Area of a Sector Calculator

What a sector is, and the one formula that sizes it

A sector is the wedge you get when two radii cut a slice out of a circle, the shape of a pizza slice or a slice of pie. It has three edges: two straight sides that both run from the center to the rim, and one curved edge that follows the circle. The angle between the two straight sides is the central angle, and it decides how much of the circle the wedge covers.

Every sector question starts from the area of the whole circle, pi times the radius squared. A sector is a fraction of that circle, and the fraction is the angle divided by 360. That gives the formula in degrees: sector area = (angle / 360) x pi x radius squared. The OpenStax Precalculus chapter on angles derives the same result from the ratio of the angle to a full turn, and the OpenStax calculus chapter on polar area shows where the radian version comes from.

In radians the formula shortens to area = 0.5 x radius squared x angle, because a full turn is 2 pi radians instead of 360 degrees. Both versions answer the same question, so pick the one that matches the angle you already have. If the angle is written with a degree sign, use the 360 version. If it is written as a decimal like 1.047, it is radians, and the 0.5 version skips a conversion step.

Degrees or radians: two versions of the same rule

Degrees are the everyday unit, and radians are the unit calculus and most programming languages use. One full turn is 360 degrees or 2 pi radians, which is about 6.2832, so one radian is about 57.2958 degrees. The arc length and sector area notes from the University of Nebraska put both forms side by side, and the conversion is the same one used for arc length.

Feeding the wrong unit into the wrong form is the most common way to get a nonsense answer. A 60 degree angle is 1.0472 radians, so the degree form gives 42.41 square inches on a radius of 9 while the radian form with the same number 60 would give 2,430 square inches, which is more than nine times the whole circle. The check is simple: if the answer is larger than pi times the radius squared, the angle went in as degrees when the formula wanted radians, or the reverse.

Read down the last column and the pattern is plain: the share is the angle divided by 360, and the area scales with it in a straight line. Doubling the angle doubles the area, which is why a 180 degree wedge is exactly half the circle and a 90 degree wedge is exactly a quarter.

Worked example: a 60 degree wedge on a 9 inch radius

Take a round table with a radius of 9 inches, which is 18 inches across, and a wedge of 60 degrees. Step one is the fraction of the circle: 60 / 360 is 0.1667, or one sixth. Step two is the area of the whole circle: pi x 9 x 9 is 254.47 square inches. Step three multiplies the two: 254.47 x 0.1667 is 42.41 square inches, and one sixth of 254.47 is the same 42.41.

The other numbers that come with the same wedge are worth having. The curved edge, the arc, is 9.42 inches, which is one sixth of the circumference of 56.55 inches. The straight line joining the two ends of the arc, the chord, is 9.00 inches, and that is no accident: a 60 degree angle makes an equilateral triangle with the two radii, so the chord equals the radius. The perimeter of the wedge is 9.42 plus 9 plus 9, which is 27.42 inches.

The segment, the small curved sliver between the chord and the arc, is the sector minus the triangle inside it. The triangle has an area of 0.5 x 9 x 9 x sin(60 degrees), which is 35.07 square inches, so the segment is 42.41 minus 35.07, or 7.34 square inches. That is the number to use when someone asks how much crust sits outside a straight cut.

Sector, segment, arc and chord: four different numbers

These four quantities come from the same two inputs and get mixed up constantly, so it helps to see them in one place. On a radius of 9 inches with a 60 degree angle:

The area and the length never convert into each other, because one is in square units and the other is in plain units. A wedge with a long arc and a short radius can have less area than one with a short arc and a long radius, since area grows with the square of the radius while the arc grows only with the radius. For any wedge up to 180 degrees the segment is smaller than the sector, and the gap between them grows as the angle grows: at 60 degrees the segment is 7.34 against a sector of 42.41, and at 180 degrees the segment is the whole half circle of 127.23, because the triangle flattens to nothing. Past 180 degrees the triangle formula changes sign and the segment becomes the major segment, which is larger than the sector.

Working backward from an area to an angle or a radius

Plenty of real questions arrive the other way around: you know the area the wedge has to cover and you need the angle or the radius that produces it. Both rearrangements come straight from the same formula. To get the angle, multiply the area by 360 and divide by pi times the radius squared: angle = 360 x area / (pi x r squared). To get the radius, solve for r: radius = the square root of (360 x area / (pi x angle)).

A landscape example shows how it works. A planting bed is a wedge with a radius of 12 feet, and the bag of mulch covers 100 square feet. The angle needed is 360 x 100 / (pi x 144), which is 36,000 / 452.39, or 79.58 degrees. Check it forward: 79.58 / 360 is 0.2211, times 452.39 is 100.0 square feet, which matches the bag.

Run it the other way for a sprinkler. A head throws 40 feet and you want a 120 degree arc, and you need the radius that would cover 50 square feet with a 120 degree pattern. That is the square root of (360 x 50 / (pi x 120)), which is the square root of 47.75, or 6.91 feet. The arithmetic is fussy by hand and instant in the calculator, which is the reason to keep the two rearrangements written down rather than memorized.

Pizza math: what a slice measures

A pizza is a circle, a slice is a sector, and the angle at the tip is the central angle. Cut a 16 inch pizza into 8 even slices and each slice has an angle of 45 degrees and a radius of 8 inches. The area is (45 / 360) x pi x 64, which is 25.13 square inches, and the crust along the outer edge is 6.28 inches. The same 45 degree cut on a 12 inch pizza gives (45 / 360) x pi x 36, or 14.14 square inches, with a crust of 4.71 inches.

The jump from 12 to 16 inches looks like a third more pizza, and it is not. Area grows with the square of the diameter, so 16 squared is 256 against 144, and 256 / 144 is 1.78. The 16 inch pie carries 78 percent more pizza than the 12 inch, which is why the price per square inch is the number to compare and not the price per pie.

Slicing changes the angle and nothing else. Twelve slices instead of eight makes each wedge 30 degrees, so each slice is (30 / 360) x pi x 64, or 16.76 square inches, and the total is unchanged. Two 30 degree slices are exactly one 60 degree slice, which is the check to use when someone swears the smaller cut gives more food.

Sectors in the yard, on the road and on a screen

Most sector problems outside a classroom come from things that sweep or spray in an arc. A sprinkler head that throws 40 feet and oscillates through 60 degrees covers (60 / 360) x pi x 1,600, which is 837.8 square feet, against 5,026.5 square feet for a head that turns a full circle. Two heads set to 60 degrees cover less than one head set to 180 degrees, and the arithmetic explains the water bill.

A windshield wiper is a sector too. A 24 inch blade sweeping 110 degrees clears (110 / 360) x pi x 576, or 552.9 square inches, and the wiped shape on the glass is that wedge. Security cameras and floodlights follow the same rule: a 60 degree lens with a 30 foot reach covers 471.2 square feet of ground, while the same light with a 120 degree spread covers twice that area at the same distance, at half the brightness per square foot.

Pie charts are the paper version. A category that takes 25 percent of the total gets a 90 degree wedge, and one that takes 12.5 percent gets 45 degrees. That is why a chart labeled only in percentages still tells you the angle, and why a chart with a wedge past 180 degrees is reporting a category that is more than half the total.

Mistakes that throw the answer off

The first mistake is using the diameter where the formula wants the radius. A 16 inch pizza has a radius of 8, and feeding 16 into the formula gives four times the correct area, 804.25 square inches instead of 201.06. Halve the diameter before you square anything.

The second is mixing degrees and radians. The 360 form needs degrees and the 0.5 form needs radians, and a 60 in the wrong place is a 57.3 times error, as the 2,430 square inch answer above shows. The third is reporting a length as an area. Arc length, chord, and perimeter are plain inches, while sector and segment are square inches, and a page that quotes 9.42 square inches for an arc has swapped the units.

The fourth is forgetting that the angle cannot pass a full turn. A 400 degree wedge is not a sector of a 400 degree circle; it is a full circle plus 40 degrees, and the area formula caps at 360. The fifth is treating the segment as the sector. On a thin wedge the two are close, and on a wide one they are not: at 180 degrees the sector is 127.23 square inches while the segment is also 127.23 only because the triangle has vanished, and at 60 degrees they differ by a factor of nearly six. Past a half circle the segment becomes the major segment and grows past the sector again.

A last habit saves trouble: write the units into the answer as you go. Square the radius first and keep the square units, then multiply by the fraction of the circle, which has no units at all. If the units come out as inches instead of square inches, the fraction was applied to a length instead of an area.