Find the area of a circular sector — the "pizza slice" shape bounded by two radii and an arc — from the radius and central angle, using A = ½ · r² · α. Enter any two of radius, central angle, and sector area, leave the third blank, and the calculator solves for it, plus the arc length and chord.
The sector area calculator works out the area of a circular sector — the pie- or pizza-slice shape formed by two radii and the arc between them. Give it the radius and the central angle and it returns the area instantly, along with the arc length and the chord. Because it works in every direction, you can also find the radius from a known area and angle, or the angle from a known area and radius. Whether you are studying geometry, laying out a garden bed, designing a fan or gauge, or slicing a pizza fairly, this tool makes the calculation quick and exact.
A sector is one of the most useful shapes in geometry, sitting between the full circle and the thin slice of an arc. This page explains what a sector is, the formulas behind the calculator, worked examples you can reproduce, how each field is solved, and answers to the questions people ask most often.
A circular sector is the portion of a circle enclosed by two radii and the arc that joins their endpoints. Picture a slice of pizza or a wedge of pie: the two straight cuts are the radii, the curved crust is the arc, and the pointed tip sits at the center of the circle. The angle at that tip, between the two radii, is the central angle, and it determines what fraction of the whole circle the sector represents.
When the central angle is small, the sector is a thin sliver; when it is 180°, the sector is a semicircle (half the circle); and when it reaches 360°, the sector is the entire circle. A sector with a central angle up to 180° is sometimes called a minor sector, and the larger remaining piece a major sector. The sector should not be confused with a segment, which is the region between a chord and its arc — the sector includes the two radii and the center, while the segment does not.
The area of a sector depends on just two things: the radius of the circle and the central angle. There are two equivalent versions of the formula, depending on whether the angle is measured in radians or degrees.
Sector area = ½ × r² × α
where r is the radius and α is the central angle in radians. This is the cleanest form of the formula and the one the calculator uses internally.
Sector area = (θ ÷ 360) × π × r²
where θ is the central angle in degrees. This version makes the idea intuitive: θ/360 is the fraction of the full circle the sector covers, and πr² is the area of the whole circle, so multiplying them gives the sector's share. If your angle is in degrees, you can also convert it to radians first by multiplying by π/180, then use the radian formula — the two give identical answers.
Take a classic example: a sector with a radius of 1 and a central angle of 90° — a quarter of a circle. First convert the angle to radians:
α = 90° × π/180° = π/2
Then apply the formula:
A = ½ × 1² × π/2 = π/4 ≈ 0.785
So the area is π/4, which is exactly a quarter of the whole circle's area (πr² = π) — just as you would expect for a 90° slice. The calculator's default values use a radius of 5 and a 90° angle, giving an area of about 19.63; press Calculate to see the full breakdown, then enter your own figures.
Beyond the area, a sector has two other important measurements that this calculator also reports.
The arc length is the length of the curved edge of the sector — the "crust" of the pizza slice. It is found with:
Arc length = r × α (α in radians)
Just as the sector area is a fraction of the circle's area, the arc length is the same fraction of the circle's full circumference (2πr).
The chord is the straight line joining the two endpoints of the arc — the base of the slice if you ignore the curve. Its length is:
Chord = 2 × r × sin(α/2)
Together, the area, arc length, and chord fully describe the size and shape of a sector, which is why the calculator shows all three at once.
The relationship between radius, central angle, and area can be rearranged, so the calculator lets you find whichever one you do not know. Enter any two and leave the third blank:
This makes the tool useful not just for finding an area, but for design and layout problems — for example, working out the radius needed to give a sector a specific area, or the angle that carves a required area out of a circle of fixed size.
Angles can be measured in degrees or radians, and both are fully supported. A degree is 1/360 of a full turn, the familiar everyday unit. A radian is the angle for which the arc length equals the radius; there are 2π (about 6.283) radians in a full circle, so 180° equals π radians and 90° equals π/2. Radians are the natural unit in higher mathematics and physics because they make formulas like the sector area and arc length beautifully simple — no extra conversion factor is needed. To convert, multiply degrees by π/180 to get radians, or multiply radians by 180/π to get degrees. The calculator handles this automatically whichever unit you pick, and always reports the angle in both.
One of the most intuitive ways to think about a sector is as a fraction of the whole circle. The central angle tells you directly what portion of the circle you have: a 90° sector is 90/360 = 1/4 of the circle, a 180° sector is half, and a 45° sector is one-eighth. Multiply that fraction by the full circle's area (πr²) and you have the sector area. This is exactly why the degree formula is written as (θ/360) × πr². The calculator shows this fraction explicitly, so you can immediately see, for instance, that a certain slice is 30% of the pizza or that a flower bed takes up an eighth of a circular lawn. Thinking in fractions also makes quick mental estimates easy: halve the angle and you halve the area, since area is directly proportional to the central angle.
For a circle of radius r, some frequently used sector angles give these areas (as a fraction of the full circle area πr²):
To get the actual area for any of these, multiply the fraction by πr². For example, a 60° sector of a circle with radius 6 has area (1/6) × π × 6² = 6π ≈ 18.85.
These two terms are often mixed up, but they describe different regions of a circle. A sector is bounded by two radii and an arc — it includes the center of the circle, like a full pizza slice. A segment is bounded by a chord and an arc — it is the region you would get by making a single straight cut across the circle, leaving out the pointed tip near the center. The segment is what remains of a sector after you remove the triangle formed by the two radii and the chord. So the area of a segment equals the sector area minus that triangle's area. This calculator focuses on the sector, but it also gives you the chord length, which is the starting point for segment calculations.
Two levers control a sector's area, and they behave very differently. The central angle affects the area linearly: double the angle and you double the area, because you are simply taking twice as large a slice of the same circle. The radius, however, affects the area quadratically, because it appears squared in the formula: double the radius and the area grows fourfold, triple it and the area grows ninefold. This is the same squared relationship that governs the area of a whole circle, and it has practical consequences. If you are sizing a fan-shaped patio and you increase its radius by half, you will need roughly 2.25 times as much paving, not 1.5 times. Keeping this square law in mind helps you estimate materials, costs, and sizes far more accurately, and it explains why small increases in radius can lead to surprisingly large increases in area.
In radians, sector area = ½ × r² × α. In degrees, sector area = (θ/360) × π × r². Both give the same result; the degree version is just the fraction of the circle times the full circle area.
A 90° sector is a quarter of the circle, so its area is ¼ × πr². For radius 1 that is π/4 ≈ 0.785; for radius 5 it is about 19.63.
Yes. The calculator is bidirectional. To find the radius, use r = √(2A/α); to find the angle, use α = 2A/r². Just enter the two values you know and leave the unknown blank.
A sector is bounded by two radii and an arc and includes the circle's center, like a pizza slice. A segment is bounded by a chord and an arc and does not include the center. A segment's area equals the sector area minus the triangle between the radii.
Both are the same fraction of the whole circle. Arc length = r × α (angle in radians), and it is to the circumference what the sector area is to the circle's total area.
Yes. Choose degrees or radians from the dropdown next to the central angle. The result always shows the angle in both units, so you can work in whichever you prefer.
This Sector Area Calculator is provided for general educational and informational purposes. It uses the standard geometric formulas for a circular sector and assumes a perfect circle. For critical engineering or construction work, verify measurements and results independently.