Segment of a Circle

A segment of a circle is the area defined by a line from one side of the circumference to the other.

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Segments are divided into major and minor segments:

• Major segments are the larger proportion of the circle.

• Minor segments are the smaller proportion of the circle.

Major and minor segment of a circle -StudySmarter Originals

When working with the area of a segment of a circle, you should always remember the formula for the area of a circle: $\mathrm{\pi }×{\mathrm{r}}^{2}$. This is the formula you use regardless of whether the angle is in radians or degrees.

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:

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

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

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

Instead of trying to remember both formulas, it might be easier to remember the area of the major segment formula as a word equation:

$MajorSegment=areaofacircle-areaofminorsegment$

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}×{r}^{2}×\left(x-\mathrm{sin}\left(x\right)\right)$$Minorsegment=\frac{1}{2}×{9}^{2}×\left(\frac{\mathrm{\pi }}{3}-Sin\left(\frac{\mathrm{\pi }}{3}\right)\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.
$Majorsegment=\left(\mathrm{\pi }×{\mathrm{r}}^{2}\right)-\left[\frac{1}{2}×{\mathrm{r}}^{2}×\left(\mathrm{x}-\mathrm{Sin}\left(\mathrm{x}\right)\right)\right]$

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

Major segment = 247 square units (3 sf)

To check, if you add both the minor and major segments together, you should get approximately the same as the area of the whole circle $\left(\mathrm{\pi }×{\mathrm{r}}^{2}\right)$. Here, $\left(\mathrm{\pi }×{9}^{2}\right)=254.47squareunits$ and minor segment + major segment = $7.34+247\approx 254.54squareunits$.

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:

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

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

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

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.

Circle B has a minor segment, and the angle at the centre defines the length of the segment. The angle is ${120}^{\circ }$ and the radius is 10 cm.

1. What is the area of the minor segment of Circle B?
2. What is the area of the major segment of Circle B?

a. Finding the minor segment of Circle B.

1. Identify all the key information required to calculate the area. Radius = 10 cm; angle at the center = ${120}^{\circ }$

2. Substitute into the formula

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

$Minorsegment=\left(\frac{120×\mathrm{\pi }}{360}-\frac{\mathrm{sin}\left(120\right)}{2}\right)×{10}^{2}$

Minor segment = 75.7 square units (3 sf)

b. Finding the major segment of Circle B.

1. Substitute the key information into the major segment formula
$Majorsegment=\left(\mathrm{\pi }×{\mathrm{r}}^{2}\right)-\left[\left(\frac{\mathrm{x}×\mathrm{\pi }}{360}-\frac{\mathrm{sin}\left(\mathrm{x}\right)}{2}\right)×{\mathrm{r}}^{2}\right]$ $Majorsegment=\left(\mathrm{\pi }×{10}^{2}\right)-\left[\left(\frac{120\mathrm{\pi }}{360}-\frac{\mathrm{sin}\left(120\right)}{2}\right)×{10}^{2}\right]$

Major segment = 239 square units (3 sf)

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:

$ArcLength=r×x$

A segment in Circle C has a radius of 7 cm with an angle of ${20}^{\circ }$. What is the arc length of this segment?

$Arclength={20}^{°}×\frac{\mathrm{\pi }}{180}×7=\frac{7\mathrm{\pi }}{9}cm$

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

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

A segment in Circle D has a radius of 5 cm with an angle of ${90}^{\circ }$. What is the arc length of this segment?

$Arclength=90×\frac{\mathrm{\pi }}{180}×5=7.85cm\left(3\mathrm{s}.\mathrm{f}\right)$

Segment of a Circle - Key takeaways

• A segment of a circle is the area bounded by the circumference and the chord. Segments can either be the major (the bigger proportion) or minor (the smaller proportion).
• To find the area of a minor segment of a circle, you either use $\frac{1}{2}×{r}^{2}×\left(x-\mathrm{sin}\left(x\right)\right)$ where the angle (x) is in radians or $\left(\frac{x×\mathrm{\pi }}{360}-\frac{\mathrm{sin}\left(x\right)}{2}\right)×{r}^{2}$ where the angle (x) is in degrees.
• To find the area of a major segment, you subtract the area of the minor segment away from the area of the circle.
• Calculating the arc length of a segment is the same as calculating the arc length of a sector. To calculate the arc length of a segment where the angle (x) is in radians, you can do $r×x$. If the angle (x) is in degrees, then you use $r×x×\frac{\mathrm{\pi }}{180}$.

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What is a segment of a circle with an example?

A segment of a circle is the area of a proportion of a circle between a chord and the circumference. Segments can either be minor (the small part) or major (the larger part).

How do you find segments of a circle?

Finding the area of a segment of a circle can be found by substituting your values into a formula, which formula you use depends on whether the angle at the centre which defines the segment is in radians or degrees.

What is the area of a segment of a circle?

The area of a segment of a circle can be broken down into major (the larger proportion) and minor (the smaller proportion). When you use the area of a segment of a circle formulas, you are calculating the minor segment area. To calculate the major area, you need to subtract the minor segment area away from the area of the circle.

What is the formula for a segment of a circle?

There are two formulas for finding the area of a minor segment of a circle. If the angle at the centre of the circle which defines the chord is in radians, then the formula you use is 1/2 ×  r ^ 2 ×  (x-sin (x)). If the angle at the centre is in degrees, you use ((X× pi)/360 - sinx/2)×  r ^ 2

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