Chords, Arcs and Sectors

Minor and major arcs

For distinct, non-diametrically opposite points, the minor arc has central angle less than $180^\circ$ and the major arc has the remaining angle greater than $180^\circ$. Diametrically opposite points divide a circle into two semicircles, each measuring $180^\circ$.

Arc measure is an angle; arc length is a distance. An arc measuring $60^\circ$ is not $60\text{ cm}$ long. A larger circle has a longer $60^\circ$ arc because the same fraction of a larger circumference is involved.

A sector is bounded by two radii and an arc. A segment is bounded by a chord and an arc. For a minor angle below $180^\circ$, the minor segment is the sector minus the central triangle.

Arc length in degrees and radians

If a central angle is $\theta$ degrees in a circle of radius $r$, then

When $\theta$ is instead measured in radians, the same relationship becomes

Do not substitute a degree value directly into $s=r\theta$. A central angle of $120^\circ$ is $2\pi/3$ radians, so with $r=6\text{ cm}$,

The major arc has angle $240^\circ$ and length $8\pi\text{ cm}$. The two lengths sum to the full circumference $12\pi\text{ cm}$.

Sector area and its boundary

The sector occupies the same angular fraction of the disk as its arc occupies of the circumference:

In radians,

For the radius-$6\text{ cm}$, $120^\circ$ sector, the area is $12\pi\text{ cm}^2$. Its perimeter is not its arc length alone. The two straight radii are also part of the boundary:

A full disk has no exposed radial edges, so treating a $360^\circ$ disk as having perimeter $2r+2\pi r$ would count an artificial cut. State whether the object is a sector with two boundary radii or the uncut disk.

Finding a chord length

Bisecting the central isosceles triangle creates a right triangle with hypotenuse $r$, half-chord $c/2$ and central half-angle $\theta/2$. Therefore

For a minor central angle between $0^\circ$ and $180^\circ$, the perpendicular centre-to-chord distance is

With $r=6\text{ cm}$ and $\theta=120^\circ$, the chord is $6\sqrt3\text{ cm}$ and the central distance is $3\text{ cm}$. Its arc is $4\pi\text{ cm}$, which is longer than the chord. The endpoints agree, but the straight segment is the shorter path.

Using the major angle in the sine chord formula gives the same positive chord for angles below $360^\circ$. The cosine distance then becomes negative; the actual perpendicular distance is its absolute value. Working with the minor central angle avoids that unnecessary sign complication.

Circular-segment area

For $0<\theta\le\pi$ in radians, the central triangle has area $r^2\sin\theta/2$. Subtract it from the sector:

In the $120^\circ$ example, the sector area is $12\pi$ and the central triangle area is $9\sqrt3$, so

The segment’s perimeter is the chord plus the arc, $6\sqrt3+4\pi\text{ cm}$. Its boundary does not include the two radii, because they lie outside that segment’s boundary.

The major segment has area equal to the full disk area minus the minor segment. Drawing and naming the required region is more reliable than applying one formula to an unspecified “shaded part.”

Annular sectors

A sector cut from a ring has outer radius $R$, inner radius $r$ and angle $\theta$ in radians. Assume $0<r<R$ and $0<\theta<2\pi$, so the piece has two radial boundary edges. Subtracting the inner sector from the outer one gives

For $R=5\text{ cm}$, $r=3\text{ cm}$ and $\theta=\pi/2$, the area is $4\pi\text{ cm}^2$. Its boundary consists of two arcs and two radial connectors. The perimeter is

Using $2R$ for the straight boundary would include the missing inner part of the disk. The connectors are each $R-r$, not $R$.

Recovering an angle or radius

If an arc is $9\text{ cm}$ long on a radius-$6\text{ cm}$ circle, its central angle is $\theta=s/r=3/2$ radians. This value is less than $\pi$, so it represents a minor arc.

If a sector angle is $\pi/3$ and its area is $8\pi\text{ cm}^2$, solve $\frac12r^2(\pi/3)=8\pi$. This gives $r^2=48$ and $r=4\sqrt3\text{ cm}$. Keep the positive root because radius is a length.

Related topics

Angles in Circles explains which arc an angle intercepts. Radians, Arc Length and Circular Motion develops radian measure and rate applications. Area and Perimeter provides the wider measurement context.