Area of a Circle in sq km: Formula & Calculator

This calculator works out the area of a circle in square kilometers when you enter its radius or diameter. You input the radius (or diameter) in kilometers, and it returns the area in km² using the standard formula. The result is the exact land or surface area covered by a circle of that size.

Formula

The area of a circle is calculated with the formula:

A = π × r²

Where:

• A = area of the circle, in square kilometers (km²)
• π (pi) ≈ 3.14159, a mathematical constant
• r = radius of the circle, in kilometers (km)

If you only know the diameter (d), the radius is half of it: r = d / 2. So the formula becomes A = π × (d/2)².

Worked Example

Using the default values from the calculator above: radius = 10 km.

1. Square the radius: 10² = 100
2. Multiply by π: 100 × 3.14159 = 314.159
3. Round to 2 decimal places: 314.16 km²

The calculator shows 314.16 km² for a radius of 10 km. If you entered the diameter instead (20 km), the result is the same: π × (20/2)² = π × 100 = 314.16 km².

When to Use Square Kilometers

Square kilometers are the standard metric unit for large areas — larger than a few hectares but smaller than a continent. You would use km² for:

• City or county land area
• Forest fire perimeters or burn scars
• Lake surface area (e.g., a circular lake)
• Agricultural fields or large plots
• Radar or radio coverage zones

For smaller circles (a garden, a room, a sports field), square meters or square feet are more practical. For very large features (oceans, continents), you might use square miles or even million km².

How to Read the Result

The output tells you the total flat surface enclosed by the circle. For example, a circle with a radius of 10 km covers 314.16 km² — that’s about the size of a small city (like Malta or the island of Guam). A circle with a 1 km radius covers 3.14 km², which is roughly 314 hectares (since 1 km² = 100 hectares).

If you are comparing to a square, a circle of radius r fits inside a square of side 2r. The square’s area is (2r)² = 4r², while the circle’s area is πr² ≈ 3.14r² — so the circle fills about 78.5% of the square.

Common Mistakes and Unit Mix-ups

One frequent error is mixing units. If your radius is in meters but you want the area in km², you must convert first: divide meters by 1000 to get kilometers. For example, a radius of 500 m = 0.5 km, then area = π × 0.5² = 0.785 km² (not 785,000 km²).

Another mistake is using the diameter directly in the formula without halving it. If you plug d = 20 into π × d², you get 1256.64 km² — four times the correct answer. Always halve the diameter first.

Rounding too early can also skew results. Keep π to at least 3.14159, and round only the final answer. For most purposes, 2 decimal places is fine, but for precise land measurements, keep 4 or more.

Limits of the Formula

The formula gives the area of a perfect circle on a flat plane. Real-world surfaces are not flat — the Earth is curved. For circles larger than a few hundred kilometers, the true surface area on a sphere is slightly larger than the flat calculation. For example, a circle with a radius of 1000 km on the Earth’s surface would have a flat area of about 3,141,590 km², but the actual spherical area is about 3,146,000 km² — a difference of roughly 0.15%. For most practical purposes (city planning, agriculture, coverage maps), the flat formula is accurate enough. For global-scale circles, you’d need spherical geometry.

Alternatives and Related Measurements

If you need different units, the same formula works with any length unit:

• Radius in meters → area in m² (divide by 1,000,000 to get km²)
• Radius in miles → area in square miles (multiply by 2.58999 to get km²)
• Radius in feet → area in square feet (divide by 10,763,910 to get km²)

You might also want the circumference (2πr) or the diameter (2r) — the calculator above provides the circumference as a bonus output. If you have the area and want the radius, rearrange: r = √(A/π).

Frequently Asked Questions

How many square kilometers is a circle with a 5 km radius?

Using A = π × 5² = π × 25 ≈ 78.54 km². That’s about 7,854 hectares.

What is the area of a circle with a 1 km diameter?

Radius = 0.5 km. Area = π × 0.5² = π × 0.25 ≈ 0.7854 km², or 78.54 hectares.

How do I convert square meters to square kilometers?

Divide the number of square meters by 1,000,000 (since 1 km = 1000 m, so 1 km² = 1,000,000 m²). For example, 500,000 m² = 0.5 km².

Why is the area of a circle πr²?

This is a fundamental geometric result, proven by methods like inscribing polygons or calculus. Intuitively, π relates the circumference to the diameter, and the area is the integral of the circumference over the radius, giving πr².

Can I use this formula for an ellipse?

No. An ellipse has a different formula: A = π × a × b, where a and b are the semi-major and semi-minor axes. The circle formula only works when all points are equidistant from the center.

What if my radius is in miles but I want km²?

First convert miles to kilometers: multiply by 1.60934. Then apply the formula. For example, a 10-mile radius = 16.0934 km, area = π × 16.0934² ≈ 813.5 km².

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in r | Radius | km | 10 | 0
in d | Diameter | km | 20 | 0
out area | Area | km² | pi * r^2 | 2
out area_d | Area from diameter | km² | pi * (d/2)^2 | 2
out circ | Circumference | km | 2 * pi * r | 2