Circle segment - perimeter and area
Online calculator — enter the values and get the result instantly, with the formula and a worked example.
A = area
P = perimeter
α = angle
r = radius
t = chord
A circle segment is the region of a circle bounded by a chord and the arc that connects the same two endpoints. In other words, if you draw a straight line across a circle, it slices off a curved piece on each side, and each of those pieces is a segment. Any chord creates two segments: the smaller one is called the minor segment and the larger one the major segment, and together they fill the whole disc. Unlike a circular sector, a segment does not reach the centre of the circle, so its shape is defined entirely by the chord and the arc. Its size depends on just two quantities, the radius and the central angle spanning the arc, and from these the chord length, arc length, and the height of the segment (the sagitta) all follow. A useful way to picture it is as a circular sector with the triangle to the centre cut away, which is exactly how its area is found. Circle segments appear surprisingly often in the real world, from the cross-section of liquid in a partly filled horizontal tank or pipe to arched windows, bridge arches, and machine parts. Being able to compute their area and perimeter therefore matters in engineering, architecture, and design whenever curved surfaces need to be measured.
Segment area
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Chord
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Arc length
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Sagitta (height)
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Perimeter
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| Radius r | – |
| Central angle α | – |
Drag the blue dot around the circle (angle α) · the orange dot sets the radius r
Worked example
The radius of the circle is 5 cm, the angle α is 60 degrees
For the calculation, we first need to calculate the circular sector using the formula A =(πr²*α)/360, thus
A = (3.14*5²*60)/360 = (3.14*25*60)/360 = 13.08 cm²
So, the area of the circular sector is 13.08 cm².
To further calculate the area of the circular segment, we need to subtract the area of the triangle from the circular sector.
The area of the triangle is calculated using the sine of the angle, where the sine of the angle is the ratio of the opposite side to the hypotenuse. The hypotenuse in our case is the radius, and the angle needed for the sine is half of α, i.e., 30 degrees.
So,
sin 30 degrees = (1/2 of the chord) / hypotenuse (radius)
0.5 = (1/2 of the chord) / 5
0.5 * 5 = (1/2 of the chord)
2.5 cm = (1/2 of the chord), so the entire chord is 5 cm
We calculate the height h using the Pythagorean theorem
c² = a² + b²
5 squared (radius) = 2.5 squared + b²
25 = 6.25 + b²
18.75 = b²
4.33 = b, so the height
We calculate the triangle using the formula
A = (a * h) / 2
A = (5 * 4.33) / 2
A = 10.82 cm²
The area of the circular segment is then equal to the area of the circular sector minus the area of the triangle, thus 13.08 - 10.82 equals 2.21 cm²
We calculate the perimeter of the circular segment using the formula
O = 2πr · α / 360°
O = (2*3.14*60)/360
O = 5.24 cm
