Area of a Circle Calculator

The area of a circle tells you how much two-dimensional space is enclosed by its curved edge. This Area of a Circle Calculator finds circle area from the radius, diameter or circumference, and it can also work backward from a known area to calculate the radius, diameter and circumference.

For more detailed geometry, switch to Advanced Circle mode to calculate the area of a semicircle, quarter circle, circular sector or annulus. Results include the circle area formula, substituted values, circumference or perimeter, exact π form, decimal answers and conversions between common metric and imperial square units.

How to Use the Area of a Circle Calculator

Choose the circle measurement you already know: radius, diameter, circumference or area. Enter the value, select the unit and choose how many decimal places you want in the answer.

If you enter a radius, the calculator applies the standard circle area formula directly. If you enter a diameter, it first divides the diameter by two to find the radius. If circumference is known, the calculator works backward using the circumference formula. If area is already known, it solves for radius and then calculates the remaining circle measurements.

The result shows the area first because this page is designed specifically as an area of a circle calculator. It also returns radius, diameter, circumference, area in terms of π, converted square units and a step-by-step explanation of the calculation.

Switch to Advanced Circle when you are working with only part of a circle. Advanced mode calculates semicircles, quarter circles, sectors and annuli.

If you need to calculate other two-dimensional shapes such as rectangles, squares, triangles, trapezoids or ellipses, use our Area Calculator.

What Is the Area of a Circle?

The area of a circle is the amount of two-dimensional space enclosed inside its circumference.

Area is different from circumference. Area measures the space inside the circle, while circumference measures the distance around the curved edge.

Because area measures two dimensions, the result is expressed in square units. A circle measured in centimetres has an area in square centimetres. A circle measured in feet has an area in square feet.

The standard formula is:

The letter r represents the radius, and π is pi, approximately 3.141592653589793.

Khan Academy describes circle area as the amount of space the circle covers and uses the same πr² formula. See its area of a circle lesson.

Circle Area Formula

The main circle area formula is one of the most important equations in geometry:

In the formula:

• A = area of the circle
• π = pi, approximately 3.14159
• r = radius
• r² = radius multiplied by itself

The radius is squared before multiplying by π. This is important because doubling a circle’s radius does not merely double its area.

If radius doubles, r² becomes four times as large. The circle therefore has four times the area.

This squared relationship is why small changes in radius can create surprisingly large changes in area.

Area of a Circle Formula

The phrases area of a circle formula, area of circle formula, circle area formula and formula for area of a circle all refer to the same equation: A = πr².

The form of the equation can change when the problem gives diameter or circumference instead of radius.

From radius:

From diameter:

From circumference:

All three equations calculate the same area. They simply begin with different known measurements.

How to Find the Area of a Circle

To find the area of a circle when you know the radius, square the radius and multiply by π.

Suppose the radius is 6 cm.

The exact answer is 36π square centimetres. Using a decimal value of π gives approximately 113.10 cm².

If you are checking the squaring, multiplication or other arithmetic manually, our Simple Math Calculator can help with the intermediate calculations.

Area of a Circle With Radius

Radius is the simplest measurement to use when calculating circle area because it can be inserted directly into A = πr².

The radius is the distance from the centre of the circle to any point along the circumference.

For a circle with radius 10:

The unit depends on the original measurement. A radius of 10 inches gives approximately 314.16 square inches. A radius of 10 metres gives approximately 314.16 square metres.

Area of a Circle With Diameter

A very common circle problem gives the diameter rather than the radius.

The diameter runs all the way across a circle through its centre. It is twice the radius.

Once radius is known, use A = πr².

For a circle with diameter 12 cm:

You can also use the direct diameter formula:

Both methods produce the same result.

Area of a Circle From Circumference

If you know the circumference but do not know the radius or diameter, you can still calculate the circle’s area.

Start with the circumference formula:

Rearrange it to find radius:

You can then substitute that radius into A = πr².

The equations can also be combined into one direct formula:

For a circumference of 31.4159 cm, the radius is approximately 5 cm and the area is approximately 78.54 cm².

How to Find Radius From the Area of a Circle

Sometimes the area is known and the radius is the missing measurement.

Start with:

Divide both sides by π:

Take the square root:

If a circle has an area of 314.159 square centimetres:

This calculator includes Area as a known-value option, so it performs the inverse calculation automatically.

For equations where you need to rearrange formulas or solve a broader unknown-variable problem, use our Algebra Calculator.

How to Find Diameter From the Area of a Circle

Once radius is calculated from area, diameter is simply twice the radius.

For an area of 78.54 square units:

The same relationship works whether the units are millimetres, inches, feet, metres or another consistent length unit.

Circumference of a Circle

Circumference is the distance around the outside of a circle.

It is the circular equivalent of perimeter.

The two common circumference formulas are:

Because diameter equals twice the radius, these formulas are equivalent.

For a radius of 5 cm:

Math Is Fun also describes circumference as π multiplied by diameter, with radius equal to half the diameter. See its circle reference.

Radius, Diameter and Circumference of a Circle

Radius, diameter, circumference and area are all connected.

Knowing any one of these measurements is enough to calculate the other three.

Why Is Pi Used to Calculate the Area of a Circle?

Pi, written π, is the constant ratio between a circle’s circumference and its diameter.

The decimal representation begins 3.1415926535 and continues without terminating or repeating.

Pi appears in both circumference and area formulas because it describes the geometry common to every circle regardless of size.

For ordinary hand calculations, π is sometimes rounded to 3.14. A calculator can use a much more precise value.

This calculator uses the JavaScript Math.PI constant rather than limiting calculations to 3.14.

Area of a Circle Using 3.14

School exercises sometimes specifically instruct students to use 3.14 for π.

For radius 7:

Using a more precise value of π produces approximately 153.938. The difference comes entirely from rounding π.

When a problem asks for an exact answer, leave the result in terms of π, such as 49π.

Area of a Semicircle

A semicircle is exactly half of a full circle.

Its area is therefore half of πr²:

For a radius of 4:

Be careful when calculating the perimeter of a semicircle. The boundary includes the curved half-circumference plus the straight diameter.

Area of a Quarter Circle

A quarter circle contains one-fourth of the area of its parent circle.

For radius 8:

A quarter circle is equivalent to a 90-degree circular sector.

Area of a Circle Sector

A sector is a wedge-shaped portion of a circle formed by two radii and the arc between them.

If the central angle is measured in degrees, calculate the fraction of the full circle by dividing the angle by 360.

For radius 10 and an angle of 72 degrees:

The sector represents 20% of the full circle. If you need to calculate an angle or portion as a percentage separately, use our Percentage Calculator.

Area of an Annulus or Circular Ring

An annulus is the ring-shaped region between two circles that share the same centre.

Calculate the outer circle area and subtract the inner circle area.

Uppercase R represents the outer radius, while lowercase r represents the inner radius.

For an outer radius of 8 and inner radius of 5:

This calculation is useful for circular borders, washers, rings, round walkways, pipe cross sections and similar shapes.

Circle Area and Square Units

One of the most common geometry mistakes is writing an area answer using ordinary linear units.

A radius measures length, so it might be 5 cm. Area measures two dimensions, so the final answer is in cm².

Area conversion also requires squaring the length conversion factor.

One foot equals 12 inches, but one square foot does not equal 12 square inches.

The calculator handles these area conversions automatically.

Square Feet of a Circle

Construction, landscaping and home-improvement projects often need the area of a circular space in square feet.

If a round patio has a radius of 10 feet:

If the measurement was taken in inches instead, convert the measurements or convert the final area into square feet correctly.

Our full Area Calculator also handles rectangular square footage, multiple geometric shapes and additional area conversions.

Area of a Circle Examples

Example 1: Radius of 3

For radius 3:

Example 2: Radius of 5

Example 3: Radius of 10

Example 4: Diameter of 20

A diameter of 20 gives a radius of 10, so the area is again approximately 314.16 square units.

Example 5: Circumference of 100

First calculate radius:

Then calculate area:

Why Area Increases Faster Than Circumference

Circumference changes directly with radius because C = 2πr. Double the radius and the circumference doubles.

Area changes with radius squared because A = πr². Double the radius and area increases by four times.

Triple the radius and area increases by nine times.

This relationship is visible in the calculator’s circle-size comparison table.

Area of Circle vs. Circumference

Area and circumference describe different properties and therefore use different units.

A circle with radius 5 cm has:

• Radius = 5 cm
• Diameter = 10 cm
• Circumference ≈ 31.42 cm
• Area ≈ 78.54 cm²

Circumference is a length, while area is a surface measurement.

Do not compare 31.42 cm directly with 78.54 cm² as though they were the same type of quantity.

Area of a Circle From Diameter Formula

If diameter is the measurement you already have, you do not technically need to calculate radius as a separate written step.

Since r = d/2:

Squaring the fraction gives:

This direct form is convenient in manufacturing, pipe sizing, construction and other situations where a circular object is normally measured across its full width.

Circle Area From Circumference Formula

Combining C = 2πr with A = πr² gives a direct formula based only on circumference:

This is useful when a flexible tape has been wrapped around a circular object and circumference is easier to measure than diameter.

For example, trees, tanks, columns and pipes may be easier to measure around than straight through.

Real-World Uses for Circle Area

Circle area is not limited to geometry homework. The same formula is used whenever the amount of surface inside a circular boundary matters.

• Round patios and concrete pads
• Garden beds
• Swimming pools
• Round tables
• Pipe and tube cross sections
• Storage tanks
• Wheels and discs
• Pizza and food portions
• Manufacturing parts
• Architecture and design
• Landscaping
• Painting and coating estimates

Once a circular area becomes the base of a three-dimensional cylinder, multiplying that area by height gives cylinder volume.

For tanks, pipes, drums and cylinders, use our Volume Calculator to continue from area into three-dimensional volume.

Area of a Circular Garden

Suppose a round garden has a radius of 12 feet.

If soil, mulch or grass coverage is sold according to square footage, the theoretical surface area is approximately 452.39 square feet.

Real purchasing estimates may require extra material for waste, overlap, uneven boundaries or installation conditions.

Area of a Round Patio

A round patio with a 20-foot diameter has a radius of 10 feet.

This gives the surface footprint. Material estimates for concrete, gravel or soil may also require depth, which turns the problem into a volume calculation.

Circle Area and Ratios

Similar circles have predictable area ratios.

If one circle has twice the radius of another circle, the ratio of their radii is 2:1.

The ratio of their areas is the square of the radius ratio:

For more ratio calculations, scale ratios and proportional relationships, use our Ratio Calculator.

Circle Area as a Percentage of a Square

Imagine a circle fitted exactly inside a square so that the circle’s diameter equals the square’s side length.

Let the circle diameter be d. The square has area d².

The circle has area:

Divide the circle area by the square area:

The inscribed circle therefore covers approximately 78.54% of the square’s area.

Common Area of a Circle Mistakes

Using Diameter as Radius

This is one of the biggest circle-area errors.

If diameter is 10, the radius is 5. Using 10 as the radius calculates the area of a circle with diameter 20 instead.

Because radius is squared, the incorrect result is four times the correct area.

Forgetting to Square the Radius

The formula is πr², not πr.

Multiplying π by radius alone does not calculate circle area.

Confusing Area With Circumference

Area uses A = πr². Circumference uses C = 2πr or C = πd.

Area uses square units; circumference uses ordinary length units.

Writing the Wrong Units

A circle measured in metres has circumference in metres but area in square metres.

Mixing Units

If part of a geometry problem uses centimetres and another measurement uses metres, convert the measurements before applying a formula that combines them.

Rounding Too Early

Using 3.14 for π or rounding radius too early can introduce additional error.

Keep extra precision during intermediate calculations and round the final answer unless the problem specifically asks you to use a particular value of π.

Why the Calculator Shows Exact π Form

Some school problems ask for an exact answer instead of a rounded decimal.

If radius is 6, area can be written exactly as 36π square units.

The decimal 113.097… is an approximation because π has infinitely many decimal digits.

Showing both forms makes it easier to match the answer format required by a teacher, textbook or test.

Area vs. Surface Area vs. Volume

Area measures a two-dimensional region. Surface area measures the combined outside surfaces of a three-dimensional object. Volume measures the space contained inside a three-dimensional object.

A circle itself has area. A sphere has surface area and volume. A cylinder has circular base areas, a curved surface area and volume.

This distinction matters because the units change:

• Length → units such as cm or ft
• Area → square units such as cm² or ft²
• Volume → cubic units such as cm³ or ft³

The Area Calculator handles additional surface-area problems, while the Volume Calculator handles three-dimensional capacity and volume.

Area of a Circle Calculator Frequently Asked Questions

Final Thoughts

The area of a circle is built around one simple relationship: A = πr². Once you know the radius, the calculation is straightforward. If you only know diameter or circumference, those measurements can be converted back to radius first.

Keep the difference between radius and diameter clear, square the radius before multiplying by π and write the final answer in square units. Those three habits prevent most circle-area mistakes.

The Advanced Circle modes extend the same geometry to semicircles, quarter circles, sectors and circular rings. Each calculation begins with the full circle or a difference between circles, then uses the appropriate fraction or subtraction.

When circle area is only one part of a larger problem, continue with the related calculators on the site. The Area Calculator handles additional two-dimensional shapes, while the Volume Calculator can use a circular base to calculate cylinders, tanks and other three-dimensional objects.