Calculating...
Enter any circle measurement above and click Calculate.
Circle Anatomy Diagram
Every measurement you need — visualised in one diagram. Hover each label to understand what it measures.
Radius (r)
Distance from the centre O to any point on the circle. Every other formula depends on r.
Diameter (d = 2r)
Longest chord through the centre. Exactly twice the radius.
Circumference (C = 2πr)
The full perimeter — the curve itself. Wrapping a string around the circle gives you C.
Area (A = πr²)
The enclosed region. Quadruples when radius doubles because area scales with r².
Sector & Arc (θ = angle)
A "pizza slice" cut by central angle θ. Arc length L = rθ, sector area = ½r²θ (θ in radians).
Circle Formulas at a Glance
All four measurements derive from a single value — the radius. Everything else is algebraic rearrangement.
The total perimeter — the distance around the circle once.
Solving for r: r = C / (2π)
The enclosed surface — proportional to the square of the radius.
Solving for r: r = √(A / π)
The longest chord — passes through the centre.
Solving for r: r = d / 2
θ in radians. The calculator converts degrees automatically.
All measurements are functions of radius. Know any one → calculate all.
Circle Reference Table
Common radii and their corresponding circle measurements (values rounded to 4 decimal places).
| Radius (r) | Diameter (d) | Circumference (C) | Area (A) |
|---|---|---|---|
| 1 | 2 | 6.2832 | 3.1416 |
| 2 | 4 | 12.5664 | 12.5664 |
| 5 | 10 | 31.4159 | 78.5398 |
| 10 | 20 | 62.8319 | 314.1593 |
| 25 | 50 | 157.0796 | 1963.4954 |
| 100 | 200 | 628.3185 | 31415.9265 |
How to Calculate Circle Measurements
A circle is defined by a single measurement: the radius — the distance from its centre to any point on its edge. Every other property (diameter, circumference, area) follows algebraically from the radius, which is why the calculator lets you start from any of them.
Starting from the radius
If you know the radius r, the three remaining measurements are direct formulas: multiply by 2 for the diameter, multiply by 2π for the circumference, and multiply πr² for the area. This is the most common starting point in engineering, design, and geometry problems.
Starting from the diameter
The diameter d is the longest possible straight line through a circle, passing exactly through the centre. Divide by 2 to recover the radius, then proceed as normal. Diameters are often the measured value in practical settings — pipe diameters, wheel sizes, and circular hole measurements are all typically quoted as diameter.
Starting from the circumference
The circumference C is the perimeter of the circle — the distance you would travel if you walked once around its edge. To find the radius, rearrange C = 2πr to give r = C / (2π). This is useful when you have a tape measure wrapped around a circular object but no way to measure across it.
Starting from the area
If you know the area A, invert the formula A = πr² to find the radius: r = √(A / π). Area-first calculations appear frequently in land measurement (circular plots), irrigation system design, and manufacturing (circular blanks, gaskets).
Sector, Arc Length, and Chord
A sector is a "pizza slice" cut from a circle by two radii and the arc between them. Defined by the radius and central angle.
Arc Length
The curved edge of the sector. θ must be in radians (multiply degrees by π/180).
Sector Area
The area of the pizza-slice region. Equal to the full circle area times the fraction θ / (2π).
Chord Length
The straight-line distance between the two endpoints of the arc — the base of the triangle formed by the two radii.
Frequently Asked Questions
What is π (pi) and why does it appear in circle formulas?
π (pi) is the ratio of any circle's circumference to its diameter — approximately 3.14159265. It is irrational (never-ending, non-repeating), so calculators store many decimal places for precision. Because C = πd by definition, and A = πr² follows from integration, π appears in every circle formula.
How do I find the radius from the circumference?
Rearrange C = 2πr to give r = C / (2π). For example, if C = 62.83, then r = 62.83 / (2 × 3.14159) ≈ 10. The calculator does this automatically when you select the Circumference tab.
What is the difference between arc length and circumference?
Circumference is the full perimeter of the circle (the arc that subtends 360°). Arc length is the length of any partial arc, defined by a central angle θ < 360°. Arc length = (θ / 360) × C, which simplifies to rθ when θ is in radians.
How is sector area different from circle area?
Circle area is πr² — the entire enclosed region. Sector area is the fraction of that region cut out by the central angle: A_sector = (θ / 2π) × πr² = ½r²θ. A 90° sector (quarter circle) has area equal to πr² / 4.
What is a chord and how is it different from a diameter?
A chord is any straight line connecting two points on a circle. A diameter is the special chord that passes through the centre — the longest possible chord. Chord length depends on the central angle: c = 2r·sin(θ/2). When θ = 180°, c = 2r = diameter.
Can I use this calculator for circles in real-world units?
Yes — the calculator is unit-agnostic. If you enter the radius in metres, all outputs are in metres (circumference), square metres (area), and so on. Just be consistent: don't mix centimetres and inches in the same calculation.
Why does doubling the radius quadruple the area?
Area scales with the square of the radius (A = πr²). If r doubles to 2r, the area becomes π(2r)² = 4πr² — exactly four times as large. This is a fundamental property of 2-dimensional scaling: linear dimensions scale by a factor k, areas scale by k², volumes by k³.
What units should I use for the central angle?
Enter angles in degrees (0 to 360). The calculator automatically converts to radians internally before computing arc length, sector area, and chord length. Radians = degrees × π/180.
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