Circle Calculator

Whether you're working through a geometry problem, designing something round, or just trying to remember what formula goes where, a circle calculator takes the guesswork out of it. Punch in whatever measurement you already know and get the rest instantly. This page covers everything: radius, diameter, circumference, area, arc length, and sector area. The formulas are here too, so you can see exactly what's happening under the hood.

Enter Details

Radius

Result

d, C, A from r

Note — This result is an estimate. Talk to a healthcare provider for personalized guidance.

How to Use the Circle Calculator

Using the calculator is straightforward. You only need one known value to find all the others. That could be the radius, the diameter, the circumference, or the area. Pick whichever one you have, enter the number, and the calculator fills in the rest.

A few things to keep in mind before you start:

Make sure your units are consistent. Mixing inches and centimeters will throw off every result.
Radius and diameter are linear measurements. Area is square units. Circumference is also linear.
For arc length and sector area, you'll also need the central angle, either in degrees or radians.

That's really it. The math runs automatically once you enter your value, so you don't need to memorize a single formula to get a correct answer.

Calculate Radius, Diameter, Circumference, and Area

These four measurements define a circle completely. Know any one of them, and the other three follow directly.

Measurement	What It Describes	Unit Type
Radius (r)	Distance from center to edge	Linear (in, cm, ft…)
Diameter (d)	Distance across the circle through the center	Linear
Circumference (C)	Total distance around the circle	Linear
Area (A)	Space enclosed inside the circle	Square (in², cm², ft²…)

The radius is the most fundamental value. Every other measurement is derived from it, which is why most formulas are written in terms of r. If you only know the diameter, just divide it by 2 to get the radius, then you're off to the races.

Circle Formulas and Equations

There are really just a handful of formulas you need for any standard circle problem. They're not complicated once you see them laid out, and they all connect back to the radius and the constant pi (π ≈ 3.14159).

Below are the core equations broken down by what they calculate.

Radius to Diameter Conversion

The relationship between radius and diameter is the simplest one in circle geometry. The diameter is exactly twice the radius:

d = 2r

And going the other direction: r = d / 2

That's it. No pi involved, no exponents. If someone hands you a diameter of 10 inches, your radius is 5 inches. You'll use this conversion constantly because many real-world measurements (like pipe sizes or wheel widths) are given as diameter, while most formulas want the radius.

Circumference Formula

The circumference is the perimeter of a circle, the total length of its outer edge. Two equivalent formulas exist depending on what you already know:

C = 2πr (when you know the radius)
C = πd (when you know the diameter)

Both give the same result. For example, a circle with a radius of 7 cm has a circumference of 2 × π × 7 ≈ 43.98 cm. If you prefer to work with diameter, that same circle has d = 14 cm, and 14 × π gives you the same 43.98 cm.

Circumference comes up constantly in real life: calculating how far a wheel travels per rotation, figuring out how much trim to buy for a round table, or measuring a circular track.

Area Formula

The area of a circle tells you how much flat space is enclosed inside it. The formula is:

A = πr²

Square the radius, then multiply by pi. A circle with a radius of 5 feet has an area of π × 25 ≈ 78.54 square feet. Notice that area grows fast as the radius increases since you're squaring it. Double the radius and you quadruple the area.

If you only know the diameter, substitute r = d/2 into the formula: A = π(d/2)², which simplifies to A = πd² / 4. Same answer, just a different starting point.

Find a Circle from Any Known Value

Sometimes you don't start with the radius. Maybe you measured the circumference of a circular pool, or you know the area of a circular garden bed from a landscape plan. No problem. You can work backwards from any value.

Given circumference: r = C / (2π) and d = C / π
Given area: r = √(A / π) and d = 2√(A / π)
Given diameter: r = d / 2, then use standard formulas

Working backwards from area is the one that trips people up most often because of the square root. If the area of a circle is 50 square meters, divide by π to get about 15.915, then take the square root to get a radius of roughly 3.99 meters. The calculator handles this automatically, but it's good to know what it's doing.

Being able to start from any known value makes circle calculations practical for all kinds of real-world situations where you can't always measure the radius directly.

Arc Length and Sector Calculations

An arc is a curved portion of a circle's circumference. A sector is the pie-slice-shaped region bounded by two radii and the arc between them. Both depend on the central angle in addition to the radius.

You can express the central angle in degrees (0° to 360°) or radians (0 to 2π). The formulas look slightly different depending on which unit you use, but they produce the same result either way.

Arc Length Formula

Arc length is the distance along the curved edge of a sector. Think of it as a piece of the full circumference.

In degrees: Arc Length = (θ / 360) × 2πr
In radians: Arc Length = θr

The radian version is notably cleaner, which is one reason radians are preferred in higher-level math. For example, an arc with a central angle of 90° (which is π/2 radians) on a circle of radius 6 has a length of (90/360) × 2π × 6 = 0.25 × 37.7 ≈ 9.42 units. Using radians: (π/2) × 6 = 9.42. Same answer, fewer steps.

Sector Area Calculation

A sector's area is a fraction of the full circle's area, determined by how large the central angle is.

In degrees: Sector Area = (θ / 360) × πr²
In radians: Sector Area = (1/2) × r² × θ

Using the same example, a 90° sector of a circle with radius 6 has an area of (90/360) × π × 36 = 0.25 × 113.1 ≈ 28.27 square units. That makes sense as exactly one-quarter of the full circle's area (which is π × 36 ≈ 113.1).

Sector calculations come up in real applications like calculating the area of a pie slice for portion sizing, figuring out paint coverage on curved surfaces, or working with circular charts and gauges.

Circle Calculation Examples

Seeing the formulas applied to actual numbers makes them click a lot faster. Here are a few worked examples covering the most common scenarios.

Example 1: Known radius of 8 cm

Diameter: 2 × 8 = 16 cm
Circumference: 2 × π × 8 ≈ 50.27 cm
Area: π × 8² ≈ 201.06 cm²

Example 2: Known circumference of 31.42 inches

Radius: 31.42 / (2π) ≈ 5 inches
Diameter: 31.42 / π ≈ 10 inches
Area: π × 5² ≈ 78.54 in²

Example 3: Arc length with r = 10 ft and θ = 45°

Arc Length: (45/360) × 2π × 10 ≈ 7.85 ft
Sector Area: (45/360) × π × 100 ≈ 39.27 ft²

These examples cover the most typical starting points. The pattern is always the same: identify what you know, pick the right formula, and solve for everything else.

Common Circle Properties

Beyond the basic measurements, circles have a few geometric properties that come up regularly in math, engineering, and design. Understanding them gives you a much clearer picture of how circles behave.

Every circle has exactly one center point, and every point on the circle is the same distance from that center (that distance being the radius). That consistency is what makes circles so useful and also what makes their math relatively clean compared to other shapes.

Radius vs Diameter

The radius and diameter are closely related but used in different contexts. The radius runs from the center of the circle to any point on its edge. The diameter runs all the way across, passing through the center, connecting two points on opposite sides.

In everyday life, diameter tends to be more practical because it's something you can physically measure with a ruler or tape measure across an object. The radius, on the other hand, is more useful in calculations because most circle formulas are built around it.

One quick way to remember it: the diameter is the longest straight line you can draw inside a circle. Any other chord (a line connecting two points on the circle without passing through the center) will be shorter.

Chord, Tangent, and Secant

These three terms describe different ways a line can relate to a circle.

Chord: A line segment connecting any two points on the circle. The diameter is a special chord that passes through the center.
Tangent: A line that touches the circle at exactly one point. It's always perpendicular to the radius at that point of contact.
Secant: A line that intersects the circle at two points, passing through it entirely.

Tangent lines show up a lot in calculus and physics, especially when dealing with circular motion or curves. Chords are useful in geometry proofs and in practical applications like calculating distances across circular structures. Secants are less common in everyday use but important in more advanced circle theorems.

Unit Circle and Coordinate Geometry

The unit circle is a circle with a radius of exactly 1, centered at the origin (0, 0) on a coordinate plane. It sounds simple, but it's one of the most important tools in trigonometry and math generally.

Because the radius is 1, every point on the unit circle has coordinates (cos θ, sin θ), where θ is the angle measured counterclockwise from the positive x-axis. This is where sine, cosine, and tangent connect to geometry in a concrete way. The x-coordinate of a point on the unit circle equals the cosine of the angle, and the y-coordinate equals the sine.

The equation of any circle on a coordinate plane is:

(x - h)² + (y - k)² = r²

Here, (h, k) is the center and r is the radius. For the unit circle specifically, h = 0, k = 0, and r = 1, so it simplifies to x² + y² = 1.

This equation becomes useful any time you need to determine whether a point lies inside, on, or outside a circle. If you plug in a point and get a value less than r², it's inside. Equal to r², it's on the circle. Greater than r², it's outside. Clean, consistent, and incredibly handy for coordinate geometry problems.