Find the area, circumference, and diameter of any circle from its radius — with every step shown. Free, exact, and mobile-friendly.
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A circle is the set of all points in a plane that are the same distance from a fixed center point. That distance is the radius (r), and nearly every measurement of a circle — its diameter, its circumference, and its area — flows from this single number. Our free Circle Calculator takes the radius and instantly returns the diameter, circumference, and area, showing each computation step so you can follow along and check your own work.
Circles appear everywhere: wheels, pipes, pizza, coins, planetary orbits, and the cross-section of a tree trunk. Whether you are a student learning geometry, a gardener sizing a round flower bed, or an engineer specifying a pipe, the three core circle formulas are tools you will use again and again. This calculator handles the arithmetic — including the irrational constant π (pi) — so you get precise answers without a scientific calculator.
Beyond the three headline measurements, the page below teaches you how the formulas work: why the circumference is 2πr, why the area is πr², and how the two are related. You will find worked examples with real numbers, practical scenarios from construction and cooking, the mistakes students most often make (like confusing radius with diameter), and answers to the most common questions about circles.
Every circle is defined by its center and its radius — the distance from the center to any point on the curve. Double the radius and you get the diameter (d = 2r), the longest straight line you can draw across the circle, passing through the center. The diameter is the measurement you most often take in real life: wrap a tape measure around a pipe and halve the reading, or measure straight across a pizza.
The constant π (pi) is the ratio of any circle’s circumference to its diameter, approximately 3.14159. It is irrational, meaning its decimal digits never repeat and never end, which is why circle calculations almost always involve rounding. For schoolwork, 3.14 is usually enough; for engineering, calculators carry pi to 10+ digits. This calculator uses JavaScript’s full-precision Math.PI, so answers are as exact as your device allows.
A beautiful fact connects the circumference and the area: if you cut a circle into many thin wedges and rearrange them, they approximate a rectangle whose width is the radius r and whose length is half the circumference (πr). Multiply them and you get πr² — the area formula. This is not a coincidence; it is the deep reason the same constant π appears in both formulas, and understanding it turns the formulas from memorized symbols into something you can reconstruct any time.
Diameter: d = 2r • Circumference: C = 2πr = πd • Area: A = πr²
Where r is the radius and π ≈ 3.14159. If you know the diameter instead of the radius, halve it first (r = d ÷ 2).
Given r = 5 cm. Diameter: d = 2 × 5 = 10 cm. Circumference: C = 2π × 5 = 10π ≈ 31.4159 cm. Area: A = π × 5² = 25π ≈ 78.5398 cm². Notice the units: length measurements (diameter, circumference) are in cm, but area is in square centimeters (cm²) because it measures a surface.
Here we know the diameter, so first find the radius: r = 70 ÷ 2 = 35 cm. Circumference: C = πd = π × 70 ≈ 219.9115 cm — this is how far the bike travels in one wheel revolution, about 2.2 meters. Area: A = π × 35² = 1225π ≈ 3848.4510 cm². The circumference-via-diameter shortcut (C = πd) saves a step whenever the diameter is the given measurement.
Take r = 4 m: C = 2π × 4 = 8π ≈ 25.1327 m and A = π × 16 = 16π ≈ 50.2655 m². Now double the radius to r = 8 m: C = 2π × 8 = 16π ≈ 50.2655 m (exactly doubled) but A = π × 64 = 64π ≈ 201.0619 m² (exactly quadrupled). This is the square-cube style scaling law of circles: circumference scales linearly with r, area scales with r². It is why a 16-inch pizza has four times the area of an 8-inch pizza, not twice — and usually a much better deal.
Maria wants a circular flower bed with a radius of 1.5 m. How much edging does she need, and how much soil to fill it 0.3 m deep? First the edging: C = 2π × 1.5 = 3π ≈ 9.4248 m, so she buys 10 m of edging. The area is A = π × 1.5² = 2.25π ≈ 7.0686 m²; multiply by the 0.3 m depth for the soil volume: 7.0686 × 0.3 ≈ 2.12 m³ of soil. One radius measurement answered both questions.
A second scenario: a plumber must choose between two pipes. Pipe X has a 10 cm diameter; pipe Y has a 14 cm diameter. The cross-sectional areas are A = π(d/2)²: pipe X gives π × 25 ≈ 78.54 cm², pipe Y gives π × 49 ≈ 153.94 cm². The 14 cm pipe carries nearly twice the water of the 10 cm pipe even though its diameter is only 40% larger — area’s r² scaling again. Sizing by diameter alone would badly underestimate the difference.
The number π is defined as the ratio C/d, and the remarkable fact is that this ratio is the same for every circle, from a coin to a planet’s orbit. Ancient civilizations noticed this: the Babylonians used 3.125, the Egyptians 3.1605, and Archimedes (c. 250 BCE) trapped π between 3.1408 and 3.1429 by inscribing polygons with up to 96 sides inside a circle. The more sides the polygon has, the closer its perimeter hugs the circle — the ancestor of the calculus idea of a limit.
Pi is irrational (proved by Lambert in 1761) and even transcendental (Lindemann, 1882), which means it cannot be the root of any polynomial with rational coefficients. A famous consequence: “squaring the circle” — constructing a square with exactly the same area as a given circle using only compass and straightedge — is mathematically impossible. Transcendence also guarantees the digits of π never fall into a repeating pattern, which is why computers have calculated trillions of digits without finding the end.
In practice you never need trillions of digits. NASA’s Jet Propulsion Laboratory uses only 15 decimal places of π for interplanetary navigation — enough to calculate the circumference of a circle spanning the observable universe with an error smaller than a hydrogen atom. For everyday work, 3.14159 (five significant figures) is plenty; this calculator keeps full double precision internally and rounds only for display.
A running track’s circular ends have a circumference of 400 m? No — suppose instead a circular fountain has circumference C = 62.83 m. Find r: from C = 2πr, r = C ÷ (2π) = 62.83 ÷ 6.2832 ≈ 10 m. Then the area is A = π × 100 ≈ 314.16 m². Rearranging the formulas is just as important as applying them forward.
A donut-shaped garden (an annulus) has outer radius 6 m and inner radius 4 m. Its area is the difference of two circle areas: π(6² − 4²) = π(36 − 16) = 20π ≈ 62.83 m². Whenever a shape is “a circle minus a circle,” subtract the areas — the π factors out neatly.
A circle has radius 12 in. Find its diameter, circumference, and area. (Answers: d = 24 in, C ≈ 75.3982 in, A ≈ 452.3893 in².)
A circular rug has an area of 50.27 ft². What is its radius? (Hint: r = √(A/π). Answer: r ≈ 4 ft.)
Two circles have radii 3 cm and 6 cm. How many times larger is the big circle’s area? (Answer: 4 times — the r² scaling law.)
A pizzeria sells an 8-inch pizza for $8 and a 16-inch pizza for $20. Which is the better deal? Compare areas: the 8-inch has radius 4, area = 16π ≈ 50.27 in², costing $8 ÷ 50.27 ≈ $0.159 per square inch. The 16-inch has radius 8, area = 64π ≈ 201.06 in², costing $20 ÷ 201.06 ≈ $0.099 per square inch. The large pizza delivers nearly 40% more food per dollar — the r² law working in the customer’s favor.
The same math governs manufacturing: doubling a pipe’s diameter quadruples its cross-section and roughly quadruples material cost, while doubling only its circumference doubles the surface to paint. Engineers who internalize which circle measurement scales linearly and which scales quadratically avoid expensive sizing mistakes.
The Circle Calculator computes a circle’s diameter (d = 2r), circumference (C = 2πr), and area (A = πr²) from the radius r, with π ≈ 3.14159. Key facts: circumference scales linearly with radius while area scales quadratically; halve any given diameter before applying the area formula; keep full pi precision until final rounding.
The area is A = πr², where r is the radius and π ≈ 3.14159. Square the radius first, then multiply by pi. For example, a circle with radius 5 cm has area π × 25 ≈ 78.54 cm².
Use C = 2πr if you know the radius, or C = πd if you know the diameter. A circle with radius 5 cm has circumference 2 × π × 5 ≈ 31.42 cm.
The radius is the distance from the center to the edge; the diameter is the distance straight across through the center, exactly twice the radius (d = 2r). Always confirm which one a problem gives you.
Pi (π) is the constant ratio of a circle’s circumference to its diameter, the same for every circle. Because both circumference and area are defined relative to the radius, π appears in both formulas: C = 2πr and A = πr².
Rearrange A = πr² to r = √(A ÷ π). For an area of 78.54 cm²: r = √(78.54 ÷ 3.14159) = √25 = 5 cm.
Rearrange C = 2πr to r = C ÷ (2π). For a circumference of 31.42 cm: r = 31.42 ÷ 6.2832 ≈ 5 cm.
No — it quadruples the area, because area depends on r². Doubling the radius exactly doubles the circumference (which depends on r linearly). This is why large pizzas are such good value.
Use the same length unit throughout (all cm, or all inches). The circumference and diameter come out in that unit; the area comes out in square units (cm², in²). Never mix units without converting first.
Cite this page as: Circle Calculator. Sigma Calculator. Retrieved October 11, 2026, from https://sigmacalculator.net/math-calculators/circle-calculator/
Last updated: October 11, 2026