Area of a Circle: Formula & Calculator

This calculator computes the area of a circle from its radius, diameter, or circumference. Enter any one of these three values, and you get the area in square units instantly, using the standard formula A = πr².

Formula

The area of a circle is the total space enclosed by its circumference. The fundamental formula is:

A = π × r²

where:

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

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

Worked Example

Using the calculator defaults: radius = 5 m.

1. Square the radius: 5² = 25 m²
2. Multiply by π: 25 × 3.14159 = 78.53975 m²
3. Rounded to 4 decimal places: 78.5398 m²

The calculator shows the same result for the radius input. For diameter = 10 m, it computes (10/2)² × π = 25π = 78.5398 m². For circumference = 31.4159 m, it computes (31.4159²) / (4π) ≈ 78.5398 m².

Why the Formula Works

Imagine cutting a circle into many thin wedges (like pizza slices) and rearranging them into a rectangle-like shape. The width of that shape is the radius (r), and the length is half the circumference (πr). Multiplying width × length gives r × πr = πr². This geometric derivation is standard and was known to ancient mathematicians; Archimedes approximated π using inscribed polygons, but the formula itself is exact for a perfect circle.

The constant π is the ratio of any circle’s circumference to its diameter, approximately 3.14159. Because π is irrational (non-repeating, non-terminating decimal), you typically round it to 3.14 or use the π button on a calculator.

When to Use It

You need the area of a circle in many practical situations:

• Construction: concrete for a circular slab, paint for a round wall, or tiles for a circular floor.
• Gardening: area of a circular garden bed for soil or mulch.
• Manufacturing: material needed for circular cutouts, discs, or gaskets.
• Sports: area of a circular running track, throwing circle, or archery target.
• Physics/Math: cross-sectional area of pipes, wires, or cylinders.

Always measure the radius (or diameter) in the same unit you want the area in. If you measure in meters, area is in square meters; if in feet, area is in square feet.

Common Mistakes and How to Avoid Them

• Using diameter instead of radius: The formula uses radius. If you have the diameter, halve it first. Forgetting this gives an area 4 times too large.
• Unit mix-ups: If you measure radius in inches but want area in square feet, convert inches to feet first (divide by 12) before squaring. Mixing units leads to nonsense results.
• Rounding π too early: Using 3.14 instead of 3.14159 introduces an error of about 0.05%. For most everyday tasks that’s fine, but for precise engineering use the π button.
• Squaring the unit: Remember that area is always in square units. 5 m radius gives 78.54 m², not 78.54 m.
• Confusing area with circumference: Area is the space inside (square units); circumference is the distance around (linear units). Don’t swap them.

Limits of the Formula

The formula A = πr² applies to perfect, flat circles in Euclidean geometry. Real-world objects might not be perfectly round (e.g., an oval or dented coin), in which case the formula gives an approximation. For very large circles (like the Earth), the curvature of the sphere means the surface area differs from a flat circle; use spherical geometry instead. Also, the formula assumes a two-dimensional shape; for a cylinder’s surface area, you need additional terms.

Frequently Asked Questions

Can I calculate area from circumference?

Yes. If you know the circumference C, use A = C² / (4π). For example, if C = 31.4159 m, then A = (31.4159²) / (4 × 3.14159) ≈ 78.54 m². This is derived by substituting r = C/(2π) into A = πr².

What if I only have the diameter?

Divide the diameter by 2 to get the radius, then apply A = πr². Or use A = π(d/2)² directly. For a diameter of 10 m, the radius is 5 m, so area is π × 25 = 78.54 m².

Why is π approximately 3.14?

π is the ratio of a circle’s circumference to its diameter. It is approximately 3.14159, but the decimal goes on forever without repeating. 3.14 is a common approximation for quick calculations; using more digits gives more precise results.

How do I convert square meters to square feet?

Multiply the area in square meters by 10.7639 (since 1 m = 3.28084 ft, so 1 m² = 3.28084² ≈ 10.7639 ft²). For 78.54 m², that’s about 845.4 ft².

Does the formula work for a semicircle?

No, for a semicircle you take half of the full circle’s area: A = (πr²)/2. The same applies to a quarter circle (πr²)/4.

What is the area of a circle with radius 1?

It is exactly π square units, approximately 3.14159. This is often used as a reference point in geometry.

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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 | 4
out area_from_d | Area (from diameter) | m² | pi * (d/2)^2 | 4
out area_from_c | Area (from circumference) | m² | c^2 / (4 * pi) | 4