Areas of Sector and Segment of a Circle
Arcs and Sectors
Most scientists in ancient Greece agreed that the Earth is a sphere. There was plenty of evidence: from ships disappearing behind the horizon at sea, to the circular motion of stars during the night.
Unfortunately, no one knew exactly how big Earth was – until around 200 BC, when the mathematician
As you can see in the diagram, an arc is a part of the
The arc between two points A and B is often written as
The smaller of the two arcs is called the minor arc, and the larger one is called the major arc. If points A and B are exactly opposite each other, both arcs have the same length and are
To find the length of an arc or the area of a sector, we need to know about the corresponding angle at the center of the circle: this is called the central angle.
Notice how the arc, sector and angle all take up the same proportion of a full circle. For example, if the central angle is , it takes up
This means that the length of the arc is also
We can express this relationship in an equation:
Now we can rearrange these equations to find whichever variable we’re interested in. For example,
| arc length | = | |
|---|---|---|
| = |
| sector area | = | |
|---|---|---|
| = |
where r is the radius of the circle, and c is the size of the central angle.
If the central angle is measured in
| arc length | = | |
|---|---|---|
| = |
| sector area | = | |
|---|---|---|
| = |
Notice how the equations become much simpler, and π cancels out everywhere. This is because, as you might recall, the definition of radians is basically the length of an arc in a circle with radius 1.
Now let’s see how we can use arcs and sectors to calculate the circumference of the Earth.
In ancient Egypt, the city of Swenet was located along the Nile river. Swenet was famous for a well with a curious property: there was one moment every year when the sunlight reached the very bottom of the well – at noon on 21 June, the day of the summer solstice. At that precise time, the bottom of the well was illuminated, but not its sides, meaning that the Sun was standing directly above the well.
Ancient Egyptians measured long distances by counting the number of steps it took to walk.
Some sources say the “Well of Eratosthenes” was on Elephantine island on the Nile river.
1. Find the area of the sector of a circle with radius 4 cm and of angle 30°.Also, find the area of the corresponding major sector (Use π = 3.14).
- Given sector is OAPB(see in fig)
- Area of the sector =
θ 360 x πr 2 - calculate the area
- and we get the answer is
cm 2 - Area of the corresponding major sector =
πr 2 - we get the answer is =
cm 2 - Therefore 46.1
cm 2
Segments
The last part of a circle that we can find the area of is called a segment, not to be confused with a line segment. A segment of a circle is the area of a circle that is bounded by a chord and the arc with the same endpoints as the chord.
Now let us take the case of the area of the segment APB of a circle with centre O and radius r.
We can see that :
Area of the segment APB = Area of the sector OAPB – Area of ∆ OAB
=
We can also observe that :
Area of the major sector OAQB =
Area of major segment AQB =
2. Find the area of the segment AYB, if radius of the circle is 21 cm and ∠AOB= 120°.
- Area of the segment AYB = Area of sector OAYB – Area of △OAB
- Now, area of the sector OAYB =
- calculate the segment then we get the answer is=
cm 2