Circle Calculator, Find Radius, Diameter, Circumference, and Area
Radius, diameter, circumference, area, arc, sector, chord, and segment
| What You Want | What You Know | Formula |
|---|---|---|
| Diameter | Radius | d = 2r |
| Radius | Diameter | r = d ÷ 2 |
| Circumference | Radius | C = 2πr |
| Area | Radius | A = πr² |
| Radius | Area | r = √(A ÷ π) |
| Arc Length | Radius + Angle | L = r × θ (radians) |
| Sector Area | Radius + Angle | A = 0.5 × r² × θ |
| Chord Length | Radius + Angle | c = 2r × sin(θ ÷ 2) |
Related Calculators
A circle is defined by a single number. Give a circle calculator any one measurement, the radius, diameter, circumference, or area, and the other three are already fixed. There is no guesswork involved, just a set of relationships that hold true for every circle.
The tool above solves all four from whatever you enter. It also handles the parts of a circle that depend on an angle, the arc, sector, chord, and segment, which come up once you move past a full circle.
Each formula below is the standard geometry relationship, shown so you can see exactly how the answer is reached rather than trusting a black box.
The Four Core Circle Measurements
Every circle problem starts with these four values. Once you know one, the rest follow.
Radius
The radius is the distance from the center of a circle to any point on its edge. It stays the same no matter which direction you measure, since every point on the edge sits an equal distance from the center.
Diameter
The diameter is a straight line that passes through the center and touches the edge on both sides. It is always exactly twice the radius, which makes it the easiest value to convert from.
Circumference
The circumference is the distance all the way around the outside of the circle. It is a length, so it gets measured in linear units like inches, feet, or centimeters.
Area
The area is the amount of space enclosed inside the circle. Because it measures a surface rather than a distance, area is always expressed in square units, such as square inches or square meters.
How to Find Radius From Diameter, Circumference, or Area
The radius is the value most other formulas depend on, so it is worth knowing all three ways to reach it.
Radius From Diameter
Divide the diameter by two. This is the most direct case, since the diameter is just two radii laid end to end.
A circle with a 10 cm diameter has a 5 cm radius.
Radius From Circumference
Divide the circumference by two times pi.
A circle with a circumference of about 31.42 cm works out to a 5 cm radius.
Radius From Area
Divide the area by pi, then take the square root of the result. The square root step undoes the squaring built into the area formula.
A circle with an area of about 78.54 cm² gives a 5 cm radius.
Radius Formulas
- From diameter: r = d ÷ 2
- From circumference: r = C ÷ (2π)
- From area: r = √(A ÷ π)
How to Find Diameter From Radius, Circumference, or Area
The diameter is often the number you measure first with a ruler, but it is just as easy to calculate from the other values.
Diameter From Radius
Multiply the radius by two. Since the diameter runs through the center from edge to edge, it is simply two radii combined.
Diameter From Circumference
Divide the circumference by pi. This works because the circumference is always pi times the diameter, so reversing it takes a single step.
Diameter From Area
There is no direct jump from area to diameter, so take it in two moves. Find the radius from the area first, then double that radius to get the diameter.
How to Find Circumference From Radius or Diameter
Circumference tells you how far it is around a circle, which matters any time you need to wrap, edge, or fence something round.
Circumference From Radius
Multiply two times pi times the radius. This is the standard form, written as C = 2πr.
Circumference From Diameter
Multiply pi times the diameter. Since the diameter is twice the radius, this shorter version, C = πd, gives the same answer with one less step.
Why Circumference and Diameter Share a Fixed Ratio
Divide any circle’s circumference by its diameter and you always get the same number. That number is pi, roughly 3.14159, and it holds true for every circle regardless of size.
This relationship is where pi comes from in the first place. It is not a rounded convenience, it is the actual ratio built into every circle.
How to Find the Area of a Circle
Area covers the space inside a circle, which is what you need for flooring, paint, turf, or any material that fills a round surface.
Area From Radius
Square the radius, then multiply by pi. The formula is A = πr².
Squaring the radius comes first, before multiplying by pi. Skipping that step is the most common area mistake.
Area From Diameter
Halve the diameter to get the radius, then apply the same area formula. Working straight from the diameter without halving it first will overstate the area significantly.
Why Area Uses Square Units
Area measures a two dimensional surface, so its unit is always squared. A radius measured in inches produces an area in square inches, and a radius in meters gives square meters.
Arc Length and Sector Area
Once you slice into a circle rather than measuring the whole thing, the central angle comes into play.
What an Arc and a Sector Are
An arc is a portion of the curved edge of the circle. A sector is the pie slice shape bounded by two radii and the arc between them.
Both depend on the central angle, the angle at the center between those two radii, not just the radius alone.
Calculating Arc Length
Multiply the radius by the central angle measured in radians. If your angle is in degrees, convert it to radians first, since the formula only works in radians.
A full turn is 2π radians, which equals 360 degrees.
Calculating Sector Area
Take half of the radius squared, then multiply by the central angle in radians. A sector that spans the full 360 degrees simply returns the area of the whole circle, which is a useful way to check the math.
Chord Length and Segment Area
A chord and a segment describe the straight cut across a circle and the piece it leaves behind.
Chord Versus Arc
A chord is the straight line joining the two ends of an arc. The arc is the curved path between those same two points.
They start and end at the same places, but the chord is always shorter, since a straight line is the shortest distance between two points.
Calculating Chord Length
Multiply two times the radius by the sine of half the central angle. This gives the straight distance across, written as c = 2r × sin(θ ÷ 2).
Calculating Segment Area
Find the sector area first, then subtract the triangle formed by the two radii and the chord. What remains is the segment, the region trapped between the chord and the arc.
A Quick Formula Reference for Every Circle Calculation
Here is every relationship covered above in one place, so you can find the right formula without scrolling back through each section.
| What You Want | What You Know | Formula |
|---|---|---|
| Diameter | Radius | d = 2r |
| Radius | Diameter | r = d ÷ 2 |
| Circumference | Radius | C = 2πr |
| Circumference | Diameter | C = πd |
| Area | Radius | A = πr² |
| Radius | Circumference | r = C ÷ (2π) |
| Radius | Area | r = √(A ÷ π) |
| Diameter | Area | d = 2 × √(A ÷ π) |
| Arc Length | Radius and angle | L = r × θ (θ in radians) |
| Sector Area | Radius and angle | A = ½ × r² × θ (θ in radians) |
| Chord Length | Radius and angle | c = 2r × sin(θ ÷ 2) |
For any formula using an angle, remember that θ has to be in radians. Multiply degrees by π ÷ 180 to convert.
Where Circle Calculations Show Up in Real Life
These formulas are not just for geometry class. They come up any time something round needs measuring, ordering, or building.
Area and Square Footage for Projects
Area is what you need when buying material to cover a round surface. Flooring for a circular room, paint for a round tabletop, or turf for a curved garden bed all start with the area formula.
Converting that area into square feet gives you the number most suppliers actually quote against.
Circumference for Real Objects
Circumference measures the distance around, which matters for anything you wrap or edge. Fencing around a circular garden or pool, edging trim, or the distance around a pipe or tank all rely on it.
Sectors and Segments in Everyday Shapes
Partial circle math shows up more often than people expect. A slice of pizza is a sector, and the curved top of an arched window can be a segment.
Pie chart wedges use the exact same sector calculation to size each slice by its angle.
Common Mistakes in Circle Calculations
Most wrong answers come from a handful of repeat errors, not from the formulas themselves.
Using Diameter Where the Formula Needs Radius
The area and circumference-from-radius formulas both expect the radius. Plugging in the diameter by mistake throws the answer off badly. Divide the diameter by two first if that is the value you have.
Forgetting to Square the Radius for Area
Area uses radius squared, not the radius on its own. Multiplying by pi without squaring first gives a much smaller number than the real area.
Mixing Degrees and Radians in Arc or Sector Math
Arc length and sector area formulas only work with the angle in radians. Entering degrees directly, without converting, produces a result that is off by a wide margin.
Reporting Area in Linear Units Instead of Square Units
Area is always in square units. Writing an area answer in plain inches instead of square inches is a small slip that changes what the number actually means.
Worked Examples
Running real numbers through the formulas makes them easier to trust and repeat on your own.
Starting From a Radius of 5
With a radius of 5 cm, double it for a diameter of 10 cm. The circumference comes to about 31.42 cm, and the area works out to roughly 78.54 cm².
Starting From a Diameter of 10
A diameter of 10 cm halves to a 5 cm radius. From there the area is about 78.54 cm² and the circumference is about 31.42 cm, matching the first example exactly.
Starting From an Area of 154
Given an area of 154 cm², divide by pi and take the square root to get a radius of about 7 cm. Doubling that gives a diameter close to 14 cm, and the circumference lands near 44 cm.
A 90 Degree Sector of a 7 cm Circle
For a 7 cm radius and a 90 degree central angle, the arc length is about 11 cm and the sector area is about 38.48 cm². The chord across the sector measures about 9.9 cm, and the segment between the chord and arc covers about 13.98 cm².
Frequently Asked Questions About Circles
How do you find the radius from the area
Divide the area by pi, then take the square root of that result. For example, an area of 78.54 cm² gives a radius of 5 cm.
How do you find the diameter from the circumference
Divide the circumference by pi. Since circumference equals pi times diameter, this reverses the relationship in one step.
What is the difference between radius and diameter
The radius runs from the center to the edge. The diameter runs all the way across through the center, making it exactly twice the radius.
What is the difference between a chord and an arc
A chord is the straight line between two points on the circle. An arc is the curved path between those same two points, and it is always longer than the chord.
How many radians are in a full circle
A full circle contains 2π radians, which equals 360 degrees. Half a circle is π radians, or 180 degrees.
Why does the area formula use pi
Pi is the fixed ratio that ties a circle’s dimensions together. It appears in the area formula because the space inside a circle scales with the square of the radius, adjusted by that same constant.
Conclusion
A circle calculator turns one known measurement into a complete picture. Enter a radius, diameter, circumference, or area, and every other value is already determined by the fixed relationships between them.
For anything beyond a full circle, the arc, sector, chord, and segment tabs handle the parts that depend on a central angle. Just keep your units consistent and convert degrees to radians before any angle-based calculation, and the results will hold up every time.