Circle Area Calculator

A circle’s area comes up in more shapes than just the full circle: pie-slice sectors, curved segments, semicircles, quarter circles, and rings (annuli) all use variations on the same underlying formula. This calculator covers all six at once — and for the full circle, it works bidirectionally, so you can enter radius, diameter, circumference, or area, and it solves for the other three automatically.

Pick the shape or mode you need below and enter what you know.

Full Circle: Radius, Diameter, Circumference, or Area

Area = π × Radius²

Where:
Radius = the distance from the center of the circle to its edge

Since radius, diameter, circumference, and area are all mathematically tied together, this mode lets you type in whichever one you already know, and it solves for the other three:

Diameter = 2 × Radius
Circumference = 2π × Radius
Radius (from diameter) = Diameter ÷ 2
Radius (from circumference) = Circumference ÷ (2π)
Radius (from area) = √(Area ÷ π)

Worked Example: A 10-ft Diameter Circle

Radius = 10 ÷ 2 = 5 ft. Area = π × 5² = π × 25 ≈ 78.54 square feet.

Sector Area (Pie-Slice)

Sector Area = (Angle ÷ 360) × π × Radius²

Where:
Angle = the sector’s central angle, in degrees

A sector is the pie-slice-shaped region between two radii and the arc connecting them. Since a full circle is 360°, the fraction of the circle’s total area a sector covers is simply its angle divided by 360.

Worked Example: A 90° Sector

A 90° sector of a circle with an 8-ft radius: Area = (90 ÷ 360) × π × 8² = 0.25 × π × 64 ≈ 50.27 square feet — exactly a quarter of the full circle’s area, since 90° is a quarter of 360°.

Segment Area

Segment Area = 0.5 × Radius² × (θ − sin θ), with θ in radians

A segment is the region between a chord and the arc it cuts off — visually, a sector with the triangular part (formed by the two radii and the chord) subtracted out. That’s exactly what the formula does: it starts from the sector’s area and subtracts the triangle’s area, expressed via the sine term.

Worked Example: A 90° Segment

Using the same 8-ft-radius circle: θ = 90° = π/2 ≈ 1.5708 radians. Area = 0.5 × 8² × (1.5708 − sin(1.5708)) = 32 × (1.5708 − 1) ≈ 32 × 0.5708 ≈ 18.27 square feet — noticeably smaller than the 50.27-square-foot sector, since the segment excludes the triangular wedge.

Semicircle and Quarter Circle Area

Semicircle (Half Circle)

Semicircle Area = (π × Radius²) ÷ 2

A semicircle is exactly half a full circle, so its area is exactly half of πr². Its perimeter also includes the straight diameter edge: Perimeter = πr + 2r.

Worked Example

A semicircular window with a 3-ft radius: Area = (π × 3²) ÷ 2 = (π × 9) ÷ 2 ≈ 14.14 square feet.

Quarter Circle

Quarter Circle Area = (π × Radius²) ÷ 4

A quarter circle is exactly one-fourth of a full circle — the special case of the sector formula where the angle is fixed at 90°.

Worked Example

A quarter-circle flower bed with a 6-ft radius: Area = (π × 6²) ÷ 4 = (π × 36) ÷ 4 ≈ 28.27 square feet.

Annulus (Hollow Circle / Ring) Area

Annulus Area = π × (Outer Radius² − Inner Radius²)

An annulus is the flat ring-shaped region between two concentric circles — think of a washer, a pipe’s cross-section, or a circular running track. Its area is simply the outer circle’s area minus the inner circle’s area.

Worked Example: A Pipe Cross-Section

A pipe with a 4-inch outer radius and a 3-inch inner radius (wall thickness of 1 inch): Area = π × (4² − 3²) = π × (16 − 9) = π × 7 ≈ 21.99 square inches of cross-sectional material.

Frequently Asked Questions

How do I find the area of a circle if I only know the circumference?

Enter the circumference in the Full Circle mode above — it solves for radius (Circumference ÷ 2π), then computes the area from that radius automatically.

What’s the difference between a sector and a segment?

A sector is the full pie-slice shape bounded by two radii and an arc. A segment is only the curved sliver between a chord (a straight line connecting the two endpoints of the arc) and the arc itself — it excludes the triangular part that the two radii and the chord would otherwise enclose.

How do I calculate the area of a ring or washer shape?

Use the Annulus mode: subtract the inner radius squared from the outer radius squared, then multiply by π.

Does the sector formula work if the angle is given in radians instead of degrees?

This calculator’s Sector mode expects degrees. If you have an angle in radians, convert it first (degrees = radians × 180 ÷ π) before entering it.

Why is the segment area smaller than the sector area for the same angle?

The sector includes the full pie-slice, triangular wedge and all. The segment subtracts that triangular wedge out, leaving only the curved sliver between the chord and the arc — so for any angle under 360°, the segment is always smaller than the sector.