Circle Calculator
Enter any one measurement — radius, diameter, area, or circumference — and instantly calculate all the others, plus optional sector area and arc length.
Last updated: March 2026
Enter one value — all others calculate automatically.
Optional: Sector & Arc Length
Circle Formulas Explained
All four key circle measurements are related through the radius r and the constant π ≈ 3.14159265:
- Diameter: d = 2r
- Circumference: C = 2πr = πd
- Area: A = πr²
- Radius from diameter: r = d / 2
- Radius from circumference: r = C / (2π)
- Radius from area: r = √(A / π)
This calculator detects which field you changed and derives the radius first, then computes all other values. If you change multiple fields, the last one you edited takes precedence.
The History and Magic of Pi
Pi (π) is one of the most celebrated constants in mathematics. Ancient Babylonians used 3.125 as an approximation; ancient Egyptians used 3.1605. Archimedes (c. 250 BCE) bounded π between 3 10/71 and 3 1/7 by inscribing and circumscribing polygons inside circles.
Today, π is known to trillions of decimal places. To 10 decimal places: π ≈ 3.1415926536. Despite its infinite decimal expansion, just a few digits suffice for most practical calculations — NASA uses only 15 digits for interplanetary navigation.
Pi appears not just in geometry but in probability, number theory, physics, and engineering — from the formula for a normal distribution to the area of an ellipse to Euler's famous identity eiπ + 1 = 0.
Parts of a Circle
Understanding circle vocabulary is essential for geometry:
- Radius: Distance from the center to any point on the circle.
- Diameter: Distance across the circle through the center (= 2 × radius).
- Chord: A line segment with both endpoints on the circle. The diameter is the longest chord.
- Arc: A portion of the circumference. A semicircle is an arc of 180°.
- Sector: The "pie slice" region bounded by two radii and the arc between them.
- Segment: The region between a chord and the arc it subtends.
- Tangent: A line that touches the circle at exactly one point, perpendicular to the radius at that point.
Sector Area and Arc Length
A sector with central angle θ (in degrees) from a circle of radius r has:
- Sector Area = (θ / 360) × πr²
- Arc Length = (θ / 360) × 2πr
A semicircle (θ = 180°) has area = πr²/2 and arc length = πr. A quarter circle (θ = 90°) has area = πr²/4 and arc length = πr/2.
How to Calculate Circle Measurements If...
How to find the area of a circle with radius 5?
Area = π × 5² = 25π ≈ 78.54 square units. Multiply π by the radius squared.
How to find the circumference of a circle with diameter 14?
Circumference = π × d = π × 14 ≈ 43.98 units. Or equivalently, C = 2πr = 2π × 7 ≈ 43.98.
How to find the radius from the circumference?
Divide the circumference by 2π. Example: C = 62.83 → r = 62.83 / (2π) ≈ 10 units.
How to find the area of a circle with diameter 20?
Radius = 10, Area = π × 100 ≈ 314.16 square units. Always halve the diameter to get the radius before using the area formula.
How to find the area of a semicircle with radius 6?
Area = πr²/2 = π × 36 / 2 ≈ 56.55 square units. A semicircle is exactly half a full circle.
How to find the arc length of a 90° sector with radius 8?
Arc = (90/360) × 2π × 8 = (1/4) × 50.27 ≈ 12.57 units.
How to find the area of a sector with angle 120° and radius 5?
Sector area = (120/360) × π × 25 = (1/3) × 78.54 ≈ 26.18 square units.
How to find the diameter if you know the area?
r = √(Area / π), then d = 2r. Example: Area = 200 → r = √(200/π) ≈ 7.98 → d ≈ 15.96 units.
How to find the area of the unit circle?
The unit circle has radius 1, so Area = π × 1² = π ≈ 3.14159 square units.
People Also Search For
How to calculate the area of a circle step by step?
Step 1: Identify the radius r. Step 2: Square it (r²). Step 3: Multiply by π ≈ 3.14159. For radius 9: 9² = 81, 81 × π ≈ 254.47 square units.
Circumference to diameter calculator
Divide circumference by π to get diameter. d = C / π. For C = 25: d = 25 / π ≈ 7.96 units. This tool does this conversion automatically when you enter circumference.
How to find the radius of a circle from its area?
Radius = √(Area / π). This is the inverse of the area formula. Enter the area into this calculator and the radius is computed instantly.
What is the relationship between circumference and diameter?
Circumference / Diameter = π for every circle, always. This is literally the definition of pi — it is the universal ratio of circumference to diameter.
How to find the area of a circle inscribed in a square?
A circle inscribed in a square touches all four sides. Its diameter equals the square's side length. So r = side/2, and Area = π(side/2)².
What is the perimeter of a circle called?
The perimeter of a circle is called its circumference. The term comes from Latin circumferentia, meaning "to carry around." It is calculated as C = 2πr or C = πd.
Frequently Asked Questions
What are the circle area and circumference formulas?
Area = πr². Circumference = 2πr = πd. Both depend on the radius r and the constant π ≈ 3.14159.
What is pi to 10 decimal places?
π = 3.1415926536. To 15 significant figures: 3.14159265358979. For everyday calculations, 3.14159 is sufficient.
What is the area of a circle with radius 10?
Area = π × 100 ≈ 314.16 square units.
What is the difference between arc and chord?
An arc is a curved portion of the circle's circumference. A chord is the straight line connecting the two endpoints of that arc. The arc is always longer than the chord (unless the arc is a full semicircle diameter, where they are equal only at the limit).
What is the unit circle and why does it matter?
The unit circle has radius 1 centered at the origin. Points on it have coordinates (cos θ, sin θ). It is the foundation of trigonometry and is used to define sine and cosine for all angles.
How do I find the circumference of a circle with radius 3?
C = 2π × 3 = 6π ≈ 18.85 units.
Can I enter the area to find the radius?
Yes. Type the area into the Area field and the calculator computes radius, diameter, and circumference automatically.
What is the equation of a circle?
Standard form: (x − h)² + (y − k)² = r², where (h, k) is the center and r is the radius. For a circle centered at the origin: x² + y² = r².
Is the data I enter private?
Yes. All calculations are performed locally in your browser. Nothing is sent to any server.
What real-world things are circular?
Coins, wheels, pizza, CDs, watch faces, manholes, pipes, lenses, and roundabouts. The circle is one of the most efficient shapes for enclosing area per unit of perimeter (only a circle maximizes area for a given circumference).