radius r = 5 (centimeters, meters, or any unit):
What is pi (π)?
Pi is an irrational number — its decimal expansion never terminates or repeats. Its value to 20 decimal places is:
Pi appears throughout mathematics: in the formula for the volume of a sphere (4/3 πr³), in trigonometry, in the Gaussian integral, and even in probability theory. Its ubiquity comes from the deep relationship between circles and angles.
Area vs circumference, which grows faster?
Both grow as r increases, but at very different rates:
- Circumference grows linearly with r (doubling r doubles C).
- Area grows quadratically with r (doubling r quadruples A — because r is squared in the formula).
For example, a circle with r = 10 has twice the circumference of r = 5, but four times the area.
Real-world applications
- Engineering & machining: calculating the material needed to cut a circular disc, or the circumference of a wheel for gear ratio calculations.
- Architecture: finding the area of a circular room, fountain, or column cross-section.
- Gardening & landscaping: estimating how much soil or mulch fills a circular bed given its diameter.
- Physics: orbital circumference, cross-sectional area of a pipe, moment of inertia calculations.
- Everyday life: finding the size of a circular table, pizza, or pool from its diameter or from measuring around the edge (circumference).
Circumference from diameter, a quick mental calculation
Since C = πd ≈ 3.14159 × d, a useful approximation is:
For rough estimates: the circumference of a circle is about 3.14 times its diameter. A wheel with a 50 cm diameter travels roughly 157 cm per revolution.
Unit considerations
This calculator works with any consistent unit — centimeters, meters, inches, feet, or miles. Whatever unit you use for the radius, the diameter and circumference will be in the same unit, and the area will be in that unit squared. For example, if you enter a radius in centimeters, the area is in cm².
Tips for using this calculator
- To find the radius of a circular object, wrap a tape measure around it to get the circumference, then enter that value in the Circumference field.
- If you know the diameter (the widest measurement straight across), enter it directly, the calculator divides by 2 internally.
- The step-by-step notepad shows the exact arithmetic so you can verify the result by hand or copy the steps for a homework problem.
- Switch between input fields freely, just type in any field and the others update.