Area of a Circle Formula & Calculator

This calculator finds the area of a circle from its radius, diameter, or circumference. Enter any one of those three values and you get the area in square units, plus the other two values for reference. The formula is A = πr², where A is area and r is radius.

Formula

The area of a circle is given by:

A = π × r²

• A = area (square units, e.g., m², ft², cm²)
• π (pi) ≈ 3.14159 (a constant)
• r = radius (distance from center to edge, in same unit as A)

If you know the diameter d instead of the radius, use A = π × (d/2)². If you know the circumference C, use A = C² / (4π).

Worked Example

Using the default values from the calculator: radius r = 5 m.

1. Square the radius: 5² = 25
2. Multiply by π: 25 × 3.14159 = 78.53975
3. Round to two decimals: 78.54 m²

The calculator shows 78.54 m² as the area, matching the formula.

Why the Formula Works

The formula A = πr² has been known for thousands of years. One intuitive way to see it: imagine cutting a circle into many thin wedges (like pizza slices) and rearranging them into a rectangle. The width of that rectangle is the radius r, and the length is half the circumference, which is πr. So area = r × πr = πr². The more slices you cut, the closer the shape becomes to a true rectangle, confirming the formula.

Another way: the area of a circle is proportional to the square of its radius. If you double the radius, the area quadruples. This is why a 10-inch pizza is not twice as much food as a 5-inch pizza—it’s four times as much.

When to Use This Calculation

You need the area of a circle whenever you’re working with round surfaces or cross-sections. Common real-world uses include:

• Finding the area of a circular garden, patio, or rug
• Calculating the cross-sectional area of a pipe or wire
• Determining the surface area of a circular table or coin
• Estimating the area of a pizza or cake for serving sizes
• Working with circular motion or physics problems

The result is always in square units—if you input meters, you get square meters; if you input feet, you get square feet. For large areas, you may want to convert to acres or hectares, but the basic formula stays the same.

Common Mistakes and How to Avoid Them

The most frequent error is mixing up radius and diameter. The radius is half the diameter. If someone gives you a diameter of 10 cm, the radius is 5 cm—using 10 in the formula gives four times the correct area.

Another common mistake is forgetting to square the radius. Some people compute π × r instead of π × r². For r = 5, π × 5 = 15.7, which is far too small. Always square first.

Unit confusion also causes problems. If you measure the radius in inches, the area will be in square inches. To get square feet, you must convert inches to feet before squaring (or divide the square inches by 144). Never mix units in the same calculation.

Rounding too early can also skew results. If you round π to 3.14, the error is small for most purposes, but for precise engineering work, use more digits (3.14159 or the π button on a calculator).

Limitations of the Formula

The formula applies to perfect circles in a flat plane. Real-world objects may not be perfectly round, so measurements can be slightly off. For an ellipse, the formula is different (π × a × b, where a and b are the semi-axes).

Also, the formula gives the area of the circle itself, not the surface area of a sphere. If you need the surface area of a ball, use 4πr². If you need the volume of a sphere, use (4/3)πr³. These are common confusions.

For very large or very small circles, the formula still works, but measurement precision becomes critical. A tiny error in the radius gets squared, so it becomes a larger error in the area.

Alternatives and Related Formulas

If you only know the circumference, you can find the area using A = C² / (4π). If you know the area and want the radius, rearrange to r = √(A/π). The same formula is used in many geometry problems, and it’s also the basis for calculating the area of a sector (a slice of a circle) by multiplying the full area by the fraction of the circle that the sector covers.

For a circle inscribed in a square, the circle’s area is π/4 (about 78.5%) of the square’s area. This is a handy quick check: if you have a square of side length 2r, its area is 4r², and the circle inside it is πr², so the ratio is π/4.

Frequently Asked Questions

How do I find the area if I only have the diameter?

Divide the diameter by 2 to get the radius, then use A = πr². For example, a circle with a diameter of 10 cm has a radius of 5 cm, so the area is π × 25 ≈ 78.54 cm².

Can I use this formula for a semi-circle?

Yes, a semi-circle is half of a full circle, so its area is (πr²)/2. For a quarter-circle, it’s (πr²)/4.

What if I know the circumference instead?

Use the formula A = C² / (4π). For example, if the circumference is 31.4159 m, then A = (31.4159²) / (4π) ≈ 78.54 m².

Why is π approximately 3.14159?

π is the ratio of a circle’s circumference to its diameter. It’s an irrational number, meaning its decimal expansion never ends or repeats. The value 3.14159 is accurate to five decimal places, which is enough for most practical purposes.

How do I convert the area to square feet if I measured in inches?

First calculate the area in square inches (using inches for the radius), then divide by 144 to get square feet. For example, a radius of 12 inches gives area π × 144 ≈ 452.39 in², which is about 3.14 ft².

Is the area of a circle always greater than its circumference?

Not necessarily. For a circle with radius less than 2, the circumference (2πr) is larger than the area (πr²). For example, r = 1 gives area ≈ 3.14 and circumference ≈ 6.28. For r = 3, area ≈ 28.27 and circumference ≈ 18.85, so the area is larger. The crossover happens at r = 2.

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in r | Radius | m | 5 | 0
in d | Diameter | m | 10 | 0
in c | Circumference | m | 31.4159 | 0
out area | Area | m² | pi * r^2 | 2
out diameter | Diameter | m | 2 * r | 2
out circumference | Circumference | m | 2 * pi * r | 2