Circular Sectors - Calculator

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This calculator can be used to calculate circular sectors, circular segments, arcs, chords, sagittas and the perimeters associated with the circular sectors and segments.

Formulas

important variables for a circular sector (arc, chord, sagitta)
circular sector
circular sector divided into a circular segment and a triangle
diameter d = 2 ∙ r
circular sector
Asector = π ∙ r² ∙
α
360°
Asector =
d²
4
∙ π ∙
α
360°
arc length
L = r ∙ π ∙
α
180°
L =
d
2
∙ π ∙
α
180°
chord
c = 2 ∙ r ∙ sin(
α
2
)
c = d ∙ sin(
α
2
)
sagitta
s =
c
2
∙ tan(
α
4
)
area of triangle
if α ≤ 180°:
Atriangle =
c ∙ ( r − s )
2
if α > 180°:
Atriangle =
c ∙ ( s − r )
2
circular segment
if α ≤ 180°:
Asegment = Asector − Atriangle
if α > 180°:
Asegment = Asector + Atriangle
perimeter of circular sector
Psector = 2 ∙ r + L
perimeter of circular segment
Psegment = c + L

Circular sector and arc

A circular sector (also called circle sector) is a portion of the area of a circle. A circular sector is bounded by a portion of the circle line and 2 radii (lines, each of which runs from the center of the circle to a point on the circle line). The portion of the circle line is called circular arc and the angle between the two lines between the center and the points on the circle line is called central angle.

Circular sector with arc, central angle and radii

Often the central angle is less than or equal to 180°, but it can also be greater than 180°.

circular sector with 280° angle

Calculate circular sectors:

To calculate the area of a circle, you can use the following formula: A = π ∙ r²

If a circular sector is to be calculated, then the area of the entire circle must be multiplied by the proportion of the circle sector in the entire circle. For example, if the central angle is 90°, then the proportion of the circular sector in the area of the entire circle is
1
4
(resp.
90
360
). Or if the center angle is 45°, then the circle sector has a share of
1
8
(resp.
45
360
) in the entire circle. In general, a circular sector with central angle α has a proportion of
α
360°
in the entire circle. Thus, the following applies to the circular sector:

Asector = π ∙ r² ∙
α
360°

Calculate circular arcs:

If you want to calculate the circumference of a circle, you can use the following formula: C = 2 ∙ π ∙ r

If an circular arc is to be calculated, then multiply the circumference of the circle by the proportion of the arc in the entire circle. For example, if the central angle is 90°, then the arc is
1
4
(resp.
90
360
) of the total length of the circle. Thus, the circumference must be multiplied by
1
4
. Generally, the ratio is calculated by
α
360°
.

The following applies to the length L of the arc: L = 2 ∙ r ∙ π ∙
α
360°

This can be simplified:

L = r ∙ π ∙
α
180°

arc of a circular sector

Chord

The line connecting the two ends of an arc is called a chord.

chord in circular sector

The length s of the chord can be calculated with the following formula:

c = 2 ∙ r ∙ sin(
α
2
)

Circular segment and sagitta

A circular segment is the area bounded by an arc of a circle and a chord.

circular segment, arc, chord

If the central angle is less than or equal to 180°, then the circular segment is a portion of the circular sector. If the central angle is greater than 180°, then in addition to the entire circular sector, a triangle, which is not part of the circular sector, also belongs to the circular segment.

circular segment with angle greater than 180°

In both cases, of 2 sides of the triangle, the side lengths correspond to the radius of the triangle and the third side is the chord.


The distance between the point located in the center of the circle chord and the point located in the center of the circle arc is called sagitta. The sagitta can be used to calculate the circular segment.

circular segment with sagitta

If the central angle is less than 180°, then the sagitta is less than the radius. If the central angle is 180°, then the sagitta is equal to the radius and if the central angle is greater than 180°, then the sagitta is greater than the radius.

sagitta for angles greater than 180

Calculate sagitta:

If c is the length of the chord and α is the central angle, then the sagitta s is:

s =
c
2
∙ tan(
α
4
)

Calculate circular segment:

Case α < 180°:

If the central angle is smaller than 180°, then the circular sector consists of the circular segment and a triangle.

Division of a circle sector into circle segment and triangle

To calculate the circular segment you can first calculate the circular sector and the area of the triangle (how to calculate the area of the triangle is described below) and then subtract the area of the triangle from the circular sector.

Asegment = Asector - Atriangle

Case α > 180°:

If the central angle is greater than 180°, then the triangle does not belong to the circular sector. In this case, the circular segment consists of the circular sector and the triangle.

Division of circle segment into circle sector and triangle

In this case, the circular sector and the area of the triangle are added to calculate the circular segment.

Asegment = Asector + Atriangle


Calculate area of triangle:

Case α < 180°:

The area of a triangle is calculated by multiplying the length of a base side by the corresponding altitude and dividing the result by 2. In this case, the chord is the base side and the altitude is the difference between the radius and the sagitta.

Triangle of a circular sector for the calculation of the circular segment

If r is the radius, c the length of the chord, and s the sagitta, and if the central angle is less than 180°, the following is valid:

Atriangle =
c ∙ ( r − s )
2

Case α > 180°:

If the central angle is greater than 180°, then the sagitta is the sum of the radius of the circle and the altitude of the triangle (where the chord of the circle is considered as the base). The altitude of the triangle is therefore the difference between the sagitta and the radius.

the triangle is part of the circular segment that is to be calculated

If r is the radius, c the length of the chord, and s the sagitta, and if the central angle is greater than 180°, then:

Atriangle =
c ∙ ( s − r )
2

Perimeter of the circular sector

the perimeter of a circular sector is calculated by summing up radii and arc

A circular sector is bounded by the arc and 2 lines, each of whose length is equal to the radius. If r is the radius and L is the length of the arc, then Psector holds for the perimeter:

Psector = 2 ∙ r + L

Perimeter of the circular segment

the perimeter of a circular segment is calculated by adding the arc and the chord

A circular segment is bounded by an arc and a chord. If L is the length of the circular arc and c is the length of the chord, then the following applies to the perimeter of the circular segment:

Psegment = L + c


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