gizmobench

Sector Area Calculator

A sector is the wedge a slice of pie makes: two radii from the centre and the arc between them. Its area is the fraction of the circle that the angle covers, so area = (θ / 360) × π × r². This page opens on the sector shape rather than the rectangle, with an example already in the fields and the working printed underneath, for example (90 / 360) × π × 5² = 19.6349540849. The angle is the one at the centre, in degrees, anything above 0 and up to 360, where 180 is a semicircle and 360 is the whole circle. Radii go in millimetres, centimetres, metres, inches or feet, the answer comes back in that unit squared, and the rows below list the same area in six more units.

r = 5 cmθ = 90 cm
Sector · (θ / 360) π r²19.635cm²1,963.5 mm² · 0.0019635 m² · 3.04342 in²
Flat
Solid, surface area
Radius (cm)
Angle (degrees)
Unit

A slice of a circle: the angle at the centre, in degrees. 360 is the whole circle.

  • The workingarea, in cm²
    (90 / 360) × π × 5² = 19.6349540849
  • Square millimetresmm²
    1,963.49540849
  • Square metresm²
    0.00196349540849
  • Square inchesin²
    3.04342397001
  • Square feetft²
    0.0211348886806
  • Square yardsyd²
    0.00234832096452
  • Acresac
    4.851903e-7
The formula is on the stage, and the working is the first row. Every answer here is one textbook formula with your numbers put in, printed so you can check it: a circle is π r², a three-sided triangle is Heron's formula, a regular polygon is n s² / (4 tan(π/n)). The solids give surface area for a closed shape, both ends of a cylinder and the base of a cone included. Conversions use exact definitions, so a square foot is 0.09290304 m² and 43,560 of them make an acre.

Common questions

What is the formula for the area of a sector?
Area = (θ / 360) × π × r², where θ is the angle at the centre in degrees and r is the radius. The angle is the fraction of a full turn the wedge takes up, so the formula is that fraction of the whole circle's π r². The formula sits on the stage beside the answer, and the first row under it is the same formula with your numbers in place, for example (90 / 360) × π × 5² = 19.6349540849.
Is the angle in degrees or in radians?
Degrees. The field is labelled Angle (degrees) and it takes anything above 0 up to and including 360; outside that range the page says so under the fields instead of printing a number. If your angle is in radians, multiply by 180 / π first: 1.5708 radians is 90 degrees. Typed as 1.5708 it would be read as 1.5708 degrees, which for a radius of 5 gives 0.342695 rather than 19.635.
I measured straight across the circle. What do I type?
Half of it. The sector asks for the radius, the distance from the centre out to the arc, and unlike the Circle shape here it carries no radius-or-diameter switch, so a dial measured at 10 across is a radius of 5. The radius has to be greater than 0, and text that is not a number is named back to you in the message under the fields rather than quietly ignored.
How do I get a semicircle or a quarter circle?
With the angle. A quarter circle is 90 degrees, a third is 120, a semicircle is 180. A semicircle of radius 10 reads 157.08 on the stage and 157.079632679 in the working row. At 360 degrees you get the whole circle, 78.5398163397 for a radius of 5, which is exactly what the Circle shape gives for the same radius.
Does it give the arc length or the perimeter of the sector?
No. Every figure on this page is an area: the sector in the unit you typed squared, then the same area converted into six more units. The arc length is a separate calculation, r × θ × π / 180, and the perimeter of a sector adds the two straight radii to it, 2r + arc. Neither is printed here.
What about a segment, the piece cut off by a chord?
A segment is the sector minus the triangle formed by the two radii and the chord, and this page does not do the subtraction for you. Work out the sector here, then subtract ½ r² sin θ. None of the flat shapes here takes two sides and the angle between them, so that triangle is one you do by hand.
Which units can I use, and how precise is the answer?
Type the radius in millimetres, centimetres, metres, inches or feet. Measured in centimetres, a 90 degree sector of radius 5 shows 19.635 cm² on the stage, with 1,963.5 mm² · 0.0019635 m² · 3.04342 in² on the line beneath it and the full rows below giving twelve significant figures each. Conversions use exact definitions, so a square inch is exactly 6.4516 cm². The headline is six significant figures, the rows are twelve.

Every area is computed from the formula shown, exact to the precision of the numbers you type, and conversions use exact unit definitions. Measurements from the real world carry their own error.