Sector area calculator

What is a sector area calculator?

A circular sector is the wedge-shaped slice of a disc bounded by two radii and the arc between them — think of a single slice of pizza or a piece cut from a round pie. The sector area calculator finds how much flat surface that wedge covers, using just the radius of the circle and the central angle that opens between the two straight edges.

The size of a sector is simply a fraction of the whole circle. A 90° wedge is one quarter of the disc, a 180° wedge is a half, and a full 360° “sector” is the entire circle. The calculator turns that intuition into an exact number for any radius and any angle, in degrees, radians, or gradians.

Key concepts

• Radius (r) — the distance from the center of the circle to any point on its edge; both straight sides of the sector have this length.
• Central angle (θ) — the angle at the center of the circle, between the two radii that bound the sector. A larger angle means a larger slice.
• Sector area (A) — the amount of two-dimensional surface enclosed by the two radii and the arc, measured in square length units.
• Radian — the natural angle unit for the area formula. A full circle is $2\pi$ radians, or 360 degrees.

How does the calculator work?

A sector takes up the same fraction of the circle’s area as its angle takes up of a full turn. Since the area of the whole circle is $\pi r^2$, the sector area is that total scaled by the angle’s share of the circle.

Formulas

When the angle is given in degrees, the sector is the fraction $\frac{\alpha}{360}$ of the full circle:

When the angle is given in radians, the formula is even simpler:

Both express the same idea: a full turn ($360°$ or $2\pi$ radians) recovers the whole circle’s area $\pi r^2$. The calculator converts whatever angle unit you choose into the right form internally, so you can mix and match units freely.

Worked examples

Example 1: radius 10 cm, angle 90°

A quarter-circle wedge of radius 10 cm:

Example 2: radius 5, angle 2 radians

Using the radian form directly:

Example 3: full circle

For radius 4 and an angle of 360°, the sector is the entire disc:

This matches the result of the circle area calculator for the same radius.

Example 4: zero angle

For an angle of 0°, the two radii coincide and the sector collapses to a line, so the area is $0$.

Practical uses

• Design and manufacturing — cutting circular gussets, fan blades, gears, or pie-shaped panels where each piece spans a known angle.
• Sewing and pattern making — calculating the fabric in a flared skirt panel or a circular tablecloth segment; the same geometry drives the arc length calculator for the curved hem.
• Architecture and landscaping — laying out wedge-shaped garden beds, paved plaza sections, or seating fanned around a central stage.
• Data visualization — sizing slices of a pie chart so each one’s area is proportional to its angle.
• Physics and engineering — finding the swept area of a rotating arm, a windshield wiper, or a radar sweep over a given angle.

Notes

• The radius must be positive and the angle non-negative for the result to be meaningful.
• A 0° (or 0 radian) angle gives an area of 0; a 360° (or $2\pi$ radian) angle gives the full circle area $\pi r^2$.
• The result is in square units of the chosen length unit; the calculator reconverts automatically when you change the radius or area unit.
• A sector is the slice between two radii and an arc. If instead you need the area between a chord and the arc — the part beyond the straight cut — use the circle segment area calculator.