Segment of a Circle

Units for the angle of the segment of a circle

When working out the area or circumference of a segment of a circle, the angle at the centre of the circle which defines the segment can be in either radians or degrees.

• Degrees are denoted by ${}^{\circ }$ . In degrees, a full rotation is equal to ${360}^{\circ }$.
• Radians are another type of unit for angles. They are defined by the ratio of the radius of the circle to the arc length of the circle and denoted by ${}^r$. In radians, a full rotation is equal to $2{\mathrm{\pi }}^{\mathrm{r}}$.

Finding the area of a segment of a circle when the area is in radians

To find the area of a segment of a circle (the blue part), you need to know the angle at the centre where the radii brackets the chord (x) and the radius:

Triangle formed from the angle defining the segment - StudySmarter Originals

Formulas for finding the area of a segment of a circle when the angle is in radians

To find the area of a minor segment of a circle when the angle at the centre (x) is in radians, the formula is:

$Minorsegment=\frac{1}{2}\times r^2\times (x-\sin (x\left)\right)$

To find the area of a major segment of a circle when the angle at the centre is in radians, the formula is:

$Majorsegment=(\mathrm{\pi }\times {\mathrm{r}}^2)-[\frac{1}{2}\times {\mathrm{r}}^2\times (\mathrm{x}-\sin \left(\mathrm{x}\right)\left)\right]$

Circle A has a minor segment which is highlighted in pink.

1. Find the area of the minor segment.
2. Find the area of the major segment.

a. Finding the minor segment

1. Start by defining the characteristics of the segment: $Radius=9;Angle=\frac{\mathrm{\pi }}{3}$
2. Substitute into the formula:

$Minorsegment=\frac{1}{2}\times r^2\times (x-\sin (x\left)\right)$$Minorsegment=\frac{1}{2}\times 9^2\times (\frac{\mathrm{\pi }}{3}-Sin(\frac{\mathrm{\pi }}{3}\left)\right)$

Minor segment = 7.64 square units (3 sf)

b. Finding the area of the major segment

• Remember to find the major segment; you subtract the minor segment from the area of the circle.

$Majorsegment=(\mathrm{\pi }\times {\mathrm{r}}^2)-[\frac{1}{2}\times {\mathrm{r}}^2\times (\mathrm{x}-\mathrm{Sin}\left(\mathrm{x}\right)\left)\right]$

$MajorSegment=(\mathrm{\pi }\times 9^2)-[\frac{1}{2}\times 9^2\times (\frac{\mathrm{\pi }}{3}-\mathrm{Sin}\left(\frac{\mathrm{\pi }}{3}\right)\left)\right]$

Major segment = 247 square units (3 sf)

Finding the area of a segment of a circle when the angle is in degrees

You still need to know the radius and the centre of the circle, but there is now a different formula.

Formulas for finding the area of a segment of a circle when the angle is in degrees

The formula to find the minor segment of a circle, when the angle at the centre (x) is in degrees:

$Minorsegment=\left(\frac{x\times \mathrm{\pi }}{360}-\frac{\sin \left(x\right)}{2}\right)\times r^2$

To find the major segment of a circle when the angle at the centre (x) is in degrees, the formula is:

$Majorsegment=(\mathrm{\pi }\times {\mathrm{r}}^2)-\left[\left(\frac{\mathrm{x}\times \mathrm{\pi }}{360}-\frac{\sin \left(\mathrm{x}\right)}{2}\right)\times {\mathrm{r}}^2\right]$

Use the same principle as when the angle is in radians – you need to minus the minor segment from the whole area of the circle.

Arc lengths

The method to calculate the arc length of a segment is the same for calculating the arc length of a sector.

• To find the arc length when the angle at the centre (x) that defines the segment is in radians:

$ArcLength=r\times x$

• To find the arc length when the angle at the centre (x) that defines the segment is in degrees:

$ArcLength=x\times r\times \frac{\mathrm{\pi }}{180}$