Circle, its Parts and Measures
Circle
Circle is defined as path of a moving point that remains at a fixed distance from a fixed point. Circle is a closed curve in which all points on its boundary are at equal distances from that fixed point.
The fixed point is called centre of circle and fixed distance is called radius of circle.
Here in the above figure, we can see that point A is a moving point which remains at fixed distance from fixed point O.
O is center of the circle and the fixed distance between O and A is called radius of circle.
The length of boundary of circle is called its circumference.
Parts of circle
A circle has many more parts in addition to its center and radius that we shall discuss here also.
Diameter of circle
Diameter of circle is a line segment which passes through centre of circle and its end points lie on the circle.
A circle can have unlimited number of diameters and centre of circle is always a mid point of every diameter in a circle.
So we can say, all diameters of a circle always pass through center of the circle. Therefore, all diameters of a circle are concurrent and center of circle is a common point.
In given figure, AB is a line segment which has its end points A and B which lie on boundary of circle and line segment passes through center O of the circle. Therefore, AB is said to be a diameter of this circle.
In circle, the length of diameter is always double the radius of circle.
\(Diameter = 2 \times Radius\)
So in above circle
\(AB = 2 \times OA\)
A circle has infinite number of diameters.
Semi circle
The diameter of a circle always divides the circle into two equal parts. Each of these two equal parts is called as semi circle.
In above figure, we can see diameter AB divides the circle in two equal parts, one is above the diameter and another is below the diameter.
Chord of circle
A straight line with its two points lying on circle is called the chord of circle.
In the above figure, AB is the chord of circle.
Also, PQ is the chord of circle which passes through center of circle and has the maximum length if compared to other chords of circle.
Thus, PQ is also a diameter of this circle. Hence, we can say diameter is always the longest chord of circle.
Secant
A straight line which passes through circle and intersects the circle at two points is called secant of circle.
In above figure, line AB passes through circle and intersects the circle at two points A and B, therefore, AB is secant of the circle.
Tangent
A straight line which touches the circle at a point on circle is called tangent of the circle.
The point where line touches the circle is called point of contact.
In above given figure, line l touches the circle at only one point P, so line l is tangent of circle and P is the point of contact.
Arc
Arc of circle is a length of the boundary of a circle bounded by those two distinct points which lie on the circumference of circle.
In above figure, A and B are two distinct points those lie on circumference of the circle. So, the length of boundary AB that exists between points A and B, is the arc of circle. It is written as \(\overset{\Huge\frown}{AB}\).
Minor Arc and Major Arc
The distinct points on circumference of circle divides the circumference into two parts. The length of smaller part of the circumference is called minor arc and length of larger part on circumference is called major arc.
In above figure, A and B are the two distinct points and they divide the circumference into two parts ARB and ASB. The length ARB is shorter in length than the length ASB. Hence, \(\overset{\Huge\frown}{ARB}\) is called minor arc and length \(\overset{\Huge\frown}{ASB}\) is called major arc.
Sector of circle
Sector of circle is the region of circle that is bounded by an arc and two radii of the circle.
In above figure, OA and OB are radii of circle with center O and \(\overset{\Huge\frown}{AXB}\) is the arc. Therefore, OAXB is the sector of circle.
Minor Sector and Major Sector
Sector of a circle which has minor arc is called minor sector of the circle.
Sector of a circle which has major arc is called major sector of the circle.
The radius OA and radius OB divides the circle into two parts.
In above figure, region OAXB is bounded by minor arc \(\overset{\Huge\frown}{AXB}\). So, OAXB is minor sector of the circle.
The region OAYB is bounded by a major arc \(\overset{\Huge\frown}{AYB}\). So, OAYB is major sector of the circle.
Also, here, OA and OB radii make an angle at center O, \(\angle AOB\).
\(\angle AOB\) is called as angle of the sector. It is denoted by Theta \(\theta\).
Quadrant
The region of a circle which is bounded by two perpendicular radii and an arc is called a quadrant.
In above figure, radius OA and radius OB are perpendicular to each other. Here, angle of sector \(\angle AOB\) = \(90^0\)
Segment
Segment is defined as part of a circle which is bounded by an arc and a chord.
In above figure, AB is chord and APB is arc of circle. So, region enclosed by chord AB and arc \(\overset{\Huge\frown}{APB}\) is called as segment of the circle,
Minor Segment and Major Segment
When a chord divides the circle in two unequal segments, the region which includes the minor arc is called as minor segment and the region which includes major arc is called as major segment.
In above figure, chord AB divides circle into two segments. One segment is bounded by arc \(\overset{\Huge\frown}{ARB}\) and the second segment is bounded by arc \(\overset{\Huge\frown}{ASB}\). Also, Arc \(\overset{\Huge\frown}{ARB}\) is shorter in length than arc \(\overset{\Huge\frown}{ASB}\).
Hence, the segment bounded by arc \(\overset{\Huge\frown}{ARB}\) is minor segment and the segment bounded by arc \(\overset{\Huge\frown}{ASB}\) is major segment.
Moreover, major segment always includes the center O of circle.
Measurements of circle
The very common measurements that can we do for a circle are circumference of circle, area of circle, area of sector, length of arc and area of segment.
So, let’s discuss about them and see how to calculate them.
Circumference of circle
Circumference is distance around the circle. Its units are same as that of length i.e meter, centimeter, millimeter etc.
We can calculate the circumference of circle using the formula as below:
\(Circumference \; of \; circle = 2 \pi r\)
where, \(\pi\) is a constant value and r is radius of circle
The constant value of \(\pi = \frac{22}{7}\) or \(3.14\)
Calculate circumference of circle whose radius is 5cm.
Here, radius of circle = 5cm
As we know, \(Circumference \; of \; circle = 2 \pi r\)
\(\therefore\), \(Circumference \; of \; circle = 2 \pi \times 5\)
\(= 2 \times 3.14 \times 5\)
\(= 31.4 \; cm\)
What is \(\pi\)?
\(\pi\) is a value that never changes. If calculated, it always remains constant i.e. \(3.14\).
How is it calculated?
It is calculated by dividing the circumference of circle to diameter of circle.
It is a ratio of circumference and diameter of circle.
This constant value is denoted by \(\pi\) and read as pi.
\(\pi = \frac{Circumference}{Diameter}\)
Area of circle
Area of circle is a space occupied by a circle. Its units are i.e \(meter^2\), \(centimeter^2\), \(millimeter^2\) etc.
\(Area \; of \; circle = \pi r^2\)
where, \(\pi\) is a constant value and r is radius of circle
Again, the constant value of \(\pi = \frac{22}{7}\) or \(3.14\)
Calculate area of circle whose radius is 10cm.
Here, radius of circle = 10cm
As we know, \(Area \; of \; circle = \pi r^2\)
\(\therefore\), \(Area \; of \; circle = \pi \times 10^2\)
\(= 3.14 \times 100\)
\(= 314 \; cm^2\)
For the following below measurements, please refer to this figure.
Area of sector
\(Area \; of \; sector \; of \; angle \; \theta = \frac{\theta}{360} \times \pi r^2\)
Area of major sector
\(Area \; of \; major \; sector \; OADB = \pi r^2 \; – \; Area \; of \; minor \; sector \; OACB\)
Length of arc
\(Length \; of \; arc \; of \; sector \; of \; angle \; \theta = \frac{\theta}{360} \times 2 \pi r\)
Area of segment
\(Area \; of \; segment \; ACB = \pi r^2 \; – \; Area \; of \; triangle\)
Area of major segment
\(Area \; of \; major \; segment \; ADB = \pi r^2 \; – \; Area \; of \; minor \; segment \; ACB\)